Intro to Functions
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Linear Functions and Circles
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- The Linear Function - Introduction
- Graph of a Linear Function
- Meaning of the Constants in the Linear Function
- Properties of the Slope
- The Slope Formula
- Finding the Equation of a Line
- The Mutual Positions of Two Lines
- The Distance between Two Points
- The Equation of a Circle
- Exercises 1-4
- Exercises 5-8
- Exercises 9-12
- Exercises 13-16
- Exercises 17-20
- Exercises 21-24
- Exercises 25-26
- Exercise 27
- Exercise 28
- Exercises 29-30
- Exercises 31-34

Quadratic Functions
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Common Functions
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Translation and Reflection of Functions
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- Vertical Translation of Basic Functions
- Horizontal Translation of Basic Functions
- Examples
- Reflection about the X-axis
- Reflection about the Y-axis
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7 - Parts 1-4
- Exercise 8 - Parts 1-2
- Exercise 8 - Parts 3-6
- Exercise 8 - Parts 7-10

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[{"Name":"Intro to Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"What is a Function?","Duration":"8m 56s","ChapterTopicVideoID":6245,"CourseChapterTopicPlaylistID":56140,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6245.jpeg","UploadDate":"2020-09-30T13:10:51.0030000","DurationForVideoObject":"PT8M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.940","Text":"In this clip, we\u0027ll learn what a Function is in Mathematics not in everyday life."},{"Start":"00:05.940 ","End":"00:10.335","Text":"To help you understand the meaning of a function in mathematics,"},{"Start":"00:10.335 ","End":"00:12.735","Text":"we\u0027ll sketch a little house here."},{"Start":"00:12.735 ","End":"00:15.000","Text":"This is the base of the house."},{"Start":"00:15.000 ","End":"00:18.705","Text":"It\u0027s a square with side of length x,"},{"Start":"00:18.705 ","End":"00:20.865","Text":"that\u0027s x, and this is x."},{"Start":"00:20.865 ","End":"00:24.195","Text":"The house has a roof in the shape of a triangle."},{"Start":"00:24.195 ","End":"00:29.963","Text":"The height of this triangle of the roof is 4,"},{"Start":"00:29.963 ","End":"00:31.500","Text":"the house has an area."},{"Start":"00:31.500 ","End":"00:36.115","Text":"Let\u0027s let y denote the area of the house."},{"Start":"00:36.115 ","End":"00:37.830","Text":"As we said already,"},{"Start":"00:37.830 ","End":"00:44.335","Text":"x denotes the length of the side of the base of the house,"},{"Start":"00:44.335 ","End":"00:49.415","Text":"y is the area of the whole house which is made up of 2 parts."},{"Start":"00:49.415 ","End":"00:55.580","Text":"There\u0027s the base here under the triangular roof."},{"Start":"00:55.580 ","End":"01:05.120","Text":"I can write a formula for y. I can say that y is equal to the area of this square base,"},{"Start":"01:05.120 ","End":"01:09.085","Text":"which is x times x, which is x^2."},{"Start":"01:09.085 ","End":"01:11.390","Text":"That\u0027s the base, and then there\u0027s the roof,"},{"Start":"01:11.390 ","End":"01:12.935","Text":"which is a triangle."},{"Start":"01:12.935 ","End":"01:18.245","Text":"The formula for the area of a triangle is 1/2 base times height."},{"Start":"01:18.245 ","End":"01:22.430","Text":"We get 1/2 times x."},{"Start":"01:22.430 ","End":"01:26.225","Text":"The base of the triangle is also x times the height,"},{"Start":"01:26.225 ","End":"01:28.090","Text":"which is 4,"},{"Start":"01:28.090 ","End":"01:31.695","Text":"1/2 times x times 4 is simply 2x."},{"Start":"01:31.695 ","End":"01:35.025","Text":"I\u0027ve expressed y in terms of x."},{"Start":"01:35.025 ","End":"01:38.180","Text":"In this case, y depends on x."},{"Start":"01:38.180 ","End":"01:43.835","Text":"Each value of x determines the corresponding value of y."},{"Start":"01:43.835 ","End":"01:45.890","Text":"Let\u0027s see how this works."},{"Start":"01:45.890 ","End":"01:52.190","Text":"If x=1, then y will equal from this formula,"},{"Start":"01:52.190 ","End":"01:56.635","Text":"1^2 plus twice 1 will be 3."},{"Start":"01:56.635 ","End":"02:01.385","Text":"In other words, if the base of the house has a side of length 1,"},{"Start":"02:01.385 ","End":"02:05.580","Text":"the area of the whole house will be 3."},{"Start":"02:05.680 ","End":"02:08.300","Text":"If, for example,"},{"Start":"02:08.300 ","End":"02:10.690","Text":"x is equal to 2,"},{"Start":"02:10.690 ","End":"02:18.285","Text":"then y=2^2 plus twice 2 is 8, 4 plus 4."},{"Start":"02:18.285 ","End":"02:24.665","Text":"If the house has base of length 2 the area of the house will be 8."},{"Start":"02:24.665 ","End":"02:30.690","Text":"Similarly, if say x=5,"},{"Start":"02:30.690 ","End":"02:35.225","Text":"then we\u0027ll get y=5^2 is 25 plus twice 5, that\u0027s 10,"},{"Start":"02:35.225 ","End":"02:40.255","Text":"then y=35, and so on."},{"Start":"02:40.255 ","End":"02:44.739","Text":"We see that y depends on x."},{"Start":"02:44.739 ","End":"02:52.100","Text":"That is the variable y depends on the variable x in such a way that to"},{"Start":"02:52.100 ","End":"02:59.552","Text":"every value of x the corresponds exactly 1 value of y. I\u0027ll write this down shortly,"},{"Start":"02:59.552 ","End":"03:04.340","Text":"but first I\u0027d like to illustrate with the help of a couple of circles,"},{"Start":"03:04.340 ","End":"03:06.885","Text":"well, more ovals and circles."},{"Start":"03:06.885 ","End":"03:11.059","Text":"This one represents the values of x,"},{"Start":"03:11.059 ","End":"03:14.525","Text":"and this one represents the values of y."},{"Start":"03:14.525 ","End":"03:16.250","Text":"The value of x,"},{"Start":"03:16.250 ","End":"03:17.975","Text":"which is 1,"},{"Start":"03:17.975 ","End":"03:23.155","Text":"has a corresponding value of y, which is 3."},{"Start":"03:23.155 ","End":"03:31.260","Text":"The x that is 2 get the corresponding y, which is 8."},{"Start":"03:31.260 ","End":"03:36.950","Text":"The x that is 5 is going to get a corresponding y,"},{"Start":"03:36.950 ","End":"03:40.415","Text":"which is 35, and so on."},{"Start":"03:40.415 ","End":"03:42.725","Text":"To each value of x,"},{"Start":"03:42.725 ","End":"03:47.135","Text":"the corresponds exactly 1 value of y."},{"Start":"03:47.135 ","End":"03:51.050","Text":"This brings us to a formal definition of function,"},{"Start":"03:51.050 ","End":"03:56.015","Text":"which I\u0027ll say out loud first and then I\u0027ll write it down. It goes as follows."},{"Start":"03:56.015 ","End":"04:01.084","Text":"If a variable y depends on the variable x,"},{"Start":"04:01.084 ","End":"04:02.615","Text":"like we see here,"},{"Start":"04:02.615 ","End":"04:10.085","Text":"in such a way that for each value of x that corresponds to exactly 1 value of y,"},{"Start":"04:10.085 ","End":"04:14.180","Text":"then we say that y is a function of x."},{"Start":"04:14.180 ","End":"04:20.785","Text":"In this case, the area of the house is a function of the length of the side of the base."},{"Start":"04:20.785 ","End":"04:23.604","Text":"Let\u0027s write this down properly."},{"Start":"04:23.604 ","End":"04:30.650","Text":"A variable y is called a function of another variable x."},{"Start":"04:30.650 ","End":"04:35.990","Text":"If y depends on x in such a way that for each value of x,"},{"Start":"04:35.990 ","End":"04:41.500","Text":"that corresponds exactly 1 value of y."},{"Start":"04:41.500 ","End":"04:45.695","Text":"Meanwhile, we\u0027ve had just one example of a function."},{"Start":"04:45.695 ","End":"04:48.320","Text":"This was the function that we had."},{"Start":"04:48.320 ","End":"04:52.460","Text":"It came out a problem about a house and"},{"Start":"04:52.460 ","End":"04:56.540","Text":"the length of the side of the base and the area of the house."},{"Start":"04:56.540 ","End":"04:59.951","Text":"But, a function doesn\u0027t have to come out of a story,"},{"Start":"04:59.951 ","End":"05:02.300","Text":"it could just be totally abstract."},{"Start":"05:02.300 ","End":"05:11.430","Text":"I could write y=x plus 1 over x minus 1,"},{"Start":"05:11.430 ","End":"05:17.960","Text":"totally abstractly, and that would be a function because for each x,"},{"Start":"05:17.960 ","End":"05:20.915","Text":"for example, x=0,"},{"Start":"05:20.915 ","End":"05:24.440","Text":"I can get exactly 1 value of y."},{"Start":"05:24.440 ","End":"05:30.850","Text":"In this case, 0 plus 1 over 0 minus 1 is minus 1."},{"Start":"05:30.850 ","End":"05:34.070","Text":"Or if x=2 in another example,"},{"Start":"05:34.070 ","End":"05:38.645","Text":"I would get y=2 plus 1 over 2 minus 1,"},{"Start":"05:38.645 ","End":"05:40.205","Text":"which is 3 over 1,"},{"Start":"05:40.205 ","End":"05:43.535","Text":"which would be 3, and so on."},{"Start":"05:43.535 ","End":"05:46.445","Text":"What I would like to point out is that"},{"Start":"05:46.445 ","End":"05:51.110","Text":"not every time that you have a dependency between y and x,"},{"Start":"05:51.110 ","End":"05:53.975","Text":"and even an equation involving y and x,"},{"Start":"05:53.975 ","End":"05:56.780","Text":"does it mean that y is a function of x?"},{"Start":"05:56.780 ","End":"05:59.990","Text":"I wanted to look at the following example."},{"Start":"05:59.990 ","End":"06:05.234","Text":"Let\u0027s take y^2=x."},{"Start":"06:05.234 ","End":"06:08.330","Text":"Certainly, y is dependent on x,"},{"Start":"06:08.330 ","End":"06:10.820","Text":"but is y a function of x?"},{"Start":"06:10.820 ","End":"06:12.995","Text":"Let\u0027s take, for instance,"},{"Start":"06:12.995 ","End":"06:17.930","Text":"x=4. What is y?"},{"Start":"06:17.930 ","End":"06:21.260","Text":"We have y^2=4."},{"Start":"06:21.260 ","End":"06:23.480","Text":"But for y^2=4,"},{"Start":"06:23.480 ","End":"06:25.235","Text":"there are two solutions."},{"Start":"06:25.235 ","End":"06:29.735","Text":"We could have y=2 and that would satisfy the equation."},{"Start":"06:29.735 ","End":"06:33.523","Text":"But, we could also have y=minus 2,"},{"Start":"06:33.523 ","End":"06:36.685","Text":"and that would also satisfy the equation."},{"Start":"06:36.685 ","End":"06:41.780","Text":"One value of x gives 2 values of y,"},{"Start":"06:41.780 ","End":"06:46.280","Text":"which contradicts the exactly 1 that is written here."},{"Start":"06:46.280 ","End":"06:49.400","Text":"In fact, I can show you another bad thing."},{"Start":"06:49.400 ","End":"06:54.545","Text":"Suppose I took x=minus 9,"},{"Start":"06:54.545 ","End":"06:56.960","Text":"then what is y equal?"},{"Start":"06:56.960 ","End":"07:00.925","Text":"Well, I have to have y^2 is minus 9,"},{"Start":"07:00.925 ","End":"07:06.140","Text":"but the square of a number can never be negative though there is no such y."},{"Start":"07:06.140 ","End":"07:09.350","Text":"For x=4, we have two values of y,"},{"Start":"07:09.350 ","End":"07:10.850","Text":"for x=minus 9,"},{"Start":"07:10.850 ","End":"07:13.040","Text":"we have no values of y."},{"Start":"07:13.040 ","End":"07:18.515","Text":"It\u0027s important that there be exactly 1 in order for it to be a function."},{"Start":"07:18.515 ","End":"07:22.220","Text":"Let\u0027s return a minute to the first example,"},{"Start":"07:22.220 ","End":"07:24.045","Text":"the one with the house."},{"Start":"07:24.045 ","End":"07:26.840","Text":"I\u0027d like to just take another couple of values here."},{"Start":"07:26.840 ","End":"07:29.285","Text":"We did x=1, 2, and 5."},{"Start":"07:29.285 ","End":"07:33.800","Text":"Let\u0027s see what happens if x=0."},{"Start":"07:33.800 ","End":"07:35.735","Text":"Something like continuing."},{"Start":"07:35.735 ","End":"07:41.160","Text":"Then y=0^2 plus twice 0 is 0."},{"Start":"07:41.160 ","End":"07:46.290","Text":"What happens if x=minus 2?"},{"Start":"07:46.290 ","End":"07:49.905","Text":"Substituting here, minus 2^2 is 4,"},{"Start":"07:49.905 ","End":"07:52.220","Text":"twice minus 2 is minus 4,"},{"Start":"07:52.220 ","End":"07:56.794","Text":"4 minus 4 means that y is also 0."},{"Start":"07:56.794 ","End":"07:59.630","Text":"My question is, is this a problem?"},{"Start":"07:59.630 ","End":"08:02.090","Text":"The answer is no, it isn\u0027t."},{"Start":"08:02.090 ","End":"08:06.140","Text":"Let me illustrate this in this diagram below."},{"Start":"08:06.140 ","End":"08:13.880","Text":"We have here the point where x is 0 and here the point where x is minus 2,"},{"Start":"08:13.880 ","End":"08:19.114","Text":"and both of them go to the same value of y, which is 0."},{"Start":"08:19.114 ","End":"08:22.130","Text":"But, this is not a problem because"},{"Start":"08:22.130 ","End":"08:27.935","Text":"our requirement is that each x should have just 1 value of y."},{"Start":"08:27.935 ","End":"08:34.415","Text":"It\u0027s possible that a single value of y can have more than 1 value of x."},{"Start":"08:34.415 ","End":"08:36.995","Text":"That is not considered a problem."},{"Start":"08:36.995 ","End":"08:43.670","Text":"Let\u0027s contrast this with the previous example of y^2=x,"},{"Start":"08:43.670 ","End":"08:47.525","Text":"where we had a single x going to two different y\u0027s."},{"Start":"08:47.525 ","End":"08:50.420","Text":"This is not allowed, and therefore,"},{"Start":"08:50.420 ","End":"08:55.590","Text":"this one is not a function."}],"ID":6257},{"Watched":false,"Name":"Notation for Functions","Duration":"7m 25s","ChapterTopicVideoID":6246,"CourseChapterTopicPlaylistID":56140,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6246.jpeg","UploadDate":"2020-09-30T13:13:31.3600000","DurationForVideoObject":"PT7M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"In this clip, I\u0027d like to show you an additional notation for functions."},{"Start":"00:04.350 ","End":"00:07.140","Text":"The old notation would go something like this."},{"Start":"00:07.140 ","End":"00:14.790","Text":"We would say y equals x^2 plus 4x plus 1,"},{"Start":"00:14.790 ","End":"00:16.920","Text":"and then y would be a function of x."},{"Start":"00:16.920 ","End":"00:18.705","Text":"In the new notation,"},{"Start":"00:18.705 ","End":"00:22.665","Text":"we will write something like f(x),"},{"Start":"00:22.665 ","End":"00:23.820","Text":"that\u0027s how it\u0027s pronounced."},{"Start":"00:23.820 ","End":"00:25.605","Text":"We say the word of,"},{"Start":"00:25.605 ","End":"00:31.050","Text":"f(x) is equal to x^2 plus 4x plus 1."},{"Start":"00:31.050 ","End":"00:33.870","Text":"For example, in the new notation,"},{"Start":"00:33.870 ","End":"00:37.700","Text":"I would say f(4)= and I would"},{"Start":"00:37.700 ","End":"00:41.530","Text":"do the same thing I would do if I was substituting x equals 4 for figuring out"},{"Start":"00:41.530 ","End":"00:51.180","Text":"y. I would say 4^2 is 16 plus 4 times 4 is another 16 plus 1 is 33. f(4)"},{"Start":"00:51.180 ","End":"00:58.130","Text":"is 33 and we would say something like the value of"},{"Start":"00:58.130 ","End":"01:07.110","Text":"the function f at the point x=4 is 33."},{"Start":"01:07.110 ","End":"01:08.420","Text":"In the old notation,"},{"Start":"01:08.420 ","End":"01:18.575","Text":"I might say something like if x=4 then y=33."},{"Start":"01:18.575 ","End":"01:21.808","Text":"We call this the old notation,"},{"Start":"01:21.808 ","End":"01:24.170","Text":"and I call this one the new notation,"},{"Start":"01:24.170 ","End":"01:26.840","Text":"but actually both are used,"},{"Start":"01:26.840 ","End":"01:29.615","Text":"it\u0027s just that for you, this is newer."},{"Start":"01:29.615 ","End":"01:32.600","Text":"Once again, for the new notation,"},{"Start":"01:32.600 ","End":"01:40.430","Text":"we write f(4)= 33 and in longhand we would say the value of the function f,"},{"Start":"01:40.430 ","End":"01:48.515","Text":"f stands for function at the point x= 4 is 33."},{"Start":"01:48.515 ","End":"01:52.910","Text":"When we say the value of the function, we mean the y."},{"Start":"01:52.910 ","End":"01:56.945","Text":"If I want to know what is f(0),"},{"Start":"01:56.945 ","End":"02:00.095","Text":"then in the old notation, it\u0027s like asking,"},{"Start":"02:00.095 ","End":"02:03.950","Text":"what is y when x is 0 and you substitute 0,"},{"Start":"02:03.950 ","End":"02:11.030","Text":"we quickly see that this equal to 1 and so the old way I would say when x is 0, y is 1."},{"Start":"02:11.030 ","End":"02:17.090","Text":"Now I would say the value of the function f at the point x=0 is 1."},{"Start":"02:17.090 ","End":"02:20.450","Text":"This new notation has some advantages."},{"Start":"02:20.450 ","End":"02:23.780","Text":"When I write f(x), I know that it\u0027s x,"},{"Start":"02:23.780 ","End":"02:27.335","Text":"which is the variable because there are situations where it\u0027s not clear."},{"Start":"02:27.335 ","End":"02:31.025","Text":"The other thing is that function names,"},{"Start":"02:31.025 ","End":"02:32.830","Text":"in this case f,"},{"Start":"02:32.830 ","End":"02:37.715","Text":"can be several letters long unlike regular variables."},{"Start":"02:37.715 ","End":"02:40.245","Text":"For example, in economics,"},{"Start":"02:40.245 ","End":"02:43.405","Text":"common function name is TC,"},{"Start":"02:43.405 ","End":"02:46.525","Text":"which stands for total cost."},{"Start":"02:46.525 ","End":"02:49.164","Text":"Everyone knows what is TC."},{"Start":"02:49.164 ","End":"02:51.685","Text":"They\u0027re similarly, names used in physics and"},{"Start":"02:51.685 ","End":"02:55.645","Text":"other branches that are familiar and they can be descriptive."},{"Start":"02:55.645 ","End":"03:03.655","Text":"In economics, TC stands for total cost, that\u0027s in economics."},{"Start":"03:03.655 ","End":"03:08.210","Text":"In mathematics, common function names are f,"},{"Start":"03:08.210 ","End":"03:10.890","Text":"g, h. Anyway,"},{"Start":"03:10.890 ","End":"03:12.420","Text":"not to get worried about that,"},{"Start":"03:12.420 ","End":"03:14.680","Text":"the point is that a function can have a name,"},{"Start":"03:14.680 ","End":"03:17.710","Text":"doesn\u0027t have to be f and the next example I\u0027m about"},{"Start":"03:17.710 ","End":"03:20.845","Text":"to give will be from economics and we\u0027ll use the TC,"},{"Start":"03:20.845 ","End":"03:23.260","Text":"which is total cost function."},{"Start":"03:23.260 ","End":"03:29.770","Text":"In this example, we\u0027re given that Danny bought some candy bars."},{"Start":"03:29.770 ","End":"03:33.850","Text":"He bought x Mars bars,"},{"Start":"03:33.850 ","End":"03:38.815","Text":"and he bought x^2 KitKats."},{"Start":"03:38.815 ","End":"03:40.540","Text":"I added some illustration here,"},{"Start":"03:40.540 ","End":"03:42.580","Text":"I\u0027m not advertising these products."},{"Start":"03:42.580 ","End":"03:46.075","Text":"Anyway, the cost of each,"},{"Start":"03:46.075 ","End":"03:47.230","Text":"I\u0027ll tell you what it is."},{"Start":"03:47.230 ","End":"03:55.375","Text":"The cost of a Mars is 40 cents,"},{"Start":"03:55.375 ","End":"04:00.700","Text":"and a KitKat costs 50 cents."},{"Start":"04:00.700 ","End":"04:02.710","Text":"I\u0027d like to explain about the x,"},{"Start":"04:02.710 ","End":"04:04.135","Text":"and the x^2,"},{"Start":"04:04.135 ","End":"04:12.680","Text":"x is a variable representing how many Mars bars Danny bought and x could be anything."},{"Start":"04:12.680 ","End":"04:15.140","Text":"Because it has to be a positive whole number of course,"},{"Start":"04:15.140 ","End":"04:21.255","Text":"like if x is 3 and he bought 3 Mars bars and x^2 is 9,"},{"Start":"04:21.255 ","End":"04:23.145","Text":"so he brought 9 KitKats."},{"Start":"04:23.145 ","End":"04:27.660","Text":"If x is 2, then he bought 2 Mars bars and 4 KitKats."},{"Start":"04:27.660 ","End":"04:32.360","Text":"If x is 5, 5 Mars bars and 5^2 is 25 KitKats and so on."},{"Start":"04:32.360 ","End":"04:34.280","Text":"Now, the question part,"},{"Start":"04:34.280 ","End":"04:36.700","Text":"there\u0027s two parts: a,"},{"Start":"04:36.700 ","End":"04:46.560","Text":"write the cost function or the total cost function for Danny\u0027s expenses."},{"Start":"04:46.630 ","End":"04:52.835","Text":"Then there\u0027s part b, where you got to figure out how much did"},{"Start":"04:52.835 ","End":"04:59.950","Text":"Danny spend if he bought 4 Mars bars?"},{"Start":"04:59.950 ","End":"05:05.150","Text":"We need to express the cost as a function of x,"},{"Start":"05:05.150 ","End":"05:06.695","Text":"number of Mars bars."},{"Start":"05:06.695 ","End":"05:08.480","Text":"Let\u0027s give the function a name,"},{"Start":"05:08.480 ","End":"05:12.195","Text":"let\u0027s call it TC."},{"Start":"05:12.195 ","End":"05:13.575","Text":"We get TC,"},{"Start":"05:13.575 ","End":"05:15.630","Text":"and it\u0027s a function of x,"},{"Start":"05:15.630 ","End":"05:17.210","Text":"and what it\u0027s going to equal."},{"Start":"05:17.210 ","End":"05:18.545","Text":"Well, let\u0027s do the math."},{"Start":"05:18.545 ","End":"05:23.233","Text":"If each Mars bar costs 40 cents,"},{"Start":"05:23.233 ","End":"05:25.875","Text":"and he bought x of them,"},{"Start":"05:25.875 ","End":"05:30.930","Text":"so this is equal to 40x if we work in cents."},{"Start":"05:30.930 ","End":"05:33.165","Text":"But, I would prefer to work in dollars,"},{"Start":"05:33.165 ","End":"05:38.060","Text":"so this 40 is a 0.4 dollars."},{"Start":"05:38.060 ","End":"05:40.460","Text":"This is the path of the Mars bars,"},{"Start":"05:40.460 ","End":"05:41.930","Text":"but he also bought KitKats."},{"Start":"05:41.930 ","End":"05:43.640","Text":"Now, how many of these did he buy?"},{"Start":"05:43.640 ","End":"05:49.511","Text":"He bought X^2 KitKats and each of them costs 50 cents or 0.5 dollars,"},{"Start":"05:49.511 ","End":"05:58.367","Text":"so we\u0027ve got an extra 0.5x^2 number of items times the cost per item for the KitKats."},{"Start":"05:58.367 ","End":"06:03.275","Text":"Altogether, this is the total cost of his candy bar expenses."},{"Start":"06:03.275 ","End":"06:07.025","Text":"This is in fact the answer to part a."},{"Start":"06:07.025 ","End":"06:09.320","Text":"Now onto part b,"},{"Start":"06:09.320 ","End":"06:12.875","Text":"how much did Danny spend if he bought 4 Mars bars?"},{"Start":"06:12.875 ","End":"06:15.800","Text":"That must mean that x is equal to 4."},{"Start":"06:15.800 ","End":"06:20.593","Text":"This means that we need TC(4),"},{"Start":"06:20.593 ","End":"06:25.755","Text":"and this is going to equal just simply substituting 4 instead of x,"},{"Start":"06:25.755 ","End":"06:36.020","Text":"so we get 0.4 times 4 plus 0.5 times 4^2."},{"Start":"06:36.020 ","End":"06:37.305","Text":"What does this equal?"},{"Start":"06:37.305 ","End":"06:42.585","Text":"4 times 0.4 is 1.6."},{"Start":"06:42.585 ","End":"06:48.180","Text":"4^2 is 16 and 16"},{"Start":"06:48.180 ","End":"06:53.605","Text":"times 0.5 is 8 because 0.5 is a half."},{"Start":"06:53.605 ","End":"06:58.355","Text":"Altogether, we have got 9.6,"},{"Start":"06:58.355 ","End":"07:00.500","Text":"and I\u0027m just reminding you that it is in dollars"},{"Start":"07:00.500 ","End":"07:04.655","Text":"and some people like to write the extra decimal place."},{"Start":"07:04.655 ","End":"07:07.760","Text":"This is the answer to part b,"},{"Start":"07:07.760 ","End":"07:14.710","Text":"and the cost function here is the answer to part a."},{"Start":"07:14.710 ","End":"07:17.990","Text":"This is an example of using a meaningful function name,"},{"Start":"07:17.990 ","End":"07:20.870","Text":"in this case TC from the world of economics,"},{"Start":"07:20.870 ","End":"07:22.400","Text":"where it means total cost."},{"Start":"07:22.400 ","End":"07:25.770","Text":"Okay, I\u0027m done with this clip."}],"ID":6258},{"Watched":false,"Name":"The Domain of Definition of a Function","Duration":"5m 53s","ChapterTopicVideoID":5104,"CourseChapterTopicPlaylistID":56140,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5104.jpeg","UploadDate":"2020-09-30T13:14:52.0100000","DurationForVideoObject":"PT5M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"In this clip, we\u0027ll talk about the domain of definition of a function."},{"Start":"00:04.680 ","End":"00:08.460","Text":"Domain of definition sometimes is just called the domain."},{"Start":"00:08.460 ","End":"00:09.930","Text":"What does it mean?"},{"Start":"00:09.930 ","End":"00:12.150","Text":"Basically, when we have a function,"},{"Start":"00:12.150 ","End":"00:16.845","Text":"the domain of definition simply means the values of x,"},{"Start":"00:16.845 ","End":"00:20.685","Text":"it\u0027s actually a set of values but just called the the values of x,"},{"Start":"00:20.685 ","End":"00:22.941","Text":"for which makes sense to say f(x),"},{"Start":"00:22.941 ","End":"00:25.724","Text":"or if we put it, in other words,"},{"Start":"00:25.724 ","End":"00:30.390","Text":"which values of x are we allowed to substitute in the function f?"},{"Start":"00:30.390 ","End":"00:32.085","Text":"Let\u0027s take an example."},{"Start":"00:32.085 ","End":"00:42.315","Text":"The first example we will take f(x) is 24 over 2 minus x."},{"Start":"00:42.315 ","End":"00:47.450","Text":"Let\u0027s see what values of x makes sense to substitute here."},{"Start":"00:47.450 ","End":"00:51.175","Text":"For example, could we put x equals 0?"},{"Start":"00:51.175 ","End":"00:55.785","Text":"Certainly, 2 minus 0 is 2,"},{"Start":"00:55.785 ","End":"00:57.345","Text":"24 over 2 is 12."},{"Start":"00:57.345 ","End":"00:59.760","Text":"Could we put x equals 1?"},{"Start":"00:59.760 ","End":"01:04.110","Text":"Yeah, 24 over 2 minus 1 is 24."},{"Start":"01:04.110 ","End":"01:06.935","Text":"We put x equals 3,"},{"Start":"01:06.935 ","End":"01:09.215","Text":"2 minus 3 is negative 1,"},{"Start":"01:09.215 ","End":"01:12.305","Text":"24 over negative 1 is minus 24."},{"Start":"01:12.305 ","End":"01:14.900","Text":"So what can we put?"},{"Start":"01:14.900 ","End":"01:20.345","Text":"If you remember, we\u0027re not allowed to divide by 0,"},{"Start":"01:20.345 ","End":"01:23.510","Text":"that\u0027s the only number we\u0027re not allowed to divide by."},{"Start":"01:23.510 ","End":"01:26.620","Text":"What happens if we put x equals 2?"},{"Start":"01:26.620 ","End":"01:28.725","Text":"If we put x equals 2,"},{"Start":"01:28.725 ","End":"01:31.050","Text":"we\u0027ll get 24 over 0,"},{"Start":"01:31.050 ","End":"01:34.150","Text":"and we\u0027re not allowed to divide by 0."},{"Start":"01:34.150 ","End":"01:36.360","Text":"If you look at it closely,"},{"Start":"01:36.360 ","End":"01:38.445","Text":"that\u0027s the only problem that can be solved."},{"Start":"01:38.445 ","End":"01:42.865","Text":"Every value of x is okay except for x equals 2."},{"Start":"01:42.865 ","End":"01:47.330","Text":"We say that the domain of definition is everything except"},{"Start":"01:47.330 ","End":"01:52.055","Text":"2 which we can write simply as x not equal to 2."},{"Start":"01:52.055 ","End":"01:55.045","Text":"Now let\u0027s take another example."},{"Start":"01:55.045 ","End":"01:56.520","Text":"As our next example,"},{"Start":"01:56.520 ","End":"02:04.260","Text":"we\u0027ll have f(x) is equal to the square root of x minus 1."},{"Start":"02:04.260 ","End":"02:06.940","Text":"We start looking at values of x,"},{"Start":"02:06.940 ","End":"02:08.725","Text":"which we could put in here."},{"Start":"02:08.725 ","End":"02:13.400","Text":"We try let\u0027s say x equals 2,"},{"Start":"02:13.400 ","End":"02:16.800","Text":"and you get 2 minus 1 is 1,"},{"Start":"02:16.800 ","End":"02:19.635","Text":"square root of 1, that\u0027s fine,"},{"Start":"02:19.635 ","End":"02:22.930","Text":"and we could take x equals 5,"},{"Start":"02:22.930 ","End":"02:26.680","Text":"5 minus 1 is 4, no problem."},{"Start":"02:26.680 ","End":"02:29.575","Text":"You could even take x equals 3."},{"Start":"02:29.575 ","End":"02:32.365","Text":"3 minus 1 is 2,"},{"Start":"02:32.365 ","End":"02:33.550","Text":"square root of 2,"},{"Start":"02:33.550 ","End":"02:34.660","Text":"which is not a whole number,"},{"Start":"02:34.660 ","End":"02:37.065","Text":"but I can say square root of 2."},{"Start":"02:37.065 ","End":"02:41.630","Text":"X equals a 100, square root of 99."},{"Start":"02:41.630 ","End":"02:44.580","Text":"Let\u0027s try x equals 0,"},{"Start":"02:44.580 ","End":"02:49.630","Text":"square root of minus 1."},{"Start":"02:49.630 ","End":"02:52.410","Text":"Seem to remember that square roots,"},{"Start":"02:52.410 ","End":"02:56.210","Text":"you can only take for positive numbers and 0,"},{"Start":"02:56.210 ","End":"02:58.100","Text":"but you can\u0027t take for negative numbers."},{"Start":"02:58.100 ","End":"02:59.360","Text":"If x is 1,"},{"Start":"02:59.360 ","End":"03:00.890","Text":"square root of 1 minus 1,"},{"Start":"03:00.890 ","End":"03:03.095","Text":"square root of 0 is 0 is okay,"},{"Start":"03:03.095 ","End":"03:05.120","Text":"x is 1 is okay,"},{"Start":"03:05.120 ","End":"03:07.315","Text":"x is 0 is not okay,"},{"Start":"03:07.315 ","End":"03:09.270","Text":"x is 2 is okay."},{"Start":"03:09.270 ","End":"03:11.495","Text":"In general, if you remember,"},{"Start":"03:11.495 ","End":"03:14.105","Text":"when can we take the square root of something,"},{"Start":"03:14.105 ","End":"03:19.430","Text":"which I\u0027ll symbolically just write as a little box,"},{"Start":"03:19.430 ","End":"03:23.520","Text":"to take the square root of x or when does it make sense,"},{"Start":"03:23.520 ","End":"03:26.525","Text":"it\u0027s when we have a positive number."},{"Start":"03:26.525 ","End":"03:29.630","Text":"But as I said, not just positive, we could even have 0."},{"Start":"03:29.630 ","End":"03:31.445","Text":"As long it\u0027s not negative,"},{"Start":"03:31.445 ","End":"03:35.635","Text":"usually called sometimes non-negative numbers when it\u0027s bigger or equal to."},{"Start":"03:35.635 ","End":"03:40.410","Text":"In this case, x minus 1 has to be non-negative."},{"Start":"03:40.410 ","End":"03:46.220","Text":"The domain of definition is x bigger or equal to"},{"Start":"03:46.220 ","End":"03:52.565","Text":"1 because that will give us x minus 1 is bigger or equal to 0 as here."},{"Start":"03:52.565 ","End":"03:54.770","Text":"There\u0027s another example."},{"Start":"03:54.770 ","End":"03:56.569","Text":"Let us take a third example."},{"Start":"03:56.569 ","End":"03:58.445","Text":"In this third example,"},{"Start":"03:58.445 ","End":"04:03.410","Text":"I\u0027m going to take the story from the previous clip,"},{"Start":"04:03.410 ","End":"04:05.210","Text":"and just in case you missed it,"},{"Start":"04:05.210 ","End":"04:08.595","Text":"that\u0027s where we drew a little house."},{"Start":"04:08.595 ","End":"04:12.620","Text":"At the end we made a computation and f(x) was the area."},{"Start":"04:12.620 ","End":"04:14.495","Text":"We actually wrote the formula,"},{"Start":"04:14.495 ","End":"04:16.130","Text":"y was over there,"},{"Start":"04:16.130 ","End":"04:18.515","Text":"or f(x), x squared plus 2 x."},{"Start":"04:18.515 ","End":"04:20.525","Text":"You might be tempted to say,"},{"Start":"04:20.525 ","End":"04:22.895","Text":"when we looking at the domain of definition,"},{"Start":"04:22.895 ","End":"04:25.985","Text":"that you\u0027re allowed to substitute every value of x."},{"Start":"04:25.985 ","End":"04:28.565","Text":"There is nothing out of bounds."},{"Start":"04:28.565 ","End":"04:31.820","Text":"Any number positive or negative, or whatever, we certainly,"},{"Start":"04:31.820 ","End":"04:38.395","Text":"we can square it and take twice of it and add but if you wrote all x,"},{"Start":"04:38.395 ","End":"04:41.095","Text":"I\u0027m putting a big question mark here."},{"Start":"04:41.095 ","End":"04:43.120","Text":"If you wrote that, you would be wrong"},{"Start":"04:43.120 ","End":"04:47.440","Text":"because this is not a purely mathematical abstract problem,"},{"Start":"04:47.440 ","End":"04:53.310","Text":"this comes from the real-world and here x represents the length of the base of the house."},{"Start":"04:53.310 ","End":"04:56.860","Text":"If x is a length of some part of the house,"},{"Start":"04:56.860 ","End":"04:58.615","Text":"then x can\u0027t be everything."},{"Start":"04:58.615 ","End":"05:01.000","Text":"X has to be strictly positive,"},{"Start":"05:01.000 ","End":"05:03.205","Text":"so the answer is no."},{"Start":"05:03.205 ","End":"05:11.485","Text":"The answer is that x is bigger than 0 because x isn\u0027t abstract,"},{"Start":"05:11.485 ","End":"05:14.934","Text":"it represents a length."},{"Start":"05:14.934 ","End":"05:17.970","Text":"This can happen often,"},{"Start":"05:17.970 ","End":"05:23.030","Text":"but mathematically there is no restriction but because of the nature of the variable,"},{"Start":"05:23.030 ","End":"05:24.469","Text":"it may be a length,"},{"Start":"05:24.469 ","End":"05:26.150","Text":"it may be a weight,"},{"Start":"05:26.150 ","End":"05:29.660","Text":"which case may be is 0 is allowed or it could even"},{"Start":"05:29.660 ","End":"05:33.380","Text":"be a temperature which allows plus or minus."},{"Start":"05:33.380 ","End":"05:36.230","Text":"But if it\u0027s a temperature in centigrade,"},{"Start":"05:36.230 ","End":"05:39.500","Text":"it has to be bigger than minus 273,"},{"Start":"05:39.500 ","End":"05:41.795","Text":"for example, which absolute 0."},{"Start":"05:41.795 ","End":"05:46.190","Text":"In other words, when you have a problem which represents a situation in the real-world,"},{"Start":"05:46.190 ","End":"05:48.500","Text":"you have not only the mathematical limitation,"},{"Start":"05:48.500 ","End":"05:50.930","Text":"but you have the real-world limitation which"},{"Start":"05:50.930 ","End":"05:54.630","Text":"determines the domain of definition of a function."}],"ID":5092},{"Watched":false,"Name":"Graphical Description of a Function","Duration":"14m 52s","ChapterTopicVideoID":5105,"CourseChapterTopicPlaylistID":56140,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5105.jpeg","UploadDate":"2020-09-30T13:17:55.5230000","DurationForVideoObject":"PT14M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.675","Text":"In this clip, we\u0027ll talk about the graphical description of a function,"},{"Start":"00:03.675 ","End":"00:07.710","Text":"which is some visual way of portraying,"},{"Start":"00:07.710 ","End":"00:13.365","Text":"presenting a function for often very useful for understanding people like visual things."},{"Start":"00:13.365 ","End":"00:16.485","Text":"Before we talk about drawing the graph of a function,"},{"Start":"00:16.485 ","End":"00:18.758","Text":"we need to go back to basics,"},{"Start":"00:18.758 ","End":"00:21.300","Text":"and talk about coordinate axes."},{"Start":"00:21.300 ","End":"00:25.110","Text":"What this involves is used to be on paper,"},{"Start":"00:25.110 ","End":"00:27.255","Text":"but now it could be anywhere,"},{"Start":"00:27.255 ","End":"00:28.895","Text":"on a computer screen."},{"Start":"00:28.895 ","End":"00:30.785","Text":"There was a plane."},{"Start":"00:30.785 ","End":"00:34.670","Text":"In the plane, there are 2 lines at right angles,"},{"Start":"00:34.670 ","End":"00:37.880","Text":"infinite, but I only have finite space here,"},{"Start":"00:37.880 ","End":"00:41.427","Text":"and the vertical one is known as the y-axis,"},{"Start":"00:41.427 ","End":"00:46.930","Text":"and the horizontal one is called the x-axis and the y-axis,"},{"Start":"00:46.930 ","End":"00:48.350","Text":"has a positive direction,"},{"Start":"00:48.350 ","End":"00:51.410","Text":"which is upwards and the x-axis has a positive direction,"},{"Start":"00:51.410 ","End":"00:53.240","Text":"which is to the right."},{"Start":"00:53.240 ","End":"00:57.080","Text":"This part of it is called the positive x-axis and this part of this is"},{"Start":"00:57.080 ","End":"01:01.255","Text":"the positive y-axis and where they meet at the crosshairs,"},{"Start":"01:01.255 ","End":"01:05.450","Text":"that\u0027s called the origin and often denoted by the letter O,"},{"Start":"01:05.450 ","End":"01:07.460","Text":"and to put numbers on here,"},{"Start":"01:07.460 ","End":"01:09.950","Text":"we first of all need a scale."},{"Start":"01:09.950 ","End":"01:17.150","Text":"What we do is we divide it up into using what are called tick marks like this."},{"Start":"01:17.150 ","End":"01:21.290","Text":"Now it doesn\u0027t have to be that each of these is one unit,"},{"Start":"01:21.290 ","End":"01:23.195","Text":"although it\u0027s commonly so,"},{"Start":"01:23.195 ","End":"01:26.245","Text":"that this might represent where x is 1,"},{"Start":"01:26.245 ","End":"01:27.960","Text":"2, 3,"},{"Start":"01:27.960 ","End":"01:29.330","Text":"4 and so on."},{"Start":"01:29.330 ","End":"01:32.120","Text":"But, this is not necessary and we could,"},{"Start":"01:32.120 ","End":"01:36.500","Text":"for example, let me get my eraser and erase these."},{"Start":"01:36.500 ","End":"01:38.390","Text":"It could be something else."},{"Start":"01:38.390 ","End":"01:41.165","Text":"It could be that we draw every 5,"},{"Start":"01:41.165 ","End":"01:43.040","Text":"maybe this is 5,"},{"Start":"01:43.040 ","End":"01:46.925","Text":"10, 15, 20 etc."},{"Start":"01:46.925 ","End":"01:52.670","Text":"In which case this would be minus 5, minus 10 etc."},{"Start":"01:52.670 ","End":"01:54.740","Text":"As long as they\u0027re equal, it could be 10, 20,"},{"Start":"01:54.740 ","End":"01:55.790","Text":"30, 2,"},{"Start":"01:55.790 ","End":"01:57.350","Text":"4, 6 and so on."},{"Start":"01:57.350 ","End":"02:00.770","Text":"There\u0027s also a vertical scale with tick marks."},{"Start":"02:00.770 ","End":"02:02.150","Text":"Now we draw them,"},{"Start":"02:02.150 ","End":"02:07.350","Text":"I don\u0027t even know if they should be drawn the same or not."},{"Start":"02:07.350 ","End":"02:10.250","Text":"I don\u0027t think they have to be equally spaced."},{"Start":"02:10.250 ","End":"02:14.185","Text":"Certainly the scale doesn\u0027t have to be the same as this scale."},{"Start":"02:14.185 ","End":"02:17.380","Text":"In other words, this could be 5, 10, 15,"},{"Start":"02:17.380 ","End":"02:19.270","Text":"whereas this might be 2,"},{"Start":"02:19.270 ","End":"02:22.325","Text":"4, 6, 8."},{"Start":"02:22.325 ","End":"02:25.220","Text":"Here too, you get negative on the other side,"},{"Start":"02:25.220 ","End":"02:26.705","Text":"this is 0 always."},{"Start":"02:26.705 ","End":"02:28.505","Text":"It could be minus 2,"},{"Start":"02:28.505 ","End":"02:30.830","Text":"minus 4, minus 6,"},{"Start":"02:30.830 ","End":"02:34.130","Text":"or it could be something completely different, 100, 200,"},{"Start":"02:34.130 ","End":"02:36.410","Text":"300 so often taken to be the same,"},{"Start":"02:36.410 ","End":"02:38.210","Text":"but very often also not."},{"Start":"02:38.210 ","End":"02:40.350","Text":"But, the consistency is what\u0027s important."},{"Start":"02:40.350 ","End":"02:42.694","Text":"They should be spaced equally,"},{"Start":"02:42.694 ","End":"02:45.380","Text":"not just graphically, the numbers as well."},{"Start":"02:45.380 ","End":"02:48.545","Text":"You can\u0027t have 5, 10 and then 20, 30, 40."},{"Start":"02:48.545 ","End":"02:50.600","Text":"Now, these are the axes, yes,"},{"Start":"02:50.600 ","End":"02:52.345","Text":"with a scale,"},{"Start":"02:52.345 ","End":"02:54.430","Text":"and what they enable us to do,"},{"Start":"02:54.430 ","End":"02:59.360","Text":"is to give any point in the plane a pair of numbers,"},{"Start":"02:59.360 ","End":"03:03.395","Text":"and conversely, every pair of numbers is a point in the plane."},{"Start":"03:03.395 ","End":"03:06.205","Text":"Let me go back to the simple 1, 2, 3."},{"Start":"03:06.205 ","End":"03:08.250","Text":"Yeah, just to make it a bit simpler,"},{"Start":"03:08.250 ","End":"03:09.510","Text":"we\u0027ll make each 1,"},{"Start":"03:09.510 ","End":"03:11.520","Text":"a single unit, 1, 2, 3, 4,"},{"Start":"03:11.520 ","End":"03:14.340","Text":"5 minus 1 minus 2 then 1, 2,"},{"Start":"03:14.340 ","End":"03:17.370","Text":"3, 4 minus 1 minus 2 for y."},{"Start":"03:17.370 ","End":"03:19.055","Text":"Let me show you an example."},{"Start":"03:19.055 ","End":"03:23.330","Text":"If I say to you the 0.3,2,"},{"Start":"03:23.330 ","End":"03:27.530","Text":"it\u0027s agreed that the first number on the 1 on the left"},{"Start":"03:27.530 ","End":"03:33.194","Text":"represents x and the other 1 represents the y."},{"Start":"03:33.194 ","End":"03:37.830","Text":"That means 3 to the East and 2, to the North,"},{"Start":"03:37.830 ","End":"03:43.160","Text":"if you like, or 3 along the positive x-axis and 2 along the y-axis."},{"Start":"03:43.160 ","End":"03:46.640","Text":"What you do in practice is that if I locate the 3,"},{"Start":"03:46.640 ","End":"03:48.815","Text":"and I locate the 2 here,"},{"Start":"03:48.815 ","End":"03:50.855","Text":"and with dotted lines,"},{"Start":"03:50.855 ","End":"03:52.515","Text":"I locate this point,"},{"Start":"03:52.515 ","End":"03:55.945","Text":"this is the point which I call 3,2."},{"Start":"03:55.945 ","End":"04:01.550","Text":"Conversely, if I\u0027m given a point in the plane,"},{"Start":"04:01.550 ","End":"04:06.305","Text":"for example, let\u0027s say this one here and you ask,"},{"Start":"04:06.305 ","End":"04:08.480","Text":"what are the coordinates of this point?"},{"Start":"04:08.480 ","End":"04:14.390","Text":"Then we drop perpendiculars to the y-axis and the x-axis,"},{"Start":"04:14.390 ","End":"04:21.920","Text":"and we see that where it hits the x-axis is the point minus 2 and x goes first,"},{"Start":"04:21.920 ","End":"04:26.020","Text":"and the y-axis at the point minus 1,"},{"Start":"04:26.020 ","End":"04:29.670","Text":"this point is labeled minus 2 minus 1,"},{"Start":"04:29.670 ","End":"04:30.950","Text":"so it\u0027s a two-way thing."},{"Start":"04:30.950 ","End":"04:32.735","Text":"Each pair of numbers,"},{"Start":"04:32.735 ","End":"04:35.000","Text":"x and y, gives me a point on the plane,"},{"Start":"04:35.000 ","End":"04:37.340","Text":"and each point on the plane gives me a pair of numbers x,"},{"Start":"04:37.340 ","End":"04:39.575","Text":"y. Here\u0027s another example."},{"Start":"04:39.575 ","End":"04:43.060","Text":"This point that\u0027s on the axis here,"},{"Start":"04:43.100 ","End":"04:46.485","Text":"across the x-axis it\u0027s how many up?"},{"Start":"04:46.485 ","End":"04:51.810","Text":"It\u0027s 4 in the x direction and nothing down in the y direction,"},{"Start":"04:51.810 ","End":"04:54.725","Text":"or if you\u0027d like the perpendicular is just here at 0."},{"Start":"04:54.725 ","End":"04:58.055","Text":"This is the 0.4,0."},{"Start":"04:58.055 ","End":"05:00.890","Text":"Now, let\u0027s take a point on the y-axis."},{"Start":"05:00.890 ","End":"05:03.905","Text":"Let\u0027s say this one,"},{"Start":"05:03.905 ","End":"05:09.860","Text":"this will be on the corresponding bit on the x-axis is 0."},{"Start":"05:09.860 ","End":"05:12.695","Text":"On the y-axis, we\u0027re up to minus 3,"},{"Start":"05:12.695 ","End":"05:17.194","Text":"so this point here is going to be 0 first,"},{"Start":"05:17.194 ","End":"05:21.155","Text":"the x-direction, and minus 3 in the y-direction."},{"Start":"05:21.155 ","End":"05:23.495","Text":"Which means that it\u0027s not up it\u0027s down."},{"Start":"05:23.495 ","End":"05:26.180","Text":"Of course, this point here is exceptional."},{"Start":"05:26.180 ","End":"05:29.615","Text":"At this point, which is the origin,"},{"Start":"05:29.615 ","End":"05:35.045","Text":"is 0,0, that\u0027s the origin."},{"Start":"05:35.045 ","End":"05:38.360","Text":"Now that we have the relation between points,"},{"Start":"05:38.360 ","End":"05:39.515","Text":"and pairs of numbers,"},{"Start":"05:39.515 ","End":"05:44.390","Text":"now we can really start to draw graphs where typically we use y, and x,"},{"Start":"05:44.390 ","End":"05:48.320","Text":"and y as a function of x and that enables us to draw the graph,"},{"Start":"05:48.320 ","End":"05:51.650","Text":"before that is something I forgot to mention, and here\u0027s the place."},{"Start":"05:51.650 ","End":"05:58.610","Text":"The 2 lines, the axes divide the plane into 4 parts,"},{"Start":"05:58.610 ","End":"06:01.055","Text":"into quarters, if you like."},{"Start":"06:01.055 ","End":"06:03.380","Text":"Each one is called a quadrant,"},{"Start":"06:03.380 ","End":"06:06.845","Text":"and this part here is called the first quadrant,"},{"Start":"06:06.845 ","End":"06:10.670","Text":"sometimes we write it as the first quadrant,"},{"Start":"06:10.670 ","End":"06:12.510","Text":"but in my days,"},{"Start":"06:12.510 ","End":"06:17.044","Text":"they used the Roman numerals in case you see that this is the first quadrant,"},{"Start":"06:17.044 ","End":"06:19.415","Text":"this is the second quadrant,"},{"Start":"06:19.415 ","End":"06:22.480","Text":"this is the third quadrant,"},{"Start":"06:22.480 ","End":"06:25.235","Text":"and this is the fourth quadrant."},{"Start":"06:25.235 ","End":"06:27.685","Text":"I\u0027ll take the example function,"},{"Start":"06:27.685 ","End":"06:29.385","Text":"one that we want to sketch,"},{"Start":"06:29.385 ","End":"06:31.365","Text":"as y equals x squared."},{"Start":"06:31.365 ","End":"06:33.835","Text":"Something very basic."},{"Start":"06:33.835 ","End":"06:37.400","Text":"What we do is partly it\u0027s an art,"},{"Start":"06:37.400 ","End":"06:39.530","Text":"but mostly it\u0027s a precise thing."},{"Start":"06:39.530 ","End":"06:42.499","Text":"Usually the art is getting the right scale."},{"Start":"06:42.499 ","End":"06:46.925","Text":"But I\u0027m also guessing how many numbers to take in the sample."},{"Start":"06:46.925 ","End":"06:50.180","Text":"But, we basically make a table of values."},{"Start":"06:50.180 ","End":"06:54.470","Text":"Most of you probably at least vaguely remember doing this thing,"},{"Start":"06:54.470 ","End":"06:56.300","Text":"and on the left column,"},{"Start":"06:56.300 ","End":"06:58.370","Text":"we call it x."},{"Start":"06:58.370 ","End":"07:00.740","Text":"Let me put values of x,"},{"Start":"07:00.740 ","End":"07:05.450","Text":"and let me compute since y is the dependent variable,"},{"Start":"07:05.450 ","End":"07:06.650","Text":"y is the function of x."},{"Start":"07:06.650 ","End":"07:13.410","Text":"We first put in the value of x and then we find its corresponding unique y."},{"Start":"07:13.410 ","End":"07:17.140","Text":"Let\u0027s say x is 0."},{"Start":"07:17.140 ","End":"07:19.015","Text":"We\u0027ll click x=1,"},{"Start":"07:19.015 ","End":"07:22.285","Text":"x=2, maybe even 3."},{"Start":"07:22.285 ","End":"07:24.265","Text":"You don\u0027t have to use it in the end."},{"Start":"07:24.265 ","End":"07:27.085","Text":"Let me compute y. Well, y is x^2,"},{"Start":"07:27.085 ","End":"07:32.456","Text":"so 0^2 is 0,"},{"Start":"07:32.456 ","End":"07:35.436","Text":"1^2 is 1,"},{"Start":"07:35.436 ","End":"07:36.622","Text":"2^2 is 4,"},{"Start":"07:36.622 ","End":"07:37.780","Text":"3^2 is 9,"},{"Start":"07:37.780 ","End":"07:40.480","Text":"and I don\u0027t think will fit very much more on this."},{"Start":"07:40.480 ","End":"07:41.995","Text":"Let me first see."},{"Start":"07:41.995 ","End":"07:44.169","Text":"Let\u0027s just draw something."},{"Start":"07:44.169 ","End":"07:47.560","Text":"You know what, I\u0027ll just put it on pause while I draw."},{"Start":"07:47.560 ","End":"07:51.475","Text":"Here I am with my coordinate axes,"},{"Start":"07:51.475 ","End":"07:53.500","Text":"and I\u0027ve got some values."},{"Start":"07:53.500 ","End":"07:59.365","Text":"I want to decide on a scale that roughly what I was going to fit in."},{"Start":"07:59.365 ","End":"08:03.100","Text":"What I notice is that if I just need these,"},{"Start":"08:03.100 ","End":"08:05.350","Text":"all I need to go is up to 9."},{"Start":"08:05.350 ","End":"08:08.785","Text":"But, I see that the y-value is generally become larger than x."},{"Start":"08:08.785 ","End":"08:12.655","Text":"Why don\u0027t I say that each unit here,"},{"Start":"08:12.655 ","End":"08:15.340","Text":"there\u0027s going to be 1 unit,"},{"Start":"08:15.340 ","End":"08:18.490","Text":"1, 2, 3, maybe 4."},{"Start":"08:18.490 ","End":"08:22.705","Text":"Here let\u0027s say that each tick is 2 units."},{"Start":"08:22.705 ","End":"08:25.855","Text":"They look the same size."},{"Start":"08:25.855 ","End":"08:31.519","Text":"If this is 1,"},{"Start":"08:31.519 ","End":"08:33.398","Text":"2, 3, 4,"},{"Start":"08:33.398 ","End":"08:35.485","Text":"just to be different."},{"Start":"08:35.485 ","End":"08:37.970","Text":"Here we\u0027ll have every 2, 2, 4, 6, 8, 10."},{"Start":"08:40.680 ","End":"08:43.360","Text":"Well, it\u0027s not just to be different because I saw I\u0027m"},{"Start":"08:43.360 ","End":"08:45.280","Text":"going to have to fit 9 in if I do it,"},{"Start":"08:45.280 ","End":"08:46.540","Text":"same scale of this,"},{"Start":"08:46.540 ","End":"08:47.755","Text":"I won\u0027t get 9 in."},{"Start":"08:47.755 ","End":"08:49.660","Text":"I doubled up."},{"Start":"08:49.660 ","End":"08:54.520","Text":"Again, across here it\u0027s going to be minus 1."},{"Start":"08:54.520 ","End":"08:56.680","Text":"This is, as I say, are supposed to be equal."},{"Start":"08:56.680 ","End":"09:01.640","Text":"It\u0027s just me minus 3, minus 4."},{"Start":"09:05.670 ","End":"09:08.440","Text":"I won\u0027t really be needing these fields."},{"Start":"09:08.440 ","End":"09:12.400","Text":"You will see because I will be squaring numbers that only came up positive,"},{"Start":"09:12.400 ","End":"09:14.170","Text":"but I didn\u0027t know that when I started,"},{"Start":"09:14.170 ","End":"09:17.590","Text":"so I\u0027m going to put in some like minus 2,"},{"Start":"09:17.590 ","End":"09:22.370","Text":"minus 4, minus 6 and so on."},{"Start":"09:24.690 ","End":"09:28.555","Text":"Also, I didn\u0027t put any negatives in here."},{"Start":"09:28.555 ","End":"09:30.685","Text":"Actually, that\u0027s going to be easy to do."},{"Start":"09:30.685 ","End":"09:34.270","Text":"Because if 0^2 is 0,"},{"Start":"09:34.270 ","End":"09:36.460","Text":"well 0 is neither positive nor negative."},{"Start":"09:36.460 ","End":"09:37.934","Text":"But if 1^2 is 1,"},{"Start":"09:37.934 ","End":"09:41.695","Text":"then minus 1^2 is also going to be 1,"},{"Start":"09:41.695 ","End":"09:44.907","Text":"minus 2^2 is also going to be 4,"},{"Start":"09:44.907 ","End":"09:50.160","Text":"and minus 3^2 is also going to be 9."},{"Start":"09:50.160 ","End":"09:54.765","Text":"As you\u0027ll see this will give me a certain symmetry."},{"Start":"09:54.765 ","End":"09:57.665","Text":"In fact, we can see it already."},{"Start":"09:57.665 ","End":"10:01.150","Text":"If we plot 0, 0,"},{"Start":"10:01.150 ","End":"10:04.780","Text":"I forgot to mention from each of these pairs of x and y,"},{"Start":"10:04.780 ","End":"10:07.555","Text":"we get the coordinates of a point."},{"Start":"10:07.555 ","End":"10:09.355","Text":"This will be 0, 0,"},{"Start":"10:09.355 ","End":"10:11.620","Text":"this will be 1,"},{"Start":"10:11.620 ","End":"10:14.102","Text":"1 this 2,"},{"Start":"10:14.102 ","End":"10:16.435","Text":"4, 3, 9."},{"Start":"10:16.435 ","End":"10:19.585","Text":"Numbers don\u0027t have to be in any particular order in the table."},{"Start":"10:19.585 ","End":"10:23.665","Text":"Each one represents a point and we plot it in whatever order we choose."},{"Start":"10:23.665 ","End":"10:27.100","Text":"We have the point minus 1, 1."},{"Start":"10:27.100 ","End":"10:28.930","Text":"That\u0027s a plus 1."},{"Start":"10:28.930 ","End":"10:39.830","Text":"Just a second."},{"Start":"10:40.980 ","End":"10:44.365","Text":"My pen is not writing."},{"Start":"10:44.365 ","End":"10:46.790","Text":"Minus 2, 4,"},{"Start":"10:47.280 ","End":"10:49.990","Text":"minus 3, 9."},{"Start":"10:49.990 ","End":"10:52.210","Text":"These are the coordinates of the points,"},{"Start":"10:52.210 ","End":"10:54.730","Text":"and then we plot them as we saw earlier."},{"Start":"10:54.730 ","End":"10:58.915","Text":"I might use a different color, it might help."},{"Start":"10:58.915 ","End":"11:02.080","Text":"Let\u0027s take blue."},{"Start":"11:02.080 ","End":"11:04.290","Text":"So 0,"},{"Start":"11:04.290 ","End":"11:12.400","Text":"0 will be here, 1, 1."},{"Start":"11:12.400 ","End":"11:13.519","Text":"You\u0027ve seen it all before,"},{"Start":"11:13.519 ","End":"11:15.670","Text":"I\u0027m not going to go into too much detail."},{"Start":"11:15.670 ","End":"11:23.390","Text":"2, 4 would be somewhere across here."},{"Start":"11:23.460 ","End":"11:26.395","Text":"I could put the dotted lines in,"},{"Start":"11:26.395 ","End":"11:28.330","Text":"but only have to erase them later."},{"Start":"11:28.330 ","End":"11:30.565","Text":"Maybe for one point, I\u0027ll do it."},{"Start":"11:30.565 ","End":"11:36.190","Text":"2, 4 is at this point,"},{"Start":"11:36.190 ","End":"11:40.030","Text":"and 3, 9 so 3 is here, 9 is here."},{"Start":"11:40.030 ","End":"11:43.990","Text":"Imagining the dotted lines, so 3,"},{"Start":"11:43.990 ","End":"11:47.005","Text":"9 will be somewhere here,"},{"Start":"11:47.005 ","End":"11:49.465","Text":"then minus 1, 1."},{"Start":"11:49.465 ","End":"11:54.055","Text":"You can see what I was talking about symmetry before."},{"Start":"11:54.055 ","End":"11:57.040","Text":"Then minus 2,"},{"Start":"11:57.040 ","End":"12:01.870","Text":"4 the symmetry this one."},{"Start":"12:01.870 ","End":"12:05.885","Text":"My crooked drawing doesn\u0027t look so symmetrical."},{"Start":"12:05.885 ","End":"12:11.480","Text":"Minus 3, 9."},{"Start":"12:12.780 ","End":"12:22.510","Text":"Then we try to plot a line through these points but it\u0027ll come out okay,"},{"Start":"12:22.510 ","End":"12:25.700","Text":"there\u0027s not much worry."},{"Start":"12:26.220 ","End":"12:29.230","Text":"It\u0027s a bit too pointed and a bit lopsided,"},{"Start":"12:29.230 ","End":"12:30.400","Text":"but you get the idea,"},{"Start":"12:30.400 ","End":"12:32.870","Text":"the idea is certainly there."},{"Start":"12:33.000 ","End":"12:40.375","Text":"This is not a very good system altogether of figuring out what the curve looks like,"},{"Start":"12:40.375 ","End":"12:42.580","Text":"the method of plotting points."},{"Start":"12:42.580 ","End":"12:47.365","Text":"I mean, how do you know how to draw the line through the points?"},{"Start":"12:47.365 ","End":"12:52.510","Text":"Isn\u0027t it possible that, for example,"},{"Start":"12:52.510 ","End":"12:57.640","Text":"it was something completely different that went through these,"},{"Start":"12:57.640 ","End":"12:59.560","Text":"maybe this point is here,"},{"Start":"12:59.560 ","End":"13:04.540","Text":"but then it goes down here,"},{"Start":"13:04.540 ","End":"13:08.395","Text":"and then up here and here."},{"Start":"13:08.395 ","End":"13:13.060","Text":"A couple of zigzags through here down to here."},{"Start":"13:13.060 ","End":"13:14.770","Text":"We don\u0027t really know."},{"Start":"13:14.770 ","End":"13:16.570","Text":"I\u0027ve drawn this in faint, but you get the idea"},{"Start":"13:16.570 ","End":"13:18.790","Text":"just because you have a certain number of points,"},{"Start":"13:18.790 ","End":"13:21.572","Text":"you don\u0027t know how it behaves in-between."},{"Start":"13:21.572 ","End":"13:23.260","Text":"Even if you draw a large number,"},{"Start":"13:23.260 ","End":"13:25.870","Text":"maybe the computer does it that way."},{"Start":"13:25.870 ","End":"13:29.200","Text":"Just lots and lots of little close to"},{"Start":"13:29.200 ","End":"13:33.590","Text":"each other points and then draw it through as best you can."},{"Start":"13:34.650 ","End":"13:42.790","Text":"Later learn how to explore and analyze functions from the equations."},{"Start":"13:42.790 ","End":"13:45.730","Text":"We\u0027ll know that it has to have a minimum here,"},{"Start":"13:45.730 ","End":"13:51.160","Text":"and curve convexity here and decrease here and so on, all these things."},{"Start":"13:51.160 ","End":"13:58.240","Text":"But, this straightforward point plotting is one way,"},{"Start":"13:58.240 ","End":"14:01.240","Text":"and it\u0027s a quick way, and usually it\u0027s roughly right,"},{"Start":"14:01.240 ","End":"14:06.100","Text":"but in principle it\u0027s not very founded."},{"Start":"14:06.100 ","End":"14:09.680","Text":"You don\u0027t know what happens between the points."},{"Start":"14:11.580 ","End":"14:15.730","Text":"This function here, remember it\u0027s y =x^2,"},{"Start":"14:15.730 ","End":"14:17.920","Text":"and also we should label the graph."},{"Start":"14:17.920 ","End":"14:19.690","Text":"I happen to be familiar with it."},{"Start":"14:19.690 ","End":"14:21.865","Text":"I know it does look something like this,"},{"Start":"14:21.865 ","End":"14:25.470","Text":"but we do need some better technique and we will learn them"},{"Start":"14:25.470 ","End":"14:30.140","Text":"for finding out what a function looks like other than lots and lots of points."},{"Start":"14:30.140 ","End":"14:40.185","Text":"You compute and plot and then interpolate or put a line through."},{"Start":"14:40.185 ","End":"14:46.070","Text":"But it\u0027s good for rough and ready for quick idea, almost always works."},{"Start":"14:46.070 ","End":"14:48.260","Text":"It gives you a rough idea."},{"Start":"14:48.260 ","End":"14:53.370","Text":"That\u0027s all I have to say at this point. That\u0027s all."}],"ID":5093},{"Watched":false,"Name":"Increase and decrease of a Function","Duration":"6m 41s","ChapterTopicVideoID":5106,"CourseChapterTopicPlaylistID":56140,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5106.jpeg","UploadDate":"2020-09-30T12:55:14.2230000","DurationForVideoObject":"PT6M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"In this clip, we\u0027ll learn about the increase and decrease of a function."},{"Start":"00:03.960 ","End":"00:07.125","Text":"Here I\u0027ve already prepared a coordinate axis,"},{"Start":"00:07.125 ","End":"00:09.960","Text":"in it I\u0027ll sketch a function,"},{"Start":"00:09.960 ","End":"00:13.605","Text":"say this is y=f(x),"},{"Start":"00:13.605 ","End":"00:16.080","Text":"and in this pair of coordinate axis,"},{"Start":"00:16.080 ","End":"00:18.330","Text":"I\u0027ll draw something that looks a bit different."},{"Start":"00:18.330 ","End":"00:19.575","Text":"Let\u0027s say this,"},{"Start":"00:19.575 ","End":"00:21.600","Text":"might also call it f(x)."},{"Start":"00:21.600 ","End":"00:26.565","Text":"Now there\u0027s a difference in these 2 graphs and I\u0027d like to demonstrate it."},{"Start":"00:26.565 ","End":"00:29.895","Text":"If you notice, as you\u0027re going from left to right,"},{"Start":"00:29.895 ","End":"00:32.580","Text":"notice that you\u0027re climbing upwards."},{"Start":"00:32.580 ","End":"00:35.135","Text":"You\u0027re going along and as you go along,"},{"Start":"00:35.135 ","End":"00:37.775","Text":"you\u0027re climbing up like upper hill or something."},{"Start":"00:37.775 ","End":"00:39.350","Text":"Whereas in this 1,"},{"Start":"00:39.350 ","End":"00:41.570","Text":"as you\u0027re going from right to left,"},{"Start":"00:41.570 ","End":"00:45.889","Text":"you are actually going downhill or climbing down."},{"Start":"00:45.889 ","End":"00:49.040","Text":"This is called increasing and this is called decreasing."},{"Start":"00:49.040 ","End":"00:51.010","Text":"Now that\u0027s not very precise,"},{"Start":"00:51.010 ","End":"00:53.469","Text":"let\u0027s try and make it a little more formal."},{"Start":"00:53.469 ","End":"00:56.255","Text":"I\u0027ve taken off the arrows and we\u0027ll try and"},{"Start":"00:56.255 ","End":"00:59.360","Text":"say something more precise than going up a hill,"},{"Start":"00:59.360 ","End":"01:00.935","Text":"going down a hill and so on."},{"Start":"01:00.935 ","End":"01:03.080","Text":"What I\u0027d like to do in the first case,"},{"Start":"01:03.080 ","End":"01:04.550","Text":"is take a couple of points,"},{"Start":"01:04.550 ","End":"01:07.865","Text":"let\u0027s say this point and this point."},{"Start":"01:07.865 ","End":"01:12.110","Text":"Each of these points has an x-coordinate and a y-coordinate."},{"Start":"01:12.110 ","End":"01:13.970","Text":"Let\u0027s say this is x_1,"},{"Start":"01:13.970 ","End":"01:16.670","Text":"y_1 and the other point will be x_2,"},{"Start":"01:16.670 ","End":"01:22.010","Text":"y_2, so here we have x_2 and here we have y_2."},{"Start":"01:22.010 ","End":"01:25.220","Text":"Now notice that when we go from x_1 to x_2,"},{"Start":"01:25.220 ","End":"01:28.175","Text":"it\u0027s in the positive direction, it\u0027s increasing,"},{"Start":"01:28.175 ","End":"01:32.335","Text":"as y goes from y_1 to y_2 it also increases,"},{"Start":"01:32.335 ","End":"01:35.735","Text":"so this function in this picture is increasing."},{"Start":"01:35.735 ","End":"01:37.640","Text":"What this means is,"},{"Start":"01:37.640 ","End":"01:40.895","Text":"besides the informal way of going up the hill and so forth,"},{"Start":"01:40.895 ","End":"01:46.065","Text":"is that if x_2 is bigger than x_1,"},{"Start":"01:46.065 ","End":"01:49.725","Text":"then y_2 is bigger than y_1."},{"Start":"01:49.725 ","End":"01:51.270","Text":"This is what I tried to show you here,"},{"Start":"01:51.270 ","End":"01:52.820","Text":"x_2 is bigger than x_1,"},{"Start":"01:52.820 ","End":"01:55.940","Text":"which means we\u0027re moving to the right and y_2 is bigger than y_1,"},{"Start":"01:55.940 ","End":"01:58.925","Text":"which means I\u0027m moving up, so as we go right we go upwards."},{"Start":"01:58.925 ","End":"02:01.650","Text":"Another way of saying this is,"},{"Start":"02:01.650 ","End":"02:04.690","Text":"that if x increases, y increases."},{"Start":"02:04.690 ","End":"02:06.095","Text":"Let\u0027s take the other graph."},{"Start":"02:06.095 ","End":"02:08.825","Text":"In this picture, we have a picture of where F"},{"Start":"02:08.825 ","End":"02:13.835","Text":"decreasing and that means again that if we take 2 points,"},{"Start":"02:13.835 ","End":"02:15.350","Text":"where this 1 is x_1,"},{"Start":"02:15.350 ","End":"02:18.875","Text":"y_1 and move to an adjacent point,"},{"Start":"02:18.875 ","End":"02:20.285","Text":"let\u0027s say this 1,"},{"Start":"02:20.285 ","End":"02:22.270","Text":"which would be x_2,"},{"Start":"02:22.270 ","End":"02:25.305","Text":"y_2, we have a reversal here,"},{"Start":"02:25.305 ","End":"02:26.610","Text":"so F is decreasing,"},{"Start":"02:26.610 ","End":"02:31.185","Text":"which means that if x_2 is bigger than x_1,"},{"Start":"02:31.185 ","End":"02:34.230","Text":"then and notice the reversal,"},{"Start":"02:34.230 ","End":"02:36.270","Text":"here y_2 was bigger than y_1,"},{"Start":"02:36.270 ","End":"02:38.780","Text":"here y_2 is less than y_1."},{"Start":"02:38.780 ","End":"02:44.120","Text":"This formal thing says basically that if x increases,"},{"Start":"02:44.120 ","End":"02:48.200","Text":"then y decreases the opposite."},{"Start":"02:48.200 ","End":"02:49.395","Text":"The difference is,"},{"Start":"02:49.395 ","End":"02:54.170","Text":"that here if x_2 is bigger than x_1 then y_2 is also bigger than y_1,"},{"Start":"02:54.170 ","End":"02:58.430","Text":"whereas here y_2 less than y_1 and then that\u0027s slightly less formal,"},{"Start":"02:58.430 ","End":"02:59.990","Text":"here if x increases,"},{"Start":"02:59.990 ","End":"03:01.835","Text":"then y increases,"},{"Start":"03:01.835 ","End":"03:03.655","Text":"but in the other case,"},{"Start":"03:03.655 ","End":"03:07.230","Text":"if x increases, then y decreases."},{"Start":"03:07.230 ","End":"03:12.795","Text":"I could even say that y is f(x) or just f even."},{"Start":"03:12.795 ","End":"03:15.720","Text":"That\u0027s why f is called increasing and decreasing,"},{"Start":"03:15.720 ","End":"03:17.150","Text":"because if x increases,"},{"Start":"03:17.150 ","End":"03:18.710","Text":"which is a natural way to go,"},{"Start":"03:18.710 ","End":"03:21.980","Text":"then here y increases and here y decreases."},{"Start":"03:21.980 ","End":"03:23.540","Text":"That\u0027s for this part."},{"Start":"03:23.540 ","End":"03:29.750","Text":"Next we\u0027ll be talking about how single function can be also increasing and decreasing,"},{"Start":"03:29.750 ","End":"03:31.550","Text":"but in different parts."},{"Start":"03:31.550 ","End":"03:32.870","Text":"While you weren\u0027t looking,"},{"Start":"03:32.870 ","End":"03:34.550","Text":"I drew a little sketch,"},{"Start":"03:34.550 ","End":"03:38.270","Text":"this is going to be an example of a graph or"},{"Start":"03:38.270 ","End":"03:43.055","Text":"a function which has both increasing and decreasing parts."},{"Start":"03:43.055 ","End":"03:46.085","Text":"We\u0027ll call this y=f(x)."},{"Start":"03:46.085 ","End":"03:47.944","Text":"Just looking at the picture,"},{"Start":"03:47.944 ","End":"03:52.130","Text":"you can see that the graph starts by going up,"},{"Start":"03:52.130 ","End":"03:54.605","Text":"as we\u0027re going all the time to the right,"},{"Start":"03:54.605 ","End":"03:56.460","Text":"but as we go to the right here,"},{"Start":"03:56.460 ","End":"03:59.490","Text":"the y increases up to a certain point,"},{"Start":"03:59.490 ","End":"04:01.748","Text":"then it starts going down again,"},{"Start":"04:01.748 ","End":"04:04.255","Text":"and then we start going up again."},{"Start":"04:04.255 ","End":"04:10.084","Text":"Here, we\u0027re increasing up to a certain point,"},{"Start":"04:10.084 ","End":"04:13.700","Text":"which happens to be 1, 1 itself,"},{"Start":"04:13.700 ","End":"04:16.340","Text":"we can\u0027t say it\u0027s neither increasing nor increasing,"},{"Start":"04:16.340 ","End":"04:17.810","Text":"but after the 1,"},{"Start":"04:17.810 ","End":"04:20.020","Text":"we start downwards,"},{"Start":"04:20.020 ","End":"04:23.505","Text":"till we get to where x=4."},{"Start":"04:23.505 ","End":"04:25.403","Text":"So there\u0027s decreasing,"},{"Start":"04:25.403 ","End":"04:29.985","Text":"and then we start increasing again from 4 to,"},{"Start":"04:29.985 ","End":"04:33.005","Text":"in general this function continues indefinitely."},{"Start":"04:33.005 ","End":"04:39.440","Text":"Think I\u0027ll just emphasize it a bit with the arrows there here we going up and up and up"},{"Start":"04:39.440 ","End":"04:46.540","Text":"and here we\u0027re going down and down and here we\u0027re going back up and up and so on."},{"Start":"04:46.540 ","End":"04:50.375","Text":"Now, typical question might be as follows,"},{"Start":"04:50.375 ","End":"04:55.355","Text":"find the intervals of increase"},{"Start":"04:55.355 ","End":"05:00.740","Text":"and decrease of the function or just say of F. First of all,"},{"Start":"05:00.740 ","End":"05:02.660","Text":"we\u0027ll look at the increase and we\u0027ll see that"},{"Start":"05:02.660 ","End":"05:06.410","Text":"the increase is all the way down from wherever,"},{"Start":"05:06.410 ","End":"05:08.419","Text":"some people would say minus infinity,"},{"Start":"05:08.419 ","End":"05:10.505","Text":"but it\u0027s just from to wherever,"},{"Start":"05:10.505 ","End":"05:13.610","Text":"up to the point where X=1."},{"Start":"05:13.610 ","End":"05:15.185","Text":"Now what concerns us,"},{"Start":"05:15.185 ","End":"05:17.060","Text":"is actually not on the graph,"},{"Start":"05:17.060 ","End":"05:19.070","Text":"but on the x-axis,"},{"Start":"05:19.070 ","End":"05:21.095","Text":"so really what I say, the yellow,"},{"Start":"05:21.095 ","End":"05:25.886","Text":"I mean the shaded part up to here."},{"Start":"05:25.886 ","End":"05:27.230","Text":"Here it\u0027s still increasing,"},{"Start":"05:27.230 ","End":"05:29.710","Text":"so I\u0027ll put some more lines like this,"},{"Start":"05:29.710 ","End":"05:33.095","Text":"the decrease will be all the way from 1-4."},{"Start":"05:33.095 ","End":"05:35.465","Text":"What we would write as an answer,"},{"Start":"05:35.465 ","End":"05:41.485","Text":"is that the interval of increase is x is less than 1,"},{"Start":"05:41.485 ","End":"05:45.275","Text":"together with, some people write or and some people write and."},{"Start":"05:45.275 ","End":"05:51.020","Text":"I like to write or because each particular x can either be less than 1 or,"},{"Start":"05:51.020 ","End":"05:53.975","Text":"in this part here, x is bigger than 4."},{"Start":"05:53.975 ","End":"05:58.125","Text":"For the decrease, the decrease is between 1 and 4,"},{"Start":"05:58.125 ","End":"06:02.835","Text":"so 1 is less than x is less than 4,"},{"Start":"06:02.835 ","End":"06:04.550","Text":"means x is somewhere in here."},{"Start":"06:04.550 ","End":"06:07.010","Text":"Notice I use strict inequalities,"},{"Start":"06:07.010 ","End":"06:10.355","Text":"because it\u0027s definitely increasing when x is less than 1,"},{"Start":"06:10.355 ","End":"06:11.900","Text":"at 1 itself,"},{"Start":"06:11.900 ","End":"06:16.490","Text":"it\u0027s the border between the increase and decrease and it\u0027s neither this nor that,"},{"Start":"06:16.490 ","End":"06:17.855","Text":"so it\u0027s not included."},{"Start":"06:17.855 ","End":"06:19.880","Text":"Similarly with the 4,"},{"Start":"06:19.880 ","End":"06:22.240","Text":"it\u0027s between a decrease and an increase,"},{"Start":"06:22.240 ","End":"06:23.660","Text":"so I\u0027m not including it,"},{"Start":"06:23.660 ","End":"06:26.864","Text":"but between 1 and 4 will definitely decreasing."},{"Start":"06:26.864 ","End":"06:29.300","Text":"Then again, when x is bigger than 4,"},{"Start":"06:29.300 ","End":"06:32.420","Text":"this part here would definitely increasing again."},{"Start":"06:32.420 ","End":"06:34.850","Text":"Actually there are 2 intervals of increase,"},{"Start":"06:34.850 ","End":"06:38.180","Text":"the 2 yellow bits and 1 interval of decrease."},{"Start":"06:38.180 ","End":"06:42.330","Text":"That\u0027s how we would write it and that answers the question."}],"ID":5094},{"Watched":false,"Name":"Positivity and Negativity of a Function","Duration":"4m 47s","ChapterTopicVideoID":5107,"CourseChapterTopicPlaylistID":56140,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5107.jpeg","UploadDate":"2020-09-30T13:02:49.5230000","DurationForVideoObject":"PT4M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.630","Text":"In this clip, we\u0027ll learn about the concept of positivity and negativity of a function."},{"Start":"00:06.630 ","End":"00:08.849","Text":"I\u0027ve brought with me a couple of coordinate"},{"Start":"00:08.849 ","End":"00:11.850","Text":"axis to demonstrate this 1 will be for the positivity,"},{"Start":"00:11.850 ","End":"00:13.620","Text":"this 1 for the negativity."},{"Start":"00:13.620 ","End":"00:17.400","Text":"Here I\u0027ll sketch something like the following."},{"Start":"00:17.400 ","End":"00:21.465","Text":"Well, if you look at any given point here on the graph,"},{"Start":"00:21.465 ","End":"00:23.490","Text":"the y is positive."},{"Start":"00:23.490 ","End":"00:25.950","Text":"It\u0027s above the 0 line."},{"Start":"00:25.950 ","End":"00:29.010","Text":"So this is positive, this is positive, positive, positive."},{"Start":"00:29.010 ","End":"00:33.270","Text":"Every x I choose will give me a positive y so that"},{"Start":"00:33.270 ","End":"00:38.175","Text":"whenever I have that f(x) is bigger than 0,"},{"Start":"00:38.175 ","End":"00:40.430","Text":"but not just for a particular x,"},{"Start":"00:40.430 ","End":"00:42.604","Text":"but for all x,"},{"Start":"00:42.604 ","End":"00:45.755","Text":"then f is called positive."},{"Start":"00:45.755 ","End":"00:48.500","Text":"I mean, a positive function,"},{"Start":"00:48.500 ","End":"00:51.500","Text":"not that each individual x is positive."},{"Start":"00:51.500 ","End":"00:54.350","Text":"A number we know what it means for it to be positive,"},{"Start":"00:54.350 ","End":"00:56.255","Text":"but a function to be positive,"},{"Start":"00:56.255 ","End":"01:00.695","Text":"we say that when f is positive for all x."},{"Start":"01:00.695 ","End":"01:04.835","Text":"Now graphically, it has an obvious graphical meaning,"},{"Start":"01:04.835 ","End":"01:07.985","Text":"it means above the x-axis."},{"Start":"01:07.985 ","End":"01:12.155","Text":"See the whole function is floating above the x-axis."},{"Start":"01:12.155 ","End":"01:13.640","Text":"It\u0027s always positive."},{"Start":"01:13.640 ","End":"01:14.780","Text":"Now, likewise,"},{"Start":"01:14.780 ","End":"01:16.730","Text":"you can guess what I\u0027m going to do here."},{"Start":"01:16.730 ","End":"01:18.605","Text":"I\u0027m going to do the opposite."},{"Start":"01:18.605 ","End":"01:23.960","Text":"Notice that every given point here is negative, it\u0027s below 0."},{"Start":"01:23.960 ","End":"01:26.375","Text":"All of the points are negative."},{"Start":"01:26.375 ","End":"01:31.835","Text":"We have here is that f(x) is strictly negative,"},{"Start":"01:31.835 ","End":"01:34.835","Text":"less than 0 for all x."},{"Start":"01:34.835 ","End":"01:39.395","Text":"Then we say under these circumstances that f is negative,"},{"Start":"01:39.395 ","End":"01:43.535","Text":"we introduced 2 new concepts for a function,"},{"Start":"01:43.535 ","End":"01:49.265","Text":"and those concepts are positive and negative."},{"Start":"01:49.265 ","End":"01:55.490","Text":"In this case, it also has a graphical interpretation that when f is negative,"},{"Start":"01:55.490 ","End":"02:01.415","Text":"then graphically the function f is below the x-axis."},{"Start":"02:01.415 ","End":"02:06.095","Text":"That\u0027s how easy way to spot it if you have the picture other than that positive,"},{"Start":"02:06.095 ","End":"02:12.050","Text":"every x when you substitute has to be bigger than 0 and in the other case,"},{"Start":"02:12.050 ","End":"02:16.970","Text":"and negative f(x) has to be negative or whatever x you substitute."},{"Start":"02:16.970 ","End":"02:18.995","Text":"That\u0027s, I think clear enough."},{"Start":"02:18.995 ","End":"02:21.740","Text":"The thing is, in fact,"},{"Start":"02:21.740 ","End":"02:23.840","Text":"most functions are not positive or negative."},{"Start":"02:23.840 ","End":"02:29.030","Text":"Some functions have places where they\u0027re positive and places where there are negative."},{"Start":"02:29.030 ","End":"02:32.240","Text":"The next slide we\u0027ll see that we have"},{"Start":"02:32.240 ","End":"02:35.585","Text":"the mixture of positive and negative and what we do about it."},{"Start":"02:35.585 ","End":"02:39.530","Text":"Here I\u0027m going to give you an example of the mixed variety,"},{"Start":"02:39.530 ","End":"02:41.315","Text":"which is most common."},{"Start":"02:41.315 ","End":"02:43.420","Text":"I\u0027ll draw a quick sketch."},{"Start":"02:43.420 ","End":"02:48.110","Text":"In this example, this is what we see and I\u0027ll also labeled the points."},{"Start":"02:48.110 ","End":"02:49.370","Text":"This is minus 4."},{"Start":"02:49.370 ","End":"02:51.080","Text":"Let\u0027s say this is 2,"},{"Start":"02:51.080 ","End":"02:52.880","Text":"this looks like a 4."},{"Start":"02:52.880 ","End":"02:58.900","Text":"We have a function which is sometimes above the axis and sometimes below the axis."},{"Start":"02:58.900 ","End":"03:01.130","Text":"What I\u0027m going to do, first of all,"},{"Start":"03:01.130 ","End":"03:04.055","Text":"is to highlight just so we can see it visually."},{"Start":"03:04.055 ","End":"03:06.290","Text":"Let\u0027s go for the positive bits first,"},{"Start":"03:06.290 ","End":"03:09.875","Text":"positive means above the x-axis."},{"Start":"03:09.875 ","End":"03:18.285","Text":"This looks like we have positive here and here up to but not including the 0."},{"Start":"03:18.285 ","End":"03:20.870","Text":"Look, it\u0027s above the axis here too,"},{"Start":"03:20.870 ","End":"03:23.285","Text":"so we\u0027ll highlight this also,"},{"Start":"03:23.285 ","End":"03:26.360","Text":"and that highlights the positive bits."},{"Start":"03:26.360 ","End":"03:28.880","Text":"Negative means below the axis,"},{"Start":"03:28.880 ","End":"03:31.610","Text":"so we\u0027re negative over here,"},{"Start":"03:31.610 ","End":"03:35.580","Text":"and we\u0027re negative in this part here."},{"Start":"03:35.580 ","End":"03:39.290","Text":"The only other places remaining are the zeros themselves,"},{"Start":"03:39.290 ","End":"03:41.720","Text":"which are neither positive nor negative."},{"Start":"03:41.720 ","End":"03:44.194","Text":"So we have negative 0,"},{"Start":"03:44.194 ","End":"03:50.125","Text":"positive 0, negative 0, and positive again."},{"Start":"03:50.125 ","End":"03:54.275","Text":"I would like to warn you not to confuse"},{"Start":"03:54.275 ","End":"03:59.930","Text":"positivity and negativity with increase and decrease."},{"Start":"03:59.930 ","End":"04:01.955","Text":"You have been warned,"},{"Start":"04:01.955 ","End":"04:05.585","Text":"but I will also go into it further that\u0027s a good reminder."},{"Start":"04:05.585 ","End":"04:07.610","Text":"We start up here with the function."},{"Start":"04:07.610 ","End":"04:09.080","Text":"We\u0027re going from left to right."},{"Start":"04:09.080 ","End":"04:12.365","Text":"We start off from where negative, but we\u0027re increasing."},{"Start":"04:12.365 ","End":"04:17.780","Text":"Now we continue increasing here all the way up to this point here,"},{"Start":"04:17.780 ","End":"04:20.570","Text":"which is nothing to do with negativity and positivity,"},{"Start":"04:20.570 ","End":"04:21.770","Text":"but in the case of increase,"},{"Start":"04:21.770 ","End":"04:23.060","Text":"this is where the increase stops,"},{"Start":"04:23.060 ","End":"04:25.630","Text":"the decrease starts so we\u0027re still positive."},{"Start":"04:25.630 ","End":"04:27.350","Text":"Once we\u0027re past the 0,"},{"Start":"04:27.350 ","End":"04:29.975","Text":"we\u0027re still decreasing, but now we\u0027re negative."},{"Start":"04:29.975 ","End":"04:33.439","Text":"Then we get to some point here and then we\u0027re still negative,"},{"Start":"04:33.439 ","End":"04:36.370","Text":"but we\u0027re now reverse direction, we\u0027re increasing."},{"Start":"04:36.370 ","End":"04:41.780","Text":"Then continue past the 0 and now we\u0027re still increasing but were positive."},{"Start":"04:41.780 ","End":"04:44.599","Text":"I hope that clears up negativity,"},{"Start":"04:44.599 ","End":"04:48.450","Text":"positivity, and increase, and decrease."}],"ID":5095}],"Thumbnail":null,"ID":56140},{"Name":"Linear Functions and Circles","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Linear Function - Introduction","Duration":"3m 51s","ChapterTopicVideoID":5118,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5118.jpeg","UploadDate":"2020-09-30T12:51:59.6300000","DurationForVideoObject":"PT3M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this clip, we introduce the concept of a linear function."},{"Start":"00:04.260 ","End":"00:14.265","Text":"Linear function is a function of the form y or f(x)=mx plus n,"},{"Start":"00:14.265 ","End":"00:17.985","Text":"where m and n are just some numbers, some constants."},{"Start":"00:17.985 ","End":"00:22.810","Text":"Often in the literature one sees it written as say,"},{"Start":"00:22.810 ","End":"00:25.274","Text":"y=ax plus b,"},{"Start":"00:25.274 ","End":"00:29.462","Text":"very common letters and I\u0027ve seen all sorts of hybrid combinations."},{"Start":"00:29.462 ","End":"00:33.090","Text":"I\u0027ve seen y=mx plus b,"},{"Start":"00:33.090 ","End":"00:34.695","Text":"I\u0027ll give you an example."},{"Start":"00:34.695 ","End":"00:41.070","Text":"A linear function could be y=4x plus 10."},{"Start":"00:41.070 ","End":"00:42.440","Text":"There\u0027s a specific example."},{"Start":"00:42.440 ","End":"00:45.655","Text":"It doesn\u0027t really matter what letters we use in the general."},{"Start":"00:45.655 ","End":"00:47.820","Text":"Of course, in this case if we\u0027re referring to,"},{"Start":"00:47.820 ","End":"00:51.130","Text":"we\u0027ll use this as the model just to keep it standard."},{"Start":"00:51.130 ","End":"00:57.840","Text":"In this case, we would have that m=4 and n=10."},{"Start":"00:57.840 ","End":"00:59.550","Text":"Any pair of numbers we give for m and n,"},{"Start":"00:59.550 ","End":"01:01.140","Text":"we get a linear function."},{"Start":"01:01.140 ","End":"01:06.765","Text":"Here\u0027s another example, y=minus 10x."},{"Start":"01:06.765 ","End":"01:08.730","Text":"What do we get here?"},{"Start":"01:08.730 ","End":"01:10.395","Text":"What is m and what is n?"},{"Start":"01:10.395 ","End":"01:15.690","Text":"Here, m=minus 10 minus 10x and what about the n?"},{"Start":"01:15.690 ","End":"01:18.570","Text":"Well, there is no n. In other words,"},{"Start":"01:18.570 ","End":"01:24.550","Text":"n=0 and yet another example, y=8."},{"Start":"01:24.830 ","End":"01:27.995","Text":"Here, we don\u0027t have anything with x,"},{"Start":"01:27.995 ","End":"01:33.795","Text":"so that means that m is 0 and in this case, n is 8."},{"Start":"01:33.795 ","End":"01:37.875","Text":"You see if I write minus 10x plus 0 as here,"},{"Start":"01:37.875 ","End":"01:39.450","Text":"and that\u0027s minus 10x."},{"Start":"01:39.450 ","End":"01:41.250","Text":"If I write 0x plus 8,"},{"Start":"01:41.250 ","End":"01:42.570","Text":"there\u0027s no need for the 0x,"},{"Start":"01:42.570 ","End":"01:44.205","Text":"so it\u0027s just 8."},{"Start":"01:44.205 ","End":"01:46.560","Text":"There could be m or n,"},{"Start":"01:46.560 ","End":"01:49.970","Text":"could be 0 so it looks like a term is missing."},{"Start":"01:49.970 ","End":"01:56.270","Text":"In fact, you can get more extreme situations with almost nothing left in the equation."},{"Start":"01:56.270 ","End":"02:01.760","Text":"For example, y=0 is also a linear equation."},{"Start":"02:01.760 ","End":"02:08.240","Text":"It\u0027s just that m and n are both 0 because if I put m and n both 0 here,"},{"Start":"02:08.240 ","End":"02:11.540","Text":"I just get y=0 and this too is a linear equation."},{"Start":"02:11.540 ","End":"02:14.330","Text":"Now sometimes something that doesn\u0027t appear to be"},{"Start":"02:14.330 ","End":"02:18.860","Text":"a linear equation is in fact or can be brought into a linear form."},{"Start":"02:18.860 ","End":"02:25.925","Text":"For example, I could take the equation 4y plus 2x=10."},{"Start":"02:25.925 ","End":"02:28.805","Text":"It\u0027s an equation but it\u0027s not a function,"},{"Start":"02:28.805 ","End":"02:31.880","Text":"but it turns out that you can actually make y as"},{"Start":"02:31.880 ","End":"02:35.919","Text":"a linear function of x just by doing a bit of algebra."},{"Start":"02:35.919 ","End":"02:40.880","Text":"Let\u0027s see. If we bring the 2x to the other side,"},{"Start":"02:40.880 ","End":"02:43.920","Text":"we will get that"},{"Start":"02:46.390 ","End":"02:51.270","Text":"4y=minus 2x plus 10,"},{"Start":"02:51.270 ","End":"02:53.295","Text":"and then dividing by 4,"},{"Start":"02:53.295 ","End":"02:57.580","Text":"we get that y=minus 2 over 4."},{"Start":"02:57.580 ","End":"02:59.438","Text":"Let\u0027s cancel the fraction,"},{"Start":"02:59.438 ","End":"03:04.645","Text":"that will be minus 1/2x plus 10 over 4,"},{"Start":"03:04.645 ","End":"03:07.890","Text":"10 over 4 is 5 over 2 or 2 1/2."},{"Start":"03:07.890 ","End":"03:10.425","Text":"I\u0027ll write it as 5 over 2."},{"Start":"03:10.425 ","End":"03:15.210","Text":"This equation actually represented y as"},{"Start":"03:15.210 ","End":"03:19.410","Text":"a linear function of x and we can say that and point"},{"Start":"03:19.410 ","End":"03:28.785","Text":"out that m=minus 1/2 and that n=5 over 2."},{"Start":"03:28.785 ","End":"03:30.815","Text":"We haven\u0027t done very much so far."},{"Start":"03:30.815 ","End":"03:36.505","Text":"We\u0027ve just shown essentially what a linear function is and this is the form,"},{"Start":"03:36.505 ","End":"03:42.620","Text":"mx plus n general linear function in x with,"},{"Start":"03:42.620 ","End":"03:46.400","Text":"as I say, some other textbooks and so on using different letters,"},{"Start":"03:46.400 ","End":"03:47.975","Text":"but we don\u0027t let that worry us."},{"Start":"03:47.975 ","End":"03:50.190","Text":"Okay, that\u0027s it for now."}],"ID":5096},{"Watched":false,"Name":"Graph of a Linear Function","Duration":"11m 16s","ChapterTopicVideoID":5119,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5119.jpeg","UploadDate":"2020-10-06T08:34:02.4830000","DurationForVideoObject":"PT11M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.145","Text":"In this section, we\u0027ll learn how to make a sketch of the graph of a linear function,"},{"Start":"00:05.145 ","End":"00:07.725","Text":"and I\u0027ll start with an example."},{"Start":"00:07.725 ","End":"00:13.755","Text":"The example will be y=4x plus 2,"},{"Start":"00:13.755 ","End":"00:15.990","Text":"which is obviously a linear function."},{"Start":"00:15.990 ","End":"00:20.295","Text":"It\u0027s of the form y=mx plus n. I\u0027m told,"},{"Start":"00:20.295 ","End":"00:24.615","Text":"and I know from geometry that if you have 2 points on a line,"},{"Start":"00:24.615 ","End":"00:27.690","Text":"then you can completely determine the line,"},{"Start":"00:27.690 ","End":"00:29.535","Text":"and that\u0027s what we\u0027ll do here."},{"Start":"00:29.535 ","End":"00:31.260","Text":"To get the graph of a linear function,"},{"Start":"00:31.260 ","End":"00:35.920","Text":"all we need is to find 2 points on the graph and join a straight line through them."},{"Start":"00:35.920 ","End":"00:39.155","Text":"In this example, I can make a little table,"},{"Start":"00:39.155 ","End":"00:41.930","Text":"and compute a pair of values of x and y,"},{"Start":"00:41.930 ","End":"00:47.570","Text":"but wait I\u0027ll also need a co-pair of coordinate axis in order to sketch the graph."},{"Start":"00:47.570 ","End":"00:50.285","Text":"Here\u0027s the axis,"},{"Start":"00:50.285 ","End":"00:53.940","Text":"label them y and x,"},{"Start":"00:53.940 ","End":"00:55.785","Text":"and as for the values,"},{"Start":"00:55.785 ","End":"01:03.585","Text":"and let\u0027s take minus 1 and 1 here and here minus 1, and 1."},{"Start":"01:03.585 ","End":"01:10.110","Text":"The corresponding y\u0027s will be 4 times minus 1 plus 2 is minus 2."},{"Start":"01:10.110 ","End":"01:14.745","Text":"And for 1, I\u0027ll get 4 times 1 plus 2 is 6."},{"Start":"01:14.745 ","End":"01:17.670","Text":"I think I\u0027d better redraw the scale."},{"Start":"01:17.670 ","End":"01:21.378","Text":"Here I have the 0.1,"},{"Start":"01:21.378 ","End":"01:22.650","Text":"6, see,"},{"Start":"01:22.650 ","End":"01:25.720","Text":"and here\u0027s the point minus 1."},{"Start":"01:25.720 ","End":"01:28.186","Text":"Wait a minute, that doesn\u0027t look like minus 2."},{"Start":"01:28.186 ","End":"01:29.720","Text":"That looks a bit better,"},{"Start":"01:29.720 ","End":"01:34.400","Text":"and to join a line through the pair of points and here we are,"},{"Start":"01:34.400 ","End":"01:35.855","Text":"and then we have a rough sketch."},{"Start":"01:35.855 ","End":"01:39.845","Text":"Once again, we have an equation of a line."},{"Start":"01:39.845 ","End":"01:42.230","Text":"We just take 2 values of x,"},{"Start":"01:42.230 ","End":"01:44.750","Text":"compute the corresponding values of y,"},{"Start":"01:44.750 ","End":"01:47.960","Text":"we get 2 points and we join them with a straight line."},{"Start":"01:47.960 ","End":"01:49.835","Text":"That\u0027s all there is to it."},{"Start":"01:49.835 ","End":"01:53.840","Text":"In this section, we learn a slightly different method than the"},{"Start":"01:53.840 ","End":"01:57.665","Text":"previous of drawing the graph of a linear function."},{"Start":"01:57.665 ","End":"02:00.545","Text":"Instead of using any old 2 points,"},{"Start":"02:00.545 ","End":"02:02.615","Text":"we\u0027re going to use 2 special points,"},{"Start":"02:02.615 ","End":"02:06.740","Text":"which will be the intersection of the graph with the axis."},{"Start":"02:06.740 ","End":"02:08.570","Text":"I\u0027ll just show you what I mean."},{"Start":"02:08.570 ","End":"02:11.450","Text":"I\u0027m going to use the same example as before,"},{"Start":"02:11.450 ","End":"02:12.800","Text":"Y=4x plus 2,"},{"Start":"02:12.800 ","End":"02:14.945","Text":"and here I have a coordinate axis,"},{"Start":"02:14.945 ","End":"02:20.490","Text":"but instead of just letting x and y have any 2 arbitrary values,"},{"Start":"02:20.490 ","End":"02:22.425","Text":"I\u0027m going to take special values."},{"Start":"02:22.425 ","End":"02:27.635","Text":"The first point I\u0027m going to take is where the graph cuts the y-axis."},{"Start":"02:27.635 ","End":"02:30.770","Text":"Now, what\u0027s special about the y-axis?"},{"Start":"02:30.770 ","End":"02:34.555","Text":"The thing is that x is 0 all along the y-axis."},{"Start":"02:34.555 ","End":"02:37.135","Text":"To find the intersection with the y-axis,"},{"Start":"02:37.135 ","End":"02:39.105","Text":"all I have to do is let x=0,"},{"Start":"02:39.105 ","End":"02:41.560","Text":"and I\u0027ll get when x is 0,"},{"Start":"02:41.560 ","End":"02:45.430","Text":"y=4 times 0 plus 2 is 2."},{"Start":"02:45.430 ","End":"02:47.980","Text":"So I get the 0, 2."},{"Start":"02:47.980 ","End":"02:49.490","Text":"Let\u0027s say it\u0027s here,"},{"Start":"02:49.490 ","End":"02:51.130","Text":"0, 2,"},{"Start":"02:51.130 ","End":"02:52.340","Text":"and that will be 1 point."},{"Start":"02:52.340 ","End":"02:56.630","Text":"The other point is where the graph cuts the x-axis."},{"Start":"02:56.630 ","End":"02:59.060","Text":"Now what\u0027s special about the x-axis?"},{"Start":"02:59.060 ","End":"03:01.430","Text":"This time y=0."},{"Start":"03:01.430 ","End":"03:03.320","Text":"So I put 0 here,"},{"Start":"03:03.320 ","End":"03:04.880","Text":"and instead of a substitution,"},{"Start":"03:04.880 ","End":"03:06.815","Text":"I have an equation to solve."},{"Start":"03:06.815 ","End":"03:11.715","Text":"I would get 0=4x plus 2."},{"Start":"03:11.715 ","End":"03:13.200","Text":"You can do this in your head."},{"Start":"03:13.200 ","End":"03:17.325","Text":"You bring in 2 to the other side is minus 2 divided by 4,"},{"Start":"03:17.325 ","End":"03:19.540","Text":"so it\u0027s minus 1/2."},{"Start":"03:19.540 ","End":"03:24.290","Text":"The other point we get is minus 1/2, 0 write it,"},{"Start":"03:24.290 ","End":"03:29.720","Text":"minus 1/2, 0 and this is the intersection with the x-axis."},{"Start":"03:29.720 ","End":"03:32.870","Text":"As usual, we just take a ruler,"},{"Start":"03:32.870 ","End":"03:38.315","Text":"and draw the line between the 2 points. There we are."},{"Start":"03:38.315 ","End":"03:40.550","Text":"So very much the same as before,"},{"Start":"03:40.550 ","End":"03:42.950","Text":"except instead of using 2 general points,"},{"Start":"03:42.950 ","End":"03:47.415","Text":"we use the intersection of the line with the axis."},{"Start":"03:47.415 ","End":"03:50.750","Text":"These are considered special points or points of interest,"},{"Start":"03:50.750 ","End":"03:52.145","Text":"which is why we do this."},{"Start":"03:52.145 ","End":"03:55.595","Text":"That\u0027s about it for this part."},{"Start":"03:55.595 ","End":"03:58.490","Text":"The only thing I could add is that it is"},{"Start":"03:58.490 ","End":"04:02.270","Text":"possible that the line actually goes through the origin,"},{"Start":"04:02.270 ","End":"04:07.235","Text":"and then we are stuck using this method because we only get 1 point instead of 2 points."},{"Start":"04:07.235 ","End":"04:10.050","Text":"So we could use the previous method."},{"Start":"04:10.390 ","End":"04:13.970","Text":"That\u0027s it for this part."},{"Start":"04:13.970 ","End":"04:18.080","Text":"In this section, we\u0027re going to study some special lines in the graph,"},{"Start":"04:18.080 ","End":"04:21.484","Text":"and we\u0027re going to study 3 special lines."},{"Start":"04:21.484 ","End":"04:23.720","Text":"Lines that are parallel to the x-axis,"},{"Start":"04:23.720 ","End":"04:25.685","Text":"lines parallel to the y-axis,"},{"Start":"04:25.685 ","End":"04:27.100","Text":"and lines through the origin."},{"Start":"04:27.100 ","End":"04:30.940","Text":"We\u0027ll start with the first lines parallel to the x-axis."},{"Start":"04:30.940 ","End":"04:33.640","Text":"Here\u0027s a pair of axis."},{"Start":"04:33.640 ","End":"04:36.725","Text":"What does the line parallel to the x-axis looks like?"},{"Start":"04:36.725 ","End":"04:39.380","Text":"Parallel to the x-axis means it\u0027s horizontal."},{"Start":"04:39.380 ","End":"04:42.720","Text":"It means it cuts through here at some point."},{"Start":"04:42.720 ","End":"04:46.245","Text":"Let\u0027s say this point is at height 4."},{"Start":"04:46.245 ","End":"04:48.680","Text":"In other words, y is 4 here."},{"Start":"04:48.680 ","End":"04:50.135","Text":"If you think about it,"},{"Start":"04:50.135 ","End":"04:53.675","Text":"since this is horizontal and parallel to the x-axis,"},{"Start":"04:53.675 ","End":"04:55.355","Text":"if y is 4 here,"},{"Start":"04:55.355 ","End":"04:57.200","Text":"everywhere along the line,"},{"Start":"04:57.200 ","End":"04:58.340","Text":"the y is going to be 4,"},{"Start":"04:58.340 ","End":"05:00.110","Text":"the height is for everywhere."},{"Start":"05:00.110 ","End":"05:01.700","Text":"Any such point will be,"},{"Start":"05:01.700 ","End":"05:03.455","Text":"I don\u0027t know what the x is,"},{"Start":"05:03.455 ","End":"05:06.200","Text":"but the y is certainly going to equal 4."},{"Start":"05:06.200 ","End":"05:08.690","Text":"In fact, that brings us to the equation of"},{"Start":"05:08.690 ","End":"05:12.275","Text":"the line because these are all the points where y is 4."},{"Start":"05:12.275 ","End":"05:17.890","Text":"The equation is y=4 in this case."},{"Start":"05:17.890 ","End":"05:27.330","Text":"In general, if this thing is at height n. Then y=n is the equation of this line."},{"Start":"05:27.330 ","End":"05:33.240","Text":"I\u0027m deliberately using letter n because the linear function is y=mx"},{"Start":"05:33.550 ","End":"05:42.545","Text":"plus n. So the thing about horizontal lines is that the m is going to be 0,"},{"Start":"05:42.545 ","End":"05:47.105","Text":"and the n is just going to be the height at which cuts the y-axis."},{"Start":"05:47.105 ","End":"05:49.490","Text":"Similarly, just to give another example,"},{"Start":"05:49.490 ","End":"05:54.035","Text":"suppose it passes through here where we have a minus 5."},{"Start":"05:54.035 ","End":"05:56.345","Text":"If we go through minus 5,"},{"Start":"05:56.345 ","End":"06:03.755","Text":"then the equation of this line just emphasis right again that here y equals 4."},{"Start":"06:03.755 ","End":"06:07.400","Text":"Here we\u0027ll have y equals minus 5,"},{"Start":"06:07.400 ","End":"06:09.275","Text":"whatever height it cuts it,"},{"Start":"06:09.275 ","End":"06:11.270","Text":"y equals that number."},{"Start":"06:11.270 ","End":"06:16.880","Text":"Again, m is 0 and n is equal to this particular height."},{"Start":"06:16.880 ","End":"06:20.180","Text":"All the values of y along this will be,"},{"Start":"06:20.180 ","End":"06:21.485","Text":"I don\u0027t know what x will be,"},{"Start":"06:21.485 ","End":"06:23.443","Text":"x can be anything you want,"},{"Start":"06:23.443 ","End":"06:28.125","Text":"but the y of it has to equal minus 5."},{"Start":"06:28.125 ","End":"06:32.930","Text":"That\u0027s basically all there is about lines parallel to the x-axis,"},{"Start":"06:32.930 ","End":"06:35.105","Text":"which are in fact horizontal lines."},{"Start":"06:35.105 ","End":"06:36.995","Text":"y equals some constant,"},{"Start":"06:36.995 ","End":"06:43.355","Text":"this n from the y=mx plus n. That\u0027s enough said about that."},{"Start":"06:43.355 ","End":"06:47.930","Text":"Next we move on to the lines parallel to the y-axis."},{"Start":"06:47.930 ","End":"06:52.985","Text":"There is some similarity between lines parallel to the y-axis,"},{"Start":"06:52.985 ","End":"06:58.250","Text":"and parallel to the x-axis because just as those were horizontal lines,"},{"Start":"06:58.250 ","End":"07:01.200","Text":"now we\u0027re going to have vertical lines."},{"Start":"07:01.200 ","End":"07:05.705","Text":"Let\u0027s take a vertical line and then it will be parallel to the y-axis."},{"Start":"07:05.705 ","End":"07:10.670","Text":"Let\u0027s say that it passes through this point where x=3."},{"Start":"07:10.670 ","End":"07:14.600","Text":"Notice that any point along this line,"},{"Start":"07:14.600 ","End":"07:20.135","Text":"will have x=3 and the general form will be 3 comma something."},{"Start":"07:20.135 ","End":"07:24.065","Text":"All these 3 comma something will determine this line."},{"Start":"07:24.065 ","End":"07:26.675","Text":"So the defining equation for this line is,"},{"Start":"07:26.675 ","End":"07:28.940","Text":"it\u0027s all the points where x=3."},{"Start":"07:28.940 ","End":"07:31.720","Text":"This is the equation, x=3."},{"Start":"07:31.720 ","End":"07:34.335","Text":"Similarly, if we pass through,"},{"Start":"07:34.335 ","End":"07:37.480","Text":"let\u0027s say here\u0027s minus 4."},{"Start":"07:37.480 ","End":"07:39.830","Text":"If we take a vertical line,"},{"Start":"07:39.830 ","End":"07:43.370","Text":"which is what it means to be parallel to the y-axis."},{"Start":"07:43.370 ","End":"07:48.680","Text":"Its equation will be x equals minus 4,"},{"Start":"07:48.680 ","End":"07:54.635","Text":"and any point along this line will be minus 4 comma something."},{"Start":"07:54.635 ","End":"07:57.110","Text":"y is not restricted at all?"},{"Start":"07:57.110 ","End":"08:00.320","Text":"Now the big thing, the big difference between"},{"Start":"08:00.320 ","End":"08:04.825","Text":"these vertical lines as opposed to the horizontal lines,"},{"Start":"08:04.825 ","End":"08:11.015","Text":"or the lines parallel to the y-axis is that they are of the form x equals a constant,"},{"Start":"08:11.015 ","End":"08:14.715","Text":"but they are not linear equations"},{"Start":"08:14.715 ","End":"08:18.845","Text":"because if this was a linear equation that I\u0027ll be able to write it in the form"},{"Start":"08:18.845 ","End":"08:24.170","Text":"y=mx plus n. There\u0027s no way I can choose m"},{"Start":"08:24.170 ","End":"08:29.360","Text":"and n in order to get the equation x=3."},{"Start":"08:29.360 ","End":"08:35.670","Text":"I mean y does not appear here."},{"Start":"08:36.230 ","End":"08:39.480","Text":"Whenever m and n and I choose,"},{"Start":"08:39.480 ","End":"08:43.100","Text":"I\u0027ll still see y, but here I don\u0027t."},{"Start":"08:43.100 ","End":"08:49.430","Text":"I\u0027ll say more, not only is it not a linear function,"},{"Start":"08:49.430 ","End":"08:52.445","Text":"it\u0027s not even a function at all."},{"Start":"08:52.445 ","End":"08:59.530","Text":"Remember for an equation to be a function,"},{"Start":"08:59.530 ","End":"09:07.535","Text":"any given value of x has to have exactly 1 value of y."},{"Start":"09:07.535 ","End":"09:11.090","Text":"But here, along this vertical line,"},{"Start":"09:11.090 ","End":"09:12.890","Text":"the same value x(x),"},{"Start":"09:12.890 ","End":"09:14.495","Text":"which namely 3,"},{"Start":"09:14.495 ","End":"09:17.090","Text":"will have many values of y here,"},{"Start":"09:17.090 ","End":"09:19.450","Text":"an infinite number in fact."},{"Start":"09:19.450 ","End":"09:24.395","Text":"As I say, this is not a linear equation or it\u0027s not a linear function,"},{"Start":"09:24.395 ","End":"09:26.870","Text":"and it\u0027s not any kind of function."},{"Start":"09:26.870 ","End":"09:32.380","Text":"Still, it does have an equation, x equals something."},{"Start":"09:32.380 ","End":"09:37.040","Text":"In general, x=3,"},{"Start":"09:37.040 ","End":"09:41.630","Text":"x=4, x equals some constant."},{"Start":"09:41.630 ","End":"09:43.370","Text":"It has an equation,"},{"Start":"09:43.370 ","End":"09:46.805","Text":"but it\u0027s not a linear function and it\u0027s not a function,"},{"Start":"09:46.805 ","End":"09:52.115","Text":"but it is interesting because it\u0027s a line and it\u0027s even a special line."},{"Start":"09:52.115 ","End":"09:54.320","Text":"There we are, we know its equation,"},{"Start":"09:54.320 ","End":"09:55.535","Text":"but it\u0027s not a function."},{"Start":"09:55.535 ","End":"09:59.900","Text":"Finally, we come to the last type of special line,"},{"Start":"09:59.900 ","End":"10:02.720","Text":"and that\u0027s lines passing through the origin."},{"Start":"10:02.720 ","End":"10:05.465","Text":"I\u0027ve sketched a few of them here."},{"Start":"10:05.465 ","End":"10:12.855","Text":"From this picture you can see y aligned to the origin is also called a ray."},{"Start":"10:12.855 ","End":"10:15.975","Text":"Line through the origin is a ray."},{"Start":"10:15.975 ","End":"10:18.335","Text":"What\u0027s its equation?"},{"Start":"10:18.335 ","End":"10:20.495","Text":"Well, if you think about it,"},{"Start":"10:20.495 ","End":"10:22.205","Text":"a ray passes through the origin."},{"Start":"10:22.205 ","End":"10:24.419","Text":"What is the origin?"},{"Start":"10:24.790 ","End":"10:27.410","Text":"The origin is 0,0,"},{"Start":"10:27.410 ","End":"10:32.210","Text":"so it goes through 0,0."},{"Start":"10:32.210 ","End":"10:36.990","Text":"If a linear equation is in general,"},{"Start":"10:36.990 ","End":"10:42.140","Text":"y=mx plus n, and I require that it goes through 0,0."},{"Start":"10:42.140 ","End":"10:44.615","Text":"If you put x is 0, and y is 0 here,"},{"Start":"10:44.615 ","End":"10:49.640","Text":"you end up with n=0 also."},{"Start":"10:49.640 ","End":"10:52.520","Text":"So ultimately, if n is 0,"},{"Start":"10:52.520 ","End":"10:57.805","Text":"we end up with a general equation y=mx."},{"Start":"10:57.805 ","End":"11:00.795","Text":"The positive m\u0027s will slope this way,"},{"Start":"11:00.795 ","End":"11:02.520","Text":"negative m\u0027s will slope this way."},{"Start":"11:02.520 ","End":"11:06.955","Text":"This might be y=2x, this might be y equals minus 3x."},{"Start":"11:06.955 ","End":"11:09.260","Text":"In general, that\u0027s the equation of array."},{"Start":"11:09.260 ","End":"11:12.500","Text":"That\u0027s all I have to say about that,"},{"Start":"11:12.500 ","End":"11:15.780","Text":"and special lines in general."}],"ID":5097},{"Watched":false,"Name":"Meaning of the Constants in the Linear Function","Duration":"7m 36s","ChapterTopicVideoID":5120,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5120.jpeg","UploadDate":"2020-10-06T08:14:17.1170000","DurationForVideoObject":"PT7M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.175","Text":"In this clip, we discuss the meaning of the constants in the linear function."},{"Start":"00:05.175 ","End":"00:11.160","Text":"Remember, the linear function is the function of the form y equals mx plus n,"},{"Start":"00:11.160 ","End":"00:16.290","Text":"or we could call it f of x if we wanted just to remember that it\u0027s a function of"},{"Start":"00:16.290 ","End":"00:22.185","Text":"x and the constants I\u0027m talking about are the m and the n I\u0027ve underlined here."},{"Start":"00:22.185 ","End":"00:25.695","Text":"They\u0027re actual values in any given case and"},{"Start":"00:25.695 ","End":"00:30.410","Text":"these 2 constants actually have some geometrical meaning."},{"Start":"00:30.410 ","End":"00:32.570","Text":"Let\u0027s go with the easier 1 first,"},{"Start":"00:32.570 ","End":"00:33.740","Text":"which is n,"},{"Start":"00:33.740 ","End":"00:41.030","Text":"and notice that if I let x equals 0 in the linear function,"},{"Start":"00:41.030 ","End":"00:45.910","Text":"then I get that y equals 0 times x plus n,"},{"Start":"00:45.910 ","End":"00:50.089","Text":"and y equals n. If I draw a little picture,"},{"Start":"00:50.089 ","End":"00:56.030","Text":"this is the y-axis and the x-axis, and this,"},{"Start":"00:56.030 ","End":"01:02.420","Text":"I\u0027d say is the function mx plus n so if I let x equals 0,"},{"Start":"01:02.420 ","End":"01:03.650","Text":"as I said y equals n,"},{"Start":"01:03.650 ","End":"01:09.163","Text":"which means that this point is the point 0,"},{"Start":"01:09.163 ","End":"01:13.610","Text":"n. Or if you like n is this particular distance here,"},{"Start":"01:13.610 ","End":"01:18.815","Text":"if I take it as positive if it\u0027s above and negative if it\u0027s below."},{"Start":"01:18.815 ","End":"01:24.260","Text":"This is also actually called often the y-intercept."},{"Start":"01:24.260 ","End":"01:29.360","Text":"I\u0027ll just write it here in case you encounter this name, the y-intercept."},{"Start":"01:29.360 ","End":"01:34.250","Text":"Or I can say briefly that n is the y-intercept,"},{"Start":"01:34.250 ","End":"01:38.030","Text":"which means it\u0027s the coordinate of y in the place where"},{"Start":"01:38.030 ","End":"01:44.360","Text":"the line cuts the y-axis so that\u0027s the significance of the n. For example,"},{"Start":"01:44.360 ","End":"01:47.795","Text":"if I had y equals 2x plus 3,"},{"Start":"01:47.795 ","End":"01:50.465","Text":"then 3 would be this distance here."},{"Start":"01:50.465 ","End":"01:53.155","Text":"This would be 0, 3."},{"Start":"01:53.155 ","End":"01:55.640","Text":"Nothing really very deep to say about this,"},{"Start":"01:55.640 ","End":"01:59.000","Text":"and that\u0027s basically it for n. Now,"},{"Start":"01:59.000 ","End":"02:02.390","Text":"let\u0027s go for the other constant m. This is a little"},{"Start":"02:02.390 ","End":"02:05.960","Text":"bit more complicated and there\u0027s a lot to say about it."},{"Start":"02:05.960 ","End":"02:11.937","Text":"In fact, I can say that m is the slope of the line;"},{"Start":"02:11.937 ","End":"02:17.374","Text":"that is the significant thing about m. However,"},{"Start":"02:17.374 ","End":"02:23.340","Text":"you might not have learnt about slope so I\u0027m now going to have to say what a slope is,"},{"Start":"02:23.340 ","End":"02:24.660","Text":"but perhaps some of you know,"},{"Start":"02:24.660 ","End":"02:31.305","Text":"but not all of you so I\u0027ll say what a slope is of this line."},{"Start":"02:31.305 ","End":"02:34.960","Text":"Well, I wrote it down and now I\u0027ll have to is explain it."},{"Start":"02:34.960 ","End":"02:42.085","Text":"First, I\u0027ll read it, a slope by how much y increases when x increases by 1 unit."},{"Start":"02:42.085 ","End":"02:44.125","Text":"Boy, that\u0027s some definition;"},{"Start":"02:44.125 ","End":"02:47.725","Text":"by how much does y increase when x increases by 1 unit."},{"Start":"02:47.725 ","End":"02:53.190","Text":"I think we\u0027d better go with an example and the example I\u0027ll take will"},{"Start":"02:53.190 ","End":"03:00.549","Text":"be y equals 2x plus 1 and I think we need a sketch."},{"Start":"03:00.549 ","End":"03:03.990","Text":"Here\u0027s a rough sketch of y equals 2x plus 1,"},{"Start":"03:03.990 ","End":"03:07.575","Text":"where of course m is equal to 2,"},{"Start":"03:07.575 ","End":"03:11.610","Text":"because this is the m. Let\u0027s see how we can"},{"Start":"03:11.610 ","End":"03:15.420","Text":"interpret this m equals 2 according to what I wrote."},{"Start":"03:15.420 ","End":"03:19.400","Text":"We\u0027re talking about increasing x by 1 unit."},{"Start":"03:19.400 ","End":"03:21.770","Text":"Let\u0027s start with an initial value of x."},{"Start":"03:21.770 ","End":"03:27.590","Text":"Let\u0027s say we take x equal 1 and we substitute y equals 2x plus 1,"},{"Start":"03:27.590 ","End":"03:33.870","Text":"which is equal to 3 so we get x equals 1, y equals 3."},{"Start":"03:33.870 ","End":"03:37.860","Text":"Now, we want to increase it by 1 unit so increasing it by"},{"Start":"03:37.860 ","End":"03:42.240","Text":"1 will give us the value of 2 here and if we"},{"Start":"03:42.240 ","End":"03:49.890","Text":"substitute x equals 2 here we get y equals twice 2 plus 1 which is 5 and 5 is somewhere"},{"Start":"03:49.890 ","End":"03:58.545","Text":"here so we get this point and notice that x has increased by 1 unit."},{"Start":"03:58.545 ","End":"04:00.230","Text":"I\u0027ll make a little triangle here,"},{"Start":"04:00.230 ","End":"04:02.615","Text":"the right triangle, close it."},{"Start":"04:02.615 ","End":"04:05.290","Text":"This is 90 degrees, of course."},{"Start":"04:05.290 ","End":"04:07.870","Text":"Here we have 1,"},{"Start":"04:07.870 ","End":"04:09.715","Text":"3, here we have 2, 5."},{"Start":"04:09.715 ","End":"04:14.295","Text":"Notice that when x increased by 1,"},{"Start":"04:14.295 ","End":"04:18.060","Text":"y increased by 2 units so I went 1 to the right,"},{"Start":"04:18.060 ","End":"04:20.180","Text":"I went 2 upwards."},{"Start":"04:20.180 ","End":"04:25.890","Text":"You can see it here also x grows from 1 to 2 and"},{"Start":"04:25.890 ","End":"04:32.565","Text":"y went from 3 to 5 so this went up 1 unit,"},{"Start":"04:32.565 ","End":"04:35.180","Text":"this went up 2 units and,"},{"Start":"04:35.180 ","End":"04:37.234","Text":"in fact, it doesn\u0027t even matter."},{"Start":"04:37.234 ","End":"04:39.290","Text":"You might think it makes a difference,"},{"Start":"04:39.290 ","End":"04:41.075","Text":"where we start doesn\u0027t matter."},{"Start":"04:41.075 ","End":"04:42.575","Text":"I mean, we could continue."},{"Start":"04:42.575 ","End":"04:44.240","Text":"Let\u0027s take another x,"},{"Start":"04:44.240 ","End":"04:49.800","Text":"let\u0027s increase it by another 1 so we will get here 3 and if we put in 3 here,"},{"Start":"04:49.800 ","End":"04:55.575","Text":"we get twice 3 plus 1 is 7 so y is now equal to 7."},{"Start":"04:55.575 ","End":"04:58.065","Text":"Somewhere here is 7."},{"Start":"04:58.065 ","End":"05:01.915","Text":"Then we can join it up here."},{"Start":"05:01.915 ","End":"05:04.743","Text":"This is the point, 3,"},{"Start":"05:04.743 ","End":"05:07.445","Text":"7 and once again,"},{"Start":"05:07.445 ","End":"05:10.535","Text":"can join it, draw a triangle."},{"Start":"05:10.535 ","End":"05:12.280","Text":"We see that here,"},{"Start":"05:12.280 ","End":"05:15.230","Text":"we increased from 2 to 3,"},{"Start":"05:15.230 ","End":"05:17.195","Text":"which is 1 unit again,"},{"Start":"05:17.195 ","End":"05:21.485","Text":"that was x and y increased from 5 to 7."},{"Start":"05:21.485 ","End":"05:23.530","Text":"That was 2 units."},{"Start":"05:23.530 ","End":"05:26.370","Text":"Here was 1 unit, here was 1 unit,"},{"Start":"05:26.370 ","End":"05:30.360","Text":"here was 2 units and on the graph we had x increased by 1,"},{"Start":"05:30.360 ","End":"05:31.550","Text":"y increased by 2."},{"Start":"05:31.550 ","End":"05:35.455","Text":"Once again, x is increasing by 1,"},{"Start":"05:35.455 ","End":"05:39.395","Text":"y increases by 2 and we could do this anywhere."},{"Start":"05:39.395 ","End":"05:43.745","Text":"Let me draw a diagram to show this general situation."},{"Start":"05:43.745 ","End":"05:47.400","Text":"Here I\u0027ve made a general sketch of y equals mx plus"},{"Start":"05:47.400 ","End":"05:51.840","Text":"n we\u0027ll get away from our specific example, y equals 2x plus 1,"},{"Start":"05:51.840 ","End":"05:55.635","Text":"where we saw that m was 2 and every time we increased x by 1,"},{"Start":"05:55.635 ","End":"05:57.170","Text":"y increased by 2,"},{"Start":"05:57.170 ","End":"06:00.650","Text":"which was the m. Now let\u0027s just draw a sketch in general."},{"Start":"06:00.650 ","End":"06:04.130","Text":"I\u0027ll use a slightly different color here, x-axis,"},{"Start":"06:04.130 ","End":"06:08.795","Text":"y-axis, y equals mx plus n. We\u0027ll take a general point,"},{"Start":"06:08.795 ","End":"06:10.190","Text":"not just 1, 2,"},{"Start":"06:10.190 ","End":"06:13.915","Text":"or 3, we\u0027ll take a general point here, which will be,"},{"Start":"06:13.915 ","End":"06:20.380","Text":"let\u0027s say the point where x is equals to a and then we\u0027ll increase a by 1."},{"Start":"06:20.380 ","End":"06:22.400","Text":"Let\u0027s say here, this is a,"},{"Start":"06:22.400 ","End":"06:27.935","Text":"here we have a plus 1 and so we take the point above it,"},{"Start":"06:27.935 ","End":"06:30.755","Text":"which is here, and then again,"},{"Start":"06:30.755 ","End":"06:38.550","Text":"complete the triangle horizontal and vertical and right angles"},{"Start":"06:38.550 ","End":"06:43.895","Text":"and if this is 1 unit and I\u0027ve increased the scale from here,"},{"Start":"06:43.895 ","End":"06:49.750","Text":"if we go across from a to a plus 1 and increase by 1 unit,"},{"Start":"06:49.750 ","End":"06:52.125","Text":"which is this 1 unit,"},{"Start":"06:52.125 ","End":"06:57.405","Text":"what happens to y is it goes from here to here,"},{"Start":"06:57.405 ","End":"06:59.195","Text":"never mind what the values are."},{"Start":"06:59.195 ","End":"07:03.405","Text":"This increases by, this is 1 unit,"},{"Start":"07:03.405 ","End":"07:06.825","Text":"this will be m units so we get a little triangle,"},{"Start":"07:06.825 ","End":"07:13.630","Text":"1 by m. That\u0027s how the situation looks in general."},{"Start":"07:13.630 ","End":"07:15.840","Text":"I can say too often,"},{"Start":"07:15.840 ","End":"07:21.770","Text":"m means by how many units y increases when x"},{"Start":"07:21.770 ","End":"07:27.710","Text":"is increased by 1 unit so we saw it in an individual example and in the general case."},{"Start":"07:27.710 ","End":"07:32.000","Text":"Coming up how to tell whether the line is increasing or"},{"Start":"07:32.000 ","End":"07:37.470","Text":"decreasing with the help of this m here."}],"ID":5098},{"Watched":false,"Name":"Properties of the Slope","Duration":"6m 5s","ChapterTopicVideoID":5121,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5121.jpeg","UploadDate":"2020-10-06T08:19:03.7530000","DurationForVideoObject":"PT6M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"Now, we\u0027re going to talk about the slope M and how"},{"Start":"00:04.020 ","End":"00:07.980","Text":"it determines whether the line is increasing or decreasing,"},{"Start":"00:07.980 ","End":"00:09.510","Text":"and even how steeply."},{"Start":"00:09.510 ","End":"00:11.670","Text":"We are going to divide into 3 cases."},{"Start":"00:11.670 ","End":"00:14.650","Text":"First case will be where m is bigger than 0,"},{"Start":"00:15.080 ","End":"00:17.460","Text":"then we\u0027ll have m less than 0,"},{"Start":"00:17.460 ","End":"00:18.900","Text":"then we\u0027ll have m equals 0."},{"Start":"00:18.900 ","End":"00:22.980","Text":"Let\u0027s start with the m bigger than 0 and what this means."},{"Start":"00:22.980 ","End":"00:25.215","Text":"When m is bigger than 0,"},{"Start":"00:25.215 ","End":"00:31.035","Text":"it means that the line is increasing,"},{"Start":"00:31.035 ","End":"00:34.380","Text":"and that\u0027s not all."},{"Start":"00:34.380 ","End":"00:37.160","Text":"The greater that m is,"},{"Start":"00:37.160 ","End":"00:38.960","Text":"the steeper the increase,"},{"Start":"00:38.960 ","End":"00:43.790","Text":"since it can increase and it can be at a steeper rate or the less steep,"},{"Start":"00:43.790 ","End":"00:45.410","Text":"I mean, just in general,"},{"Start":"00:45.410 ","End":"00:46.880","Text":"even outside of mathematics,"},{"Start":"00:46.880 ","End":"00:48.185","Text":"you can have different lines."},{"Start":"00:48.185 ","End":"00:52.310","Text":"This line increases, but this line increases more steeply,"},{"Start":"00:52.310 ","End":"00:54.140","Text":"assuming we\u0027re going this way."},{"Start":"00:54.140 ","End":"00:56.210","Text":"Likewise, when we get to decrease,"},{"Start":"00:56.210 ","End":"00:58.760","Text":"there could be a line that\u0027s decreasing,"},{"Start":"00:58.760 ","End":"01:02.075","Text":"but another line might be decreasing more steeply."},{"Start":"01:02.075 ","End":"01:05.360","Text":"That\u0027s just what I mean by this word steep."},{"Start":"01:05.360 ","End":"01:09.770","Text":"Let\u0027s draw a picture to explain this part."},{"Start":"01:09.770 ","End":"01:12.125","Text":"Here, we have a y-axis,"},{"Start":"01:12.125 ","End":"01:15.620","Text":"here we have an x-axis and let\u0027s take,"},{"Start":"01:15.620 ","End":"01:20.040","Text":"for example, the function y equals x plus 1."},{"Start":"01:21.080 ","End":"01:27.480","Text":"We can see the coefficient of m is 1 so here m equals 1."},{"Start":"01:27.480 ","End":"01:32.870","Text":"Now, the fact that this is increasing follows from the fact that m is positive,"},{"Start":"01:32.870 ","End":"01:36.230","Text":"which means that when we increase x by 1 unit,"},{"Start":"01:36.230 ","End":"01:39.290","Text":"y increases also by 1 unit,"},{"Start":"01:39.290 ","End":"01:42.590","Text":"which is a positive quantity so it increases."},{"Start":"01:42.590 ","End":"01:45.525","Text":"Only a negative increase is a decrease."},{"Start":"01:45.525 ","End":"01:51.210","Text":"However, the steepness is something else we want to be concerned about."},{"Start":"01:51.210 ","End":"01:53.465","Text":"Let\u0027s take another example,"},{"Start":"01:53.465 ","End":"01:59.465","Text":"which will be y equals 2x plus 1 is not to scale."},{"Start":"01:59.465 ","End":"02:04.685","Text":"But if I take y equals 2x plus 1,"},{"Start":"02:04.685 ","End":"02:07.360","Text":"then m is equal to 2."},{"Start":"02:07.360 ","End":"02:14.369","Text":"The line is also increasing as we go in this direction, it\u0027s increasing,"},{"Start":"02:14.369 ","End":"02:18.180","Text":"but because m is 2,"},{"Start":"02:18.180 ","End":"02:21.210","Text":"each time x increases by 1 unit,"},{"Start":"02:21.210 ","End":"02:23.085","Text":"y goes up by 2 units,"},{"Start":"02:23.085 ","End":"02:25.050","Text":"not 1, so it\u0027s steeper."},{"Start":"02:25.050 ","End":"02:27.675","Text":"For each unit across the more I go up,"},{"Start":"02:27.675 ","End":"02:29.325","Text":"the steeper the line is."},{"Start":"02:29.325 ","End":"02:31.650","Text":"These 2 examples will do."},{"Start":"02:31.650 ","End":"02:36.400","Text":"Now, we\u0027ll talk about m less than 0."},{"Start":"02:36.400 ","End":"02:38.535","Text":"We\u0027ll take the example,"},{"Start":"02:38.535 ","End":"02:42.500","Text":"y equals minus x plus 1,"},{"Start":"02:42.500 ","End":"02:47.010","Text":"which makes m equal to minus 1."},{"Start":"02:47.010 ","End":"02:50.720","Text":"Now, a minus 1 is a negative number,"},{"Start":"02:50.720 ","End":"02:57.175","Text":"which means that every time x increases by 1 unit,"},{"Start":"02:57.175 ","End":"03:00.565","Text":"y increases by minus 1 unit."},{"Start":"03:00.565 ","End":"03:04.205","Text":"Now, an increase of a minus number is a decrease."},{"Start":"03:04.205 ","End":"03:11.555","Text":"You can see here that this thing goes down indeed because when x increases, y decreases."},{"Start":"03:11.555 ","End":"03:15.165","Text":"Each 1 unit across is 1 unit down."},{"Start":"03:15.165 ","End":"03:19.460","Text":"Although it decreases, the negative slope is decreasing,"},{"Start":"03:19.460 ","End":"03:21.920","Text":"but it could decrease at different rates."},{"Start":"03:21.920 ","End":"03:25.950","Text":"If I take another line,"},{"Start":"03:25.950 ","End":"03:29.140","Text":"and let\u0027s say this was"},{"Start":"03:37.550 ","End":"03:43.880","Text":"y equals minus 2x plus 1 which means that m is equal to minus 2."},{"Start":"03:43.880 ","End":"03:45.860","Text":"The line also decreases,"},{"Start":"03:45.860 ","End":"03:51.110","Text":"but it\u0027s decreasing more steeply,"},{"Start":"03:51.110 ","End":"03:54.515","Text":"because each time I increase x by 1,"},{"Start":"03:54.515 ","End":"03:59.855","Text":"y increases by minus 2 or it means decreases by 2."},{"Start":"03:59.855 ","End":"04:03.040","Text":"It decreases more, so it\u0027s steeper."},{"Start":"04:03.040 ","End":"04:05.200","Text":"Now, let\u0027s get it written in."},{"Start":"04:05.200 ","End":"04:07.645","Text":"When m is less than 0,"},{"Start":"04:07.645 ","End":"04:14.148","Text":"the line is decreasing and the smaller m is,"},{"Start":"04:14.148 ","End":"04:17.590","Text":"the steeper the decrease."},{"Start":"04:17.590 ","End":"04:21.925","Text":"There we are. We see it illustrated here."},{"Start":"04:21.925 ","End":"04:24.055","Text":"Purple ones are increasing,"},{"Start":"04:24.055 ","End":"04:27.070","Text":"m is 1, m is 2, this is steeper."},{"Start":"04:27.070 ","End":"04:29.525","Text":"Here, in the green ones,"},{"Start":"04:29.525 ","End":"04:32.650","Text":"m is negative, the line is decreasing,"},{"Start":"04:32.650 ","End":"04:35.155","Text":"but here we have minus 1 and minus 2."},{"Start":"04:35.155 ","End":"04:36.550","Text":"This 1 is steeper."},{"Start":"04:36.550 ","End":"04:38.770","Text":"Of course, minus 2 is less than minus 1,"},{"Start":"04:38.770 ","End":"04:40.240","Text":"even though 2 is bigger than 1,"},{"Start":"04:40.240 ","End":"04:42.165","Text":"but with negative numbers, it\u0027s reversed."},{"Start":"04:42.165 ","End":"04:48.160","Text":"The final case is going to be when m equal 0."},{"Start":"04:48.160 ","End":"04:52.633","Text":"If m equals 0, then the line neither increasing nor decreasing,"},{"Start":"04:52.633 ","End":"04:54.500","Text":"the line is constant."},{"Start":"04:54.500 ","End":"05:03.485","Text":"Constant meaning horizontal or another way of putting it is parallel to x-axis."},{"Start":"05:03.485 ","End":"05:09.890","Text":"As an illustration, we\u0027ll take the function y equals 1."},{"Start":"05:10.070 ","End":"05:13.645","Text":"A constant function."},{"Start":"05:13.645 ","End":"05:15.070","Text":"Why is m equal to 0?"},{"Start":"05:15.070 ","End":"05:16.880","Text":"Because this is equal to,"},{"Start":"05:16.880 ","End":"05:20.150","Text":"if you like, y equals 0x plus 1."},{"Start":"05:20.150 ","End":"05:24.610","Text":"If you write it that way, then you see that m equals 0."},{"Start":"05:24.610 ","End":"05:29.930","Text":"We know that this is horizontal because we\u0027ve learnt already about horizontal lines,"},{"Start":"05:29.930 ","End":"05:31.145","Text":"y is a constant."},{"Start":"05:31.145 ","End":"05:35.750","Text":"I can also show it to you that in general,"},{"Start":"05:35.750 ","End":"05:42.504","Text":"if we have y equals mx plus n,"},{"Start":"05:42.504 ","End":"05:45.200","Text":"but if m is equal to 0,"},{"Start":"05:45.200 ","End":"05:48.038","Text":"then we just get y equals n,"},{"Start":"05:48.038 ","End":"05:49.845","Text":"and in our case, n is 1,"},{"Start":"05:49.845 ","End":"05:52.275","Text":"and that\u0027s a horizontal line."},{"Start":"05:52.275 ","End":"05:56.385","Text":"That takes care of the increasing,"},{"Start":"05:56.385 ","End":"06:01.800","Text":"the decreasing, and the level horizontal lines."},{"Start":"06:01.800 ","End":"06:05.460","Text":"That\u0027s all I have to say about slope."}],"ID":5099},{"Watched":false,"Name":"The Slope Formula","Duration":"4m 28s","ChapterTopicVideoID":5122,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5122.jpeg","UploadDate":"2020-10-06T08:20:31.3800000","DurationForVideoObject":"PT4M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this clip, we\u0027ll learn a very useful formula for computing the slope."},{"Start":"00:05.250 ","End":"00:07.799","Text":"We use this when we have a line,"},{"Start":"00:07.799 ","End":"00:09.585","Text":"let me just draw a line."},{"Start":"00:09.585 ","End":"00:13.395","Text":"Besides the line, we\u0027re given 2 points on the line."},{"Start":"00:13.395 ","End":"00:17.065","Text":"Let\u0027s say this 1 and this 1."},{"Start":"00:17.065 ","End":"00:18.995","Text":"We\u0027re actually given the coordinates."},{"Start":"00:18.995 ","End":"00:24.180","Text":"Let\u0027s say this 1 is x_1,"},{"Start":"00:24.760 ","End":"00:28.000","Text":"y_1, and the other 1,"},{"Start":"00:28.000 ","End":"00:38.390","Text":"the second point will be x_2, y_2."},{"Start":"00:38.390 ","End":"00:43.740","Text":"That\u0027s all we\u0027re given and we have to compute the slope m of this line."},{"Start":"00:44.110 ","End":"00:46.850","Text":"The slope which we use obviously"},{"Start":"00:46.850 ","End":"00:52.250","Text":"the letter m typically will be equal to according to the formula,"},{"Start":"00:52.250 ","End":"00:57.739","Text":"the difference between the y\u0027s over"},{"Start":"00:57.739 ","End":"01:04.645","Text":"the difference between the x\u0027s: y_2 minus y_1 over x_2 minus x_1."},{"Start":"01:04.645 ","End":"01:06.805","Text":"Let\u0027s say it\u0027s always the second minus the first."},{"Start":"01:06.805 ","End":"01:09.400","Text":"Decide on which point is first, this 1 is the first,"},{"Start":"01:09.400 ","End":"01:13.660","Text":"this 1 is the second and there\u0027s the formula."},{"Start":"01:13.660 ","End":"01:22.030","Text":"On the picture, what it is I\u0027ll use some color and let\u0027s see."},{"Start":"01:22.050 ","End":"01:30.085","Text":"The difference in the y\u0027s is y_2 minus y_1 is this length here in this direction."},{"Start":"01:30.085 ","End":"01:36.940","Text":"If we complete this to the triangle like we\u0027ve done before,"},{"Start":"01:36.940 ","End":"01:40.675","Text":"wrong color, second black,"},{"Start":"01:40.675 ","End":"01:42.650","Text":"complete it to the triangle."},{"Start":"01:42.650 ","End":"01:44.465","Text":"We have a little triangle here."},{"Start":"01:44.465 ","End":"01:49.520","Text":"Basically this part of the triangle,"},{"Start":"01:49.520 ","End":"01:51.395","Text":"the height over the base,"},{"Start":"01:51.395 ","End":"01:55.385","Text":"which we also call rise over run."},{"Start":"01:55.385 ","End":"02:01.960","Text":"This part here will be the rise."},{"Start":"02:01.960 ","End":"02:07.085","Text":"This part here in yellow will be the run,"},{"Start":"02:07.085 ","End":"02:10.055","Text":"which is also the difference between the x\u0027s,"},{"Start":"02:10.055 ","End":"02:13.504","Text":"x_2 minus x_1 in this direction."},{"Start":"02:13.504 ","End":"02:19.790","Text":"Let me write it also as rise over run."},{"Start":"02:25.410 ","End":"02:26.559","Text":"In some colleges they also call this difference in the y\u0027s,"},{"Start":"02:26.559 ","End":"02:34.000","Text":"here\u0027s another name, pardon me,"},{"Start":"02:34.000 ","End":"02:39.135","Text":"it\u0027s a Delta not an A."},{"Start":"02:39.135 ","End":"02:47.560","Text":"Delta y over delta x. Delta is the Greek letter D for difference."},{"Start":"02:47.560 ","End":"02:49.180","Text":"Difference in the y\u0027s, difference in the x\u0027s."},{"Start":"02:49.180 ","End":"02:52.510","Text":"There\u0027s many ways to label it."},{"Start":"02:52.510 ","End":"02:57.680","Text":"Essentially it\u0027s the green over the yellow."},{"Start":"02:57.960 ","End":"03:00.840","Text":"The formula is really all that is,"},{"Start":"03:00.840 ","End":"03:02.400","Text":"you just plug in the numbers,"},{"Start":"03:02.400 ","End":"03:05.760","Text":"but certainly, an example would help."},{"Start":"03:05.760 ","End":"03:12.210","Text":"Let\u0027s take the following exercise if you like."},{"Start":"03:12.790 ","End":"03:15.440","Text":"Here\u0027s the exercise."},{"Start":"03:15.440 ","End":"03:18.490","Text":"Find the slope of the line in the sketch."},{"Start":"03:18.490 ","End":"03:19.650","Text":"Here we have a sketch."},{"Start":"03:19.650 ","End":"03:21.620","Text":"Here we have the line, the 1 in blue,"},{"Start":"03:21.620 ","End":"03:26.660","Text":"and we\u0027re given the 2 points with their coordinates."},{"Start":"03:26.660 ","End":"03:30.155","Text":"All we do is look at the formula."},{"Start":"03:30.155 ","End":"03:32.330","Text":"You just have to look at this part here."},{"Start":"03:32.330 ","End":"03:35.075","Text":"The rise over run is just a mnemonic."},{"Start":"03:35.075 ","End":"03:38.775","Text":"We say that M,"},{"Start":"03:38.775 ","End":"03:42.610","Text":"the slope is equal to,"},{"Start":"03:46.970 ","End":"03:49.260","Text":"this is x_1, y_1,"},{"Start":"03:49.260 ","End":"03:50.565","Text":"this is x_2, y_2."},{"Start":"03:50.565 ","End":"03:52.230","Text":"This is the first, this is the second."},{"Start":"03:52.230 ","End":"03:53.675","Text":"If we look at this,"},{"Start":"03:53.675 ","End":"03:56.255","Text":"we can say the difference in the y\u0027s,"},{"Start":"03:56.255 ","End":"04:00.470","Text":"second minus the first is 4 minus 2."},{"Start":"04:00.470 ","End":"04:04.040","Text":"That\u0027s the difference in the y\u0027s over the difference in the x\u0027s,"},{"Start":"04:04.040 ","End":"04:06.230","Text":"the 3 minus the 1."},{"Start":"04:06.230 ","End":"04:09.170","Text":"As I said, you can get the order of the points backwards,"},{"Start":"04:09.170 ","End":"04:14.060","Text":"but you can\u0027t mix and match because if you\u0027ve got them both negative,"},{"Start":"04:14.060 ","End":"04:15.455","Text":"it would come out positive."},{"Start":"04:15.455 ","End":"04:20.450","Text":"Anyway, the answer would be 4 minus 2 is 2 over 3 minus 1,"},{"Start":"04:20.450 ","End":"04:24.440","Text":"which is also 2 and the answer is 1, the slope is 1."},{"Start":"04:24.440 ","End":"04:25.700","Text":"That\u0027s all there is to it."},{"Start":"04:25.700 ","End":"04:27.880","Text":"Done."}],"ID":5100},{"Watched":false,"Name":"Finding the Equation of a Line","Duration":"7m ","ChapterTopicVideoID":5123,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5123.jpeg","UploadDate":"2020-10-06T08:23:10.7230000","DurationForVideoObject":"PT7M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"In this clip, we\u0027ll learn how to find the equation of a line,"},{"Start":"00:04.185 ","End":"00:06.450","Text":"if we\u0027re given a point on the line,"},{"Start":"00:06.450 ","End":"00:08.145","Text":"and the slope of the line."},{"Start":"00:08.145 ","End":"00:11.925","Text":"I\u0027ll just jump right into an example."},{"Start":"00:11.925 ","End":"00:15.750","Text":"Given the line with slope equal to 4,"},{"Start":"00:15.750 ","End":"00:18.440","Text":"and which passes through the point 1,"},{"Start":"00:18.440 ","End":"00:21.810","Text":"2, we have to find the equation of the line."},{"Start":"00:21.810 ","End":"00:25.965","Text":"We know the general equation of a line is"},{"Start":"00:25.965 ","End":"00:33.390","Text":"y equals mx plus n. What\u0027s more,"},{"Start":"00:33.390 ","End":"00:36.180","Text":"we know that m is the slope."},{"Start":"00:36.180 ","End":"00:42.735","Text":"Here basically we\u0027re given that m equals 4."},{"Start":"00:42.735 ","End":"00:44.820","Text":"Even though it doesn\u0027t say m equals 4,"},{"Start":"00:44.820 ","End":"00:46.965","Text":"but it says it has slope 4."},{"Start":"00:46.965 ","End":"00:48.835","Text":"That\u0027s m equals 4."},{"Start":"00:48.835 ","End":"00:56.090","Text":"In other words, the equation is up to the form y equals 4x plus n. All we"},{"Start":"00:56.090 ","End":"00:59.750","Text":"have left to find is n. We\u0027re"},{"Start":"00:59.750 ","End":"01:04.740","Text":"given another piece of information that it passes through 1, 2."},{"Start":"01:04.810 ","End":"01:09.385","Text":"It passes through 1, 2."},{"Start":"01:09.385 ","End":"01:11.090","Text":"What does passes through mean?"},{"Start":"01:11.090 ","End":"01:14.525","Text":"It means that the point satisfies the equation of the line."},{"Start":"01:14.525 ","End":"01:17.945","Text":"This is obviously the x this is the y."},{"Start":"01:17.945 ","End":"01:23.819","Text":"We get 2 equals 4 times x,"},{"Start":"01:23.819 ","End":"01:26.800","Text":"which is 1 plus n,"},{"Start":"01:26.800 ","End":"01:32.435","Text":"which gives us that n equals minus 2."},{"Start":"01:32.435 ","End":"01:35.570","Text":"Once we have m, which is 4,"},{"Start":"01:35.570 ","End":"01:38.540","Text":"and we have n, which is minus 2,"},{"Start":"01:38.540 ","End":"01:41.465","Text":"we just have to put it into the equation."},{"Start":"01:41.465 ","End":"01:45.365","Text":"We have all the constants in the equation of the line."},{"Start":"01:45.365 ","End":"01:47.554","Text":"We get as our answer,"},{"Start":"01:47.554 ","End":"01:54.240","Text":"that y equals 4x plus minus 2,"},{"Start":"01:54.240 ","End":"01:56.040","Text":"which is just minus 2."},{"Start":"01:56.040 ","End":"01:57.585","Text":"That is our answer."},{"Start":"01:57.585 ","End":"01:59.820","Text":"We can check ourselves,"},{"Start":"01:59.820 ","End":"02:03.890","Text":"the 4 doesn\u0027t need to be checked as we know that 4 is the slope and only have"},{"Start":"02:03.890 ","End":"02:08.350","Text":"to check is to substitute 1,"},{"Start":"02:08.350 ","End":"02:10.610","Text":"2 in here, put x equals 1,"},{"Start":"02:10.610 ","End":"02:12.185","Text":"y equals 2, and you\u0027ll see it works."},{"Start":"02:12.185 ","End":"02:15.920","Text":"There is a general formula for this equation,"},{"Start":"02:15.920 ","End":"02:17.390","Text":"but it\u0027s messy,"},{"Start":"02:17.390 ","End":"02:22.880","Text":"and cumbersome, and I\u0027m not going to give it because we don\u0027t really need the formula."},{"Start":"02:22.880 ","End":"02:26.030","Text":"We have the simple technique of solving such examples,"},{"Start":"02:26.030 ","End":"02:27.800","Text":"and I\u0027ll leave it at that,"},{"Start":"02:27.800 ","End":"02:31.255","Text":"at least this is half of the clip."},{"Start":"02:31.255 ","End":"02:36.734","Text":"The other half deals with finding the equation of a line given 2 points on the line."},{"Start":"02:36.734 ","End":"02:38.180","Text":"That\u0027s next."},{"Start":"02:38.180 ","End":"02:43.544","Text":"We come to the second part of finding the equation of a line,"},{"Start":"02:43.544 ","End":"02:46.790","Text":"this time given 2 points on the line."},{"Start":"02:46.790 ","End":"02:50.569","Text":"As before, we\u0027ll just work with an example."},{"Start":"02:50.569 ","End":"02:53.030","Text":"This is the example exercise."},{"Start":"02:53.030 ","End":"02:56.530","Text":"Given a line which passes through 1,"},{"Start":"02:56.530 ","End":"02:57.735","Text":"2 and 3,"},{"Start":"02:57.735 ","End":"02:59.180","Text":"5 these are 2 points of course,"},{"Start":"02:59.180 ","End":"03:01.880","Text":"we have to find the equation of the line."},{"Start":"03:01.880 ","End":"03:03.740","Text":"I\u0027d like to mention that there is"},{"Start":"03:03.740 ","End":"03:08.530","Text":"one obvious way of solving it according to what we\u0027ve learned."},{"Start":"03:08.530 ","End":"03:13.445","Text":"Because if we\u0027re given 2 points on the line,"},{"Start":"03:13.445 ","End":"03:17.045","Text":"we can find the slope using the slope equation."},{"Start":"03:17.045 ","End":"03:21.560","Text":"If we have the slope and we have a point, I mean,"},{"Start":"03:21.560 ","End":"03:27.380","Text":"just choose any one of these 2 [inaudible] Choose the first or the second."},{"Start":"03:27.380 ","End":"03:29.825","Text":"We have a slope, and a point,"},{"Start":"03:29.825 ","End":"03:32.360","Text":"and we\u0027ve already learned in the first part,"},{"Start":"03:32.360 ","End":"03:35.585","Text":"to find the equation using a slope, and a point."},{"Start":"03:35.585 ","End":"03:40.575","Text":"I\u0027d like to suggest quite a different method than that."},{"Start":"03:40.575 ","End":"03:43.860","Text":"But for those who want to go and find the slope between"},{"Start":"03:43.860 ","End":"03:45.830","Text":"the 2 points and then use the previous method"},{"Start":"03:45.830 ","End":"03:47.990","Text":"for a slope and a point after you\u0027ve picked one."},{"Start":"03:47.990 ","End":"03:52.725","Text":"The way I want to show you involves 2 equations and 2 unknowns."},{"Start":"03:52.725 ","End":"03:54.090","Text":"What we do is this,"},{"Start":"03:54.090 ","End":"03:59.825","Text":"the equation of a line in general is"},{"Start":"03:59.825 ","End":"04:08.025","Text":"y equals mx plus n. m and n are going to be our unknowns,"},{"Start":"04:08.025 ","End":"04:10.080","Text":"and we\u0027re going to get 2 equations."},{"Start":"04:10.080 ","End":"04:12.045","Text":"We have 2 points on the line."},{"Start":"04:12.045 ","End":"04:17.790","Text":"What we can do is substitute them for y and x into here,"},{"Start":"04:17.790 ","End":"04:19.860","Text":"and then we\u0027ll get 2 equations and 2 unknowns,"},{"Start":"04:19.860 ","End":"04:21.560","Text":"m and n, I\u0027ll show you what I mean."},{"Start":"04:21.560 ","End":"04:25.700","Text":"Let\u0027s take the first point 1, 2."},{"Start":"04:25.700 ","End":"04:30.035","Text":"1, 2 gives us the y is 2,"},{"Start":"04:30.035 ","End":"04:34.835","Text":"equals m times 1 plus n,"},{"Start":"04:34.835 ","End":"04:38.855","Text":"and the second equation I get is from the 3, 5."},{"Start":"04:38.855 ","End":"04:45.525","Text":"I get 5 equals m times 3 plus n,"},{"Start":"04:45.525 ","End":"04:48.500","Text":"2 equations and 2 unknowns."},{"Start":"04:48.500 ","End":"04:51.410","Text":"It\u0027s not really a problem to solve them,"},{"Start":"04:51.410 ","End":"04:52.700","Text":"you know how to do this thing."},{"Start":"04:52.700 ","End":"04:58.745","Text":"But I\u0027ll just show you one suggested method works well for this problem."},{"Start":"04:58.745 ","End":"05:02.350","Text":"That is to isolate n in both cases."},{"Start":"05:02.350 ","End":"05:04.955","Text":"My 2 equations will become,"},{"Start":"05:04.955 ","End":"05:06.785","Text":"if I isolate n,"},{"Start":"05:06.785 ","End":"05:09.395","Text":"I will get, on the one hand,"},{"Start":"05:09.395 ","End":"05:14.210","Text":"n is going to equal minus m plus 2."},{"Start":"05:14.210 ","End":"05:17.370","Text":"On the other hand, n will equal,"},{"Start":"05:17.370 ","End":"05:22.185","Text":"if I throw this over minus 3m plus 5."},{"Start":"05:22.185 ","End":"05:24.000","Text":"The way to solve these 2,"},{"Start":"05:24.000 ","End":"05:25.490","Text":"it\u0027s again 2 equations and 2 unknowns,"},{"Start":"05:25.490 ","End":"05:29.535","Text":"but look, n is on the same on the left hand side,"},{"Start":"05:29.535 ","End":"05:34.085","Text":"what we have to do is equate this to this and we have an equation in m,"},{"Start":"05:34.085 ","End":"05:35.600","Text":"we find the solution to that,"},{"Start":"05:35.600 ","End":"05:37.390","Text":"and then substitute it in one of them."},{"Start":"05:37.390 ","End":"05:41.525","Text":"We\u0027ll get by comparing this with this,"},{"Start":"05:41.525 ","End":"05:49.830","Text":"minus m plus 2 is equal to minus 3m plus 5,"},{"Start":"05:49.830 ","End":"05:53.790","Text":"by comparing both right hand sides."},{"Start":"05:53.790 ","End":"05:56.440","Text":"We can solve this in our heads."},{"Start":"05:56.440 ","End":"05:58.360","Text":"Let\u0027s say we take the 3m over here,"},{"Start":"05:58.360 ","End":"06:01.825","Text":"so it becomes 2m equals 3,"},{"Start":"06:01.825 ","End":"06:06.190","Text":"and m is equal to 1 and 1/2."},{"Start":"06:06.190 ","End":"06:08.845","Text":"Once m is equal to 1 and 1/2,"},{"Start":"06:08.845 ","End":"06:12.985","Text":"I can put it into either one of these and get,"},{"Start":"06:12.985 ","End":"06:14.200","Text":"say on the first one,"},{"Start":"06:14.200 ","End":"06:16.290","Text":"if m is 1 and 1/2,"},{"Start":"06:16.290 ","End":"06:19.530","Text":"n equals 1 1/2."},{"Start":"06:19.530 ","End":"06:21.630","Text":"Now that we have these 2 values,"},{"Start":"06:21.630 ","End":"06:23.770","Text":"what we were asked for, is the equation of the line."},{"Start":"06:23.770 ","End":"06:27.235","Text":"The equation of a line is mx plus n, y equals."},{"Start":"06:27.235 ","End":"06:31.090","Text":"Ultimately we have y equals mx plus n,"},{"Start":"06:31.090 ","End":"06:36.205","Text":"which is 1 and 1/2 x plus n plus 1 1/2."},{"Start":"06:36.205 ","End":"06:41.315","Text":"That is the equation of the line."},{"Start":"06:41.315 ","End":"06:44.160","Text":"That\u0027s the general technique."},{"Start":"06:44.160 ","End":"06:47.825","Text":"In beginning I showed you one possible other technique,"},{"Start":"06:47.825 ","End":"06:52.130","Text":"using the 2 points to get a slope and then using a slope and a point."},{"Start":"06:52.130 ","End":"06:56.359","Text":"But this is a variant using 2 equations and 2 unknowns."},{"Start":"06:56.359 ","End":"07:00.840","Text":"That\u0027s it for finding the equation of a line."}],"ID":5101},{"Watched":false,"Name":"The Mutual Positions of Two Lines","Duration":"5m 52s","ChapterTopicVideoID":5124,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5124.jpeg","UploadDate":"2020-10-06T08:25:10.6200000","DurationForVideoObject":"PT5M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.850","Text":"In this section, I want to talk about the mutual position of 2 lines in the plane."},{"Start":"00:05.850 ","End":"00:07.815","Text":"I should mention that."},{"Start":"00:07.815 ","End":"00:13.275","Text":"First of all, I just want to give you a general descriptive thing of what I mean."},{"Start":"00:13.275 ","End":"00:16.035","Text":"Let\u0027s take, for example,"},{"Start":"00:16.035 ","End":"00:18.255","Text":"I\u0027ll take the same line."},{"Start":"00:18.255 ","End":"00:20.820","Text":"I\u0027ll take a line here. You\u0027ll see what I\u0027m doing."},{"Start":"00:20.820 ","End":"00:22.800","Text":"Then I\u0027ll compare it with another line,"},{"Start":"00:22.800 ","End":"00:25.830","Text":"let\u0027s say a line here and a line here."},{"Start":"00:25.830 ","End":"00:28.710","Text":"Now I\u0027ll go for another color."},{"Start":"00:28.710 ","End":"00:37.085","Text":"Now 2 lines generally will cross each other so these are called intersecting."},{"Start":"00:37.085 ","End":"00:41.330","Text":"If the other line is parallel to it,"},{"Start":"00:41.330 ","End":"00:44.725","Text":"then it\u0027s parallel, it\u0027s not intersecting."},{"Start":"00:44.725 ","End":"00:49.025","Text":"There is another special intersection,"},{"Start":"00:49.025 ","End":"00:58.005","Text":"which is when it intersects at 90 degrees and then it\u0027s perpendicular."},{"Start":"00:58.005 ","End":"01:01.460","Text":"Finally, this 1 which some people don\u0027t even count at"},{"Start":"01:01.460 ","End":"01:05.060","Text":"all because it\u0027s exactly the same line."},{"Start":"01:05.060 ","End":"01:10.010","Text":"But in some places it\u0027s given a name and they\u0027re called coincidents."},{"Start":"01:10.010 ","End":"01:14.540","Text":"Just write that in summary."},{"Start":"01:14.540 ","End":"01:19.610","Text":"These are intersecting under the special case of intersecting,"},{"Start":"01:19.610 ","End":"01:23.105","Text":"which is called perpendicular."},{"Start":"01:23.105 ","End":"01:27.500","Text":"Then there\u0027s non-intersecting which are parallel."},{"Start":"01:27.500 ","End":"01:30.440","Text":"Like I said, there is a bit of a controversy whether this is"},{"Start":"01:30.440 ","End":"01:33.410","Text":"a mutual position where they\u0027re exactly the same line,"},{"Start":"01:33.410 ","End":"01:34.640","Text":"but some people say yes,"},{"Start":"01:34.640 ","End":"01:38.297","Text":"so it just happens to be coincidence."},{"Start":"01:38.297 ","End":"01:39.995","Text":"1 on top of the other."},{"Start":"01:39.995 ","End":"01:45.170","Text":"In this course, we\u0027ll discount the coincident case and we\u0027ll assume right away"},{"Start":"01:45.170 ","End":"01:50.539","Text":"that we have 2 different lines and our job is to decide whether they\u0027re parallel,"},{"Start":"01:50.539 ","End":"01:53.809","Text":"intersecting generally, or perpendicular."},{"Start":"01:53.809 ","End":"01:57.560","Text":"Let\u0027s see how we go about this."},{"Start":"01:57.560 ","End":"02:04.820","Text":"Remember in general, a line is y=mx plus n. But if I\u0027m talking about 2 lines,"},{"Start":"02:04.820 ","End":"02:06.545","Text":"I have blue 1 and the red 1."},{"Start":"02:06.545 ","End":"02:09.950","Text":"Then 1 of them, I won\u0027t call them both mx plus"},{"Start":"02:09.950 ","End":"02:13.955","Text":"n. So the first line I\u0027ll call m_1x plus n_1."},{"Start":"02:13.955 ","End":"02:15.605","Text":"That\u0027s 1 line,"},{"Start":"02:15.605 ","End":"02:20.480","Text":"maybe the blue 1 and the other 1 will be"},{"Start":"02:20.480 ","End":"02:25.980","Text":"y=m_2x plus n_2,"},{"Start":"02:25.980 ","End":"02:30.950","Text":"a different m and n. I\u0027d like to know which of these cases they are."},{"Start":"02:30.950 ","End":"02:36.620","Text":"Well, we\u0027re already assuming we\u0027re just counting this as like the cheating case,"},{"Start":"02:36.620 ","End":"02:38.120","Text":"whether they\u0027re the same line,"},{"Start":"02:38.120 ","End":"02:40.510","Text":"but some people think that\u0027s okay."},{"Start":"02:40.510 ","End":"02:44.990","Text":"We\u0027re assuming that we don\u0027t have the exactly the identical numbers."},{"Start":"02:44.990 ","End":"02:50.420","Text":"It\u0027s not that m_1 is m_2 and then 1 is n_2 and they\u0027re everything\u0027s the same."},{"Start":"02:50.420 ","End":"02:51.950","Text":"Some different."},{"Start":"02:51.950 ","End":"02:54.440","Text":"Let\u0027s start with the parallel ones."},{"Start":"02:54.440 ","End":"02:57.215","Text":"How do you know when 2 lines are parallel?"},{"Start":"02:57.215 ","End":"03:01.010","Text":"Well, the simple answer is geometrically that they have the same slope."},{"Start":"03:01.010 ","End":"03:05.240","Text":"If they\u0027re parallel, they have the same slope and therefore they will not intersect."},{"Start":"03:05.240 ","End":"03:08.360","Text":"For the 2 lines to be parallel,"},{"Start":"03:08.360 ","End":"03:14.215","Text":"the condition is that m_1=m_2."},{"Start":"03:14.215 ","End":"03:19.160","Text":"What it means in simple words is that they have the same slope."},{"Start":"03:19.160 ","End":"03:27.870","Text":"Same slope means they\u0027re parallel and intersecting means that m_1 is not equal to m_2."},{"Start":"03:27.870 ","End":"03:29.045","Text":"As I said,"},{"Start":"03:29.045 ","End":"03:32.270","Text":"we\u0027re discounting this case because this is the case."},{"Start":"03:32.270 ","End":"03:34.820","Text":"But if you are interested in it,"},{"Start":"03:34.820 ","End":"03:37.130","Text":"in this case is where everything\u0027s the same."},{"Start":"03:37.130 ","End":"03:38.675","Text":"M_1 equals m_2,"},{"Start":"03:38.675 ","End":"03:41.225","Text":"but also n_1 equals n_2."},{"Start":"03:41.225 ","End":"03:45.215","Text":"We\u0027re not going to deal with this."},{"Start":"03:45.215 ","End":"03:47.930","Text":"M_1 is not equal to m_2."},{"Start":"03:47.930 ","End":"03:52.220","Text":"M_2 is intersecting and the special case of"},{"Start":"03:52.220 ","End":"03:57.155","Text":"intersecting is when it\u0027s actually 90 degrees,"},{"Start":"03:57.155 ","End":"03:59.435","Text":"which is called perpendicular,"},{"Start":"03:59.435 ","End":"04:02.565","Text":"has a rule that,"},{"Start":"04:02.565 ","End":"04:04.965","Text":"if you multiply the 2 slopes,"},{"Start":"04:04.965 ","End":"04:07.620","Text":"m_1 times m_2,"},{"Start":"04:07.620 ","End":"04:09.895","Text":"you get minus 1."},{"Start":"04:09.895 ","End":"04:14.360","Text":"But we could also write this formula which I like because it\u0027s symmetrical"},{"Start":"04:14.360 ","End":"04:19.550","Text":"as m_2 is minus 1 over m_1 and vice versa."},{"Start":"04:19.550 ","End":"04:22.430","Text":"Each 1 is the negative reciprocal of the other."},{"Start":"04:22.430 ","End":"04:25.055","Text":"Negative reciprocal or negative inverse,"},{"Start":"04:25.055 ","End":"04:27.325","Text":"meaning minus 1 over."},{"Start":"04:27.325 ","End":"04:32.590","Text":"Negative is the minus and inverse or reciprocal is 1 over."},{"Start":"04:32.590 ","End":"04:37.420","Text":"Just give maybe a small example of the use of this formula."},{"Start":"04:37.420 ","End":"04:39.980","Text":"Let\u0027s say we have 2 perpendicular lines."},{"Start":"04:39.980 ","End":"04:47.105","Text":"Again, I took blue and the red and suppose I know that this line is y equals,"},{"Start":"04:47.105 ","End":"04:54.755","Text":"let\u0027s say -0.5 (x+1)."},{"Start":"04:54.755 ","End":"04:59.690","Text":"I know it\u0027s slope,"},{"Start":"04:59.690 ","End":"05:09.470","Text":"because the slope of this 1 m is m_1 is -0.5 and if this is the second line,"},{"Start":"05:09.470 ","End":"05:12.440","Text":"then m_2 would be the negative reciprocal,"},{"Start":"05:12.440 ","End":"05:16.340","Text":"so it\u0027s minus 1 over this so it\u0027s 2."},{"Start":"05:16.340 ","End":"05:20.435","Text":"I don\u0027t know exactly the equation of the red line,"},{"Start":"05:20.435 ","End":"05:25.884","Text":"but I know it\u0027ll be y equals 2x plus something."},{"Start":"05:25.884 ","End":"05:28.790","Text":"Just say plus I plus n because I don\u0027t know what the constant is."},{"Start":"05:28.790 ","End":"05:34.250","Text":"But you can see that if you compare the m here and here,"},{"Start":"05:34.250 ","End":"05:39.660","Text":"they are negative reciprocals or the product is minus 1."},{"Start":"05:40.600 ","End":"05:47.945","Text":"That\u0027s enough on this subject and coming up next,"},{"Start":"05:47.945 ","End":"05:51.480","Text":"the intersection of 2 lines."}],"ID":5102},{"Watched":false,"Name":"The Distance between Two Points","Duration":"9m 36s","ChapterTopicVideoID":6558,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6558.jpeg","UploadDate":"2020-10-06T08:28:39.9770000","DurationForVideoObject":"PT9M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.575","Text":"In this clip, I\u0027ll be talking about the distance between 2 points."},{"Start":"00:04.575 ","End":"00:07.965","Text":"This is again in the field of analytic geometry."},{"Start":"00:07.965 ","End":"00:09.900","Text":"We\u0027re going to be given 2 points,"},{"Start":"00:09.900 ","End":"00:11.783","Text":"and their coordinates,"},{"Start":"00:11.783 ","End":"00:13.620","Text":"and we\u0027ll find the distance between them."},{"Start":"00:13.620 ","End":"00:16.985","Text":"I\u0027d like to divide it up into cases."},{"Start":"00:16.985 ","End":"00:18.620","Text":"All the cases are easy,"},{"Start":"00:18.620 ","End":"00:21.649","Text":"but there are a couple of even especially easy cases."},{"Start":"00:21.649 ","End":"00:26.300","Text":"One case will be where the one-point is directly above,"},{"Start":"00:26.300 ","End":"00:28.415","Text":"or below the other."},{"Start":"00:28.415 ","End":"00:31.490","Text":"I wrote above because if one is above the other,"},{"Start":"00:31.490 ","End":"00:36.305","Text":"then the other is below the one, so it doesn\u0027t matter."},{"Start":"00:36.305 ","End":"00:41.720","Text":"The second case will be where one point is directly to the left,"},{"Start":"00:41.720 ","End":"00:43.970","Text":"or the right of the other."},{"Start":"00:43.970 ","End":"00:45.590","Text":"I just wrote to the right."},{"Start":"00:45.590 ","End":"00:51.425","Text":"Of course, if you reverse the points then one will be to the left of the other."},{"Start":"00:51.425 ","End":"00:54.770","Text":"The third case will be the general case."},{"Start":"00:54.770 ","End":"00:57.025","Text":"No restrictions."},{"Start":"00:57.025 ","End":"01:00.215","Text":"I\u0027m going to start with some examples,"},{"Start":"01:00.215 ","End":"01:02.240","Text":"and a drawing, and you\u0027ll see what I mean."},{"Start":"01:02.240 ","End":"01:04.520","Text":"An example for the first case will be,"},{"Start":"01:04.520 ","End":"01:05.870","Text":"let\u0027s take points A and B."},{"Start":"01:05.870 ","End":"01:13.850","Text":"Let\u0027s say that A is the point 3,2 and B is the point 3,6."},{"Start":"01:13.850 ","End":"01:17.250","Text":"Let\u0027s see if we can find them on the graph."},{"Start":"01:17.980 ","End":"01:20.540","Text":"They both have the same x."},{"Start":"01:20.540 ","End":"01:23.965","Text":"If I take a dotted line here,"},{"Start":"01:23.965 ","End":"01:27.680","Text":"and I even continue it down to see where the x-coordinate is,"},{"Start":"01:27.680 ","End":"01:31.280","Text":"the x-coordinate of both of them is 3."},{"Start":"01:31.280 ","End":"01:37.160","Text":"They have the same x, that\u0027s why you could say they\u0027re vertical one to the other."},{"Start":"01:37.160 ","End":"01:39.545","Text":"I\u0027m going to even write that down."},{"Start":"01:39.545 ","End":"01:46.039","Text":"That this case is the case of the same x."},{"Start":"01:46.039 ","End":"01:49.850","Text":"That\u0027s another way of saying that one is directly above, or below the other."},{"Start":"01:49.850 ","End":"01:51.140","Text":"Look, x here is 3,"},{"Start":"01:51.140 ","End":"01:52.835","Text":"and x here is 3."},{"Start":"01:52.835 ","End":"01:55.610","Text":"Now, how do we compute the distance?"},{"Start":"01:55.610 ","End":"01:59.985","Text":"Well, let\u0027s look at the y of this point, 6."},{"Start":"01:59.985 ","End":"02:04.770","Text":"Here we would get the point 2."},{"Start":"02:04.770 ","End":"02:07.905","Text":"What we have here is a rectangle."},{"Start":"02:07.905 ","End":"02:10.370","Text":"The distance between these two would be the"},{"Start":"02:10.370 ","End":"02:13.760","Text":"same as the distance between these two points."},{"Start":"02:13.760 ","End":"02:16.820","Text":"Here, it\u0027s easy to see that they\u0027re on the axis."},{"Start":"02:16.820 ","End":"02:21.840","Text":"So I would say that the distance is 6 minus 2."},{"Start":"02:21.840 ","End":"02:24.430","Text":"Here, I know which one is above the other,"},{"Start":"02:24.430 ","End":"02:26.210","Text":"but if I somehow didn\u0027t,"},{"Start":"02:26.210 ","End":"02:28.740","Text":"I might be tempted to say, well,"},{"Start":"02:28.740 ","End":"02:33.690","Text":"2 minus 6 and I\u0027d get minus 4 for a distance."},{"Start":"02:33.690 ","End":"02:37.003","Text":"Best thing to do is just to put absolute value,"},{"Start":"02:37.003 ","End":"02:38.300","Text":"and then you\u0027re covered both ways."},{"Start":"02:38.300 ","End":"02:42.470","Text":"If I say the absolute value of the second y minus the first y,"},{"Start":"02:42.470 ","End":"02:45.130","Text":"I get absolute value of 6 minus 2,"},{"Start":"02:45.130 ","End":"02:47.550","Text":"and I\u0027ve got the answer of 4."},{"Start":"02:47.550 ","End":"02:51.500","Text":"This distance here would be 4 units."},{"Start":"02:51.500 ","End":"02:53.195","Text":"Just subtract the y\u0027s,"},{"Start":"02:53.195 ","End":"02:57.295","Text":"but make sure it comes up positive by taking absolute value."},{"Start":"02:57.295 ","End":"03:02.005","Text":"Now for the second example."},{"Start":"03:02.005 ","End":"03:05.015","Text":"That\u0027s when one is directly to the right of the other."},{"Start":"03:05.015 ","End":"03:14.880","Text":"The two points I\u0027m going to take are C and D. C will be the point minus 4,"},{"Start":"03:14.880 ","End":"03:21.925","Text":"minus 3, and D will be the point 2, minus 3."},{"Start":"03:21.925 ","End":"03:23.960","Text":"Let\u0027s plot them on the graph."},{"Start":"03:23.960 ","End":"03:26.225","Text":"This would be the point C,"},{"Start":"03:26.225 ","End":"03:35.360","Text":"this will be the point D. I\u0027ll just write the coordinates here also."},{"Start":"03:35.360 ","End":"03:39.815","Text":"How do I know that this is one directly to the right of the other?"},{"Start":"03:39.815 ","End":"03:42.530","Text":"I could call this the horizontal case."},{"Start":"03:42.530 ","End":"03:44.420","Text":"This is the vertical case. The horizontal case."},{"Start":"03:44.420 ","End":"03:48.890","Text":"The reason is because they have the same y-value."},{"Start":"03:48.890 ","End":"03:53.450","Text":"It\u0027s horizontal, they have the same y-value,"},{"Start":"03:53.450 ","End":"03:57.575","Text":"so I\u0027m going to write that in brackets, same y."},{"Start":"03:57.575 ","End":"03:59.220","Text":"Just like before,"},{"Start":"03:59.220 ","End":"04:04.940","Text":"we can find the distance by taking the point directly above,"},{"Start":"04:04.940 ","End":"04:07.980","Text":"or below on the x-axis."},{"Start":"04:07.980 ","End":"04:10.190","Text":"Because this is 2,"},{"Start":"04:10.190 ","End":"04:15.830","Text":"this would be the point 2 and this would be the point negative 4."},{"Start":"04:15.830 ","End":"04:19.830","Text":"Then I want to know what the distances from 2 to negative 4."},{"Start":"04:19.830 ","End":"04:21.825","Text":"So I just subtract them,"},{"Start":"04:21.825 ","End":"04:26.025","Text":"2 less minus 4 is 6,"},{"Start":"04:26.025 ","End":"04:28.155","Text":"and if this is between these 2 and 6,"},{"Start":"04:28.155 ","End":"04:29.975","Text":"since we have a rectangle here,"},{"Start":"04:29.975 ","End":"04:34.370","Text":"this distance will be 6 also."},{"Start":"04:34.370 ","End":"04:37.790","Text":"The general case. Let\u0027s take an example."},{"Start":"04:37.790 ","End":"04:45.770","Text":"Let\u0027s say that P is minus 3,1,"},{"Start":"04:45.770 ","End":"04:51.920","Text":"and that Q will be, say, 2,5."},{"Start":"04:51.920 ","End":"04:54.270","Text":"Let\u0027s see where they are."},{"Start":"04:54.920 ","End":"04:59.085","Text":"I\u0027ll join them with a line segment."},{"Start":"04:59.085 ","End":"05:05.740","Text":"Let\u0027s call this distance from P to Q, d for distance."},{"Start":"05:05.740 ","End":"05:09.785","Text":"The way we go about it is as follows."},{"Start":"05:09.785 ","End":"05:14.490","Text":"We complete this to a right angle triangle."},{"Start":"05:17.200 ","End":"05:19.580","Text":"If I have a right-angle triangle,"},{"Start":"05:19.580 ","End":"05:21.905","Text":"I can use Pythagoras\u0027 theorem."},{"Start":"05:21.905 ","End":"05:25.700","Text":"Now, I know the distance between this, and this."},{"Start":"05:25.700 ","End":"05:26.735","Text":"Let\u0027s give it a letter."},{"Start":"05:26.735 ","End":"05:30.800","Text":"I\u0027ll call it R for the right angle."},{"Start":"05:30.800 ","End":"05:32.825","Text":"So PQR."},{"Start":"05:32.825 ","End":"05:35.000","Text":"Now I know the distance from Q to R,"},{"Start":"05:35.000 ","End":"05:37.640","Text":"because these two points fit the first case."},{"Start":"05:37.640 ","End":"05:39.500","Text":"One is directly above the other."},{"Start":"05:39.500 ","End":"05:42.635","Text":"They have the same x, which is 2."},{"Start":"05:42.635 ","End":"05:44.510","Text":"What I have to do is just,"},{"Start":"05:44.510 ","End":"05:45.920","Text":"like the example here,"},{"Start":"05:45.920 ","End":"05:47.990","Text":"subtract the y values,"},{"Start":"05:47.990 ","End":"05:53.510","Text":"which in this case would be 5."},{"Start":"05:53.510 ","End":"05:57.895","Text":"This one is here, 1."},{"Start":"05:57.895 ","End":"06:00.675","Text":"So 5 minus 1,"},{"Start":"06:00.675 ","End":"06:04.270","Text":"that would make this 4."},{"Start":"06:04.940 ","End":"06:15.030","Text":"Similarly, this distance from here to here is,"},{"Start":"06:15.030 ","End":"06:16.635","Text":"here we have the 2,"},{"Start":"06:16.635 ","End":"06:19.500","Text":"here we have minus 3,"},{"Start":"06:19.500 ","End":"06:22.110","Text":"the distance from minus 3 to 2,"},{"Start":"06:22.110 ","End":"06:23.620","Text":"subtract them either way."},{"Start":"06:23.620 ","End":"06:26.940","Text":"Take the absolute value is 5."},{"Start":"06:27.820 ","End":"06:34.085","Text":"What we have is basically a Pythagoras triangle."},{"Start":"06:34.085 ","End":"06:36.375","Text":"This is 4, this is 5,"},{"Start":"06:36.375 ","End":"06:41.375","Text":"this is d. Just using the formula for Pythagoras,"},{"Start":"06:41.375 ","End":"06:46.010","Text":"what we can say is that d^2 is equal"},{"Start":"06:46.010 ","End":"06:52.580","Text":"to 5^2 plus 4^2,"},{"Start":"06:52.580 ","End":"06:56.805","Text":"and 5^2 plus 4^2 is 25 plus 16."},{"Start":"06:56.805 ","End":"06:59.475","Text":"25 plus 16 is 41."},{"Start":"06:59.475 ","End":"07:04.170","Text":"This distance here, d is going to equal,"},{"Start":"07:04.170 ","End":"07:06.555","Text":"I\u0027ll just write it over here,"},{"Start":"07:06.555 ","End":"07:10.000","Text":"the square root of 41."},{"Start":"07:11.780 ","End":"07:14.425","Text":"That\u0027s the answer in this case."},{"Start":"07:14.425 ","End":"07:17.650","Text":"All I want to do now is just generalize it."},{"Start":"07:17.650 ","End":"07:19.240","Text":"I have a formula,"},{"Start":"07:19.240 ","End":"07:22.100","Text":"and then I don\u0027t need all these pictures."},{"Start":"07:22.460 ","End":"07:27.565","Text":"Low and behold, the distance formula."},{"Start":"07:27.565 ","End":"07:30.655","Text":"I\u0027m just presenting it like that."},{"Start":"07:30.655 ","End":"07:35.920","Text":"The logic behind getting this formula is the same as we did here."},{"Start":"07:35.920 ","End":"07:39.670","Text":"Basically, what it says is that if we have two points,"},{"Start":"07:39.670 ","End":"07:45.490","Text":"let\u0027s say in our case we have x_1 and y_1,"},{"Start":"07:45.490 ","End":"07:48.850","Text":"and this would be x_2 and y_2,"},{"Start":"07:48.850 ","End":"07:50.995","Text":"we just mimic the logic here."},{"Start":"07:50.995 ","End":"07:53.740","Text":"We take the difference in the x\u0027s,"},{"Start":"07:53.740 ","End":"07:57.580","Text":"and we square it just like we did here,"},{"Start":"07:57.580 ","End":"08:00.250","Text":"and then we took a difference in the y\u0027s and square"},{"Start":"08:00.250 ","End":"08:04.030","Text":"it and take the square root like we did here."},{"Start":"08:04.030 ","End":"08:05.770","Text":"We took this squared plus this squared,"},{"Start":"08:05.770 ","End":"08:07.795","Text":"and finally we took the square root."},{"Start":"08:07.795 ","End":"08:12.325","Text":"Let\u0027s just see that it works out in our particular case using the formula."},{"Start":"08:12.325 ","End":"08:14.530","Text":"Let\u0027s see what we would get."},{"Start":"08:14.530 ","End":"08:21.370","Text":"D is equal to the square root of x_2 is 2,"},{"Start":"08:21.370 ","End":"08:24.040","Text":"x_1 is minus 3,"},{"Start":"08:24.040 ","End":"08:27.900","Text":"and all of this squared,"},{"Start":"08:27.900 ","End":"08:31.740","Text":"plus y_2 is 5,"},{"Start":"08:31.740 ","End":"08:34.655","Text":"less y_1 is 1,"},{"Start":"08:34.655 ","End":"08:40.000","Text":"all this squared, which is equal to the square root of 2 minus,"},{"Start":"08:40.000 ","End":"08:43.675","Text":"minus 3 is 5^2 is 25,"},{"Start":"08:43.675 ","End":"08:48.115","Text":"5 minus 1 is 4^2 is 16,"},{"Start":"08:48.115 ","End":"08:52.855","Text":"which is equal to the square root of 41."},{"Start":"08:52.855 ","End":"08:57.250","Text":"What we have here is the distance formula"},{"Start":"08:57.250 ","End":"09:02.540","Text":"and I\u0027ll just highlight it because it\u0027s very important."},{"Start":"09:05.600 ","End":"09:07.890","Text":"It works in this case."},{"Start":"09:07.890 ","End":"09:09.460","Text":"I\u0027ll tell you something more."},{"Start":"09:09.460 ","End":"09:12.610","Text":"Even in the case of when one point is directly above,"},{"Start":"09:12.610 ","End":"09:14.106","Text":"or below, or to the left,"},{"Start":"09:14.106 ","End":"09:16.255","Text":"or to the right, it still works."},{"Start":"09:16.255 ","End":"09:18.940","Text":"In fact, you can use this for any two points."},{"Start":"09:18.940 ","End":"09:23.135","Text":"Even in that peculiar case I mentioned where the two points are the same,"},{"Start":"09:23.135 ","End":"09:24.815","Text":"x_2 is equal to x_1,"},{"Start":"09:24.815 ","End":"09:26.450","Text":"and y_2 is equal to y_1,"},{"Start":"09:26.450 ","End":"09:28.700","Text":"we get square root of 0 is 0."},{"Start":"09:28.700 ","End":"09:30.365","Text":"It works for all cases."},{"Start":"09:30.365 ","End":"09:34.500","Text":"This is an absolute general formula for two points in the plane."},{"Start":"09:34.500 ","End":"09:37.620","Text":"That\u0027s it for this clip."}],"ID":6623},{"Watched":false,"Name":"The Equation of a Circle","Duration":"12m 5s","ChapterTopicVideoID":6557,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6557.jpeg","UploadDate":"2020-10-06T08:32:03.4370000","DurationForVideoObject":"PT12M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In this clip, we\u0027ll be talking about"},{"Start":"00:02.490 ","End":"00:07.515","Text":"The Equation of a Circle from the point of view of analytical geometry."},{"Start":"00:07.515 ","End":"00:10.350","Text":"Here it is, I\u0027ll read it."},{"Start":"00:10.350 ","End":"00:15.210","Text":"It says the locus of points on the plane that are a fixed distance from a center."},{"Start":"00:15.210 ","End":"00:21.150","Text":"Now the word locus is probably unfamiliar to most of you so let me rephrase that."},{"Start":"00:21.150 ","End":"00:24.450","Text":"Instead of locus, locus in geometry it just means the set"},{"Start":"00:24.450 ","End":"00:28.665","Text":"of points that satisfy something so let me rewrite that."},{"Start":"00:28.665 ","End":"00:34.410","Text":"That\u0027s better, and use the expression set of all points on the plane."},{"Start":"00:34.410 ","End":"00:37.340","Text":"There\u0027s a fixed distance from the center,"},{"Start":"00:37.340 ","End":"00:38.360","Text":"which is actually a point."},{"Start":"00:38.360 ","End":"00:39.995","Text":"I should have said that."},{"Start":"00:39.995 ","End":"00:43.835","Text":"Here\u0027s a diagram to illustrate this."},{"Start":"00:43.835 ","End":"00:46.660","Text":"Also, I mark the center as sometimes called."},{"Start":"00:46.660 ","End":"00:50.310","Text":"Here is a fixed point, and there it is."},{"Start":"00:50.310 ","End":"00:51.530","Text":"It\u0027s called the center,"},{"Start":"00:51.530 ","End":"00:53.705","Text":"but it\u0027s just some fixed point."},{"Start":"00:53.705 ","End":"01:01.055","Text":"Now, a circle is determined by 2 things but it\u0027s really 3 numbers."},{"Start":"01:01.055 ","End":"01:02.979","Text":"We need to know the center,"},{"Start":"01:02.979 ","End":"01:05.163","Text":"and we need to know the distance,"},{"Start":"01:05.163 ","End":"01:07.495","Text":"and that determines everything."},{"Start":"01:07.495 ","End":"01:11.132","Text":"The center is a point so it\u0027s a pair of numbers,"},{"Start":"01:11.132 ","End":"01:16.465","Text":"and let\u0027s use the pair of numbers a, b."},{"Start":"01:16.465 ","End":"01:20.479","Text":"In other books or texts,"},{"Start":"01:20.479 ","End":"01:22.400","Text":"it\u0027s sometimes called h,"},{"Start":"01:22.400 ","End":"01:26.075","Text":"k, and any other combination you like."},{"Start":"01:26.075 ","End":"01:28.960","Text":"The distance is,"},{"Start":"01:28.960 ","End":"01:30.920","Text":"I sometimes use the letter d,"},{"Start":"01:30.920 ","End":"01:33.650","Text":"sometimes the letter r for radius."},{"Start":"01:33.650 ","End":"01:41.365","Text":"I\u0027m going to go with r because that fixed distance is the radius of the circle."},{"Start":"01:41.365 ","End":"01:44.220","Text":"The 3 numbers we need to know a,"},{"Start":"01:44.220 ","End":"01:45.420","Text":"b and r,"},{"Start":"01:45.420 ","End":"01:46.910","Text":"a point in a distance."},{"Start":"01:46.910 ","End":"01:51.305","Text":"In this case, this would be the point a, b."},{"Start":"01:51.305 ","End":"01:53.735","Text":"In other words, if I go down here,"},{"Start":"01:53.735 ","End":"01:55.388","Text":"this is the point a,"},{"Start":"01:55.388 ","End":"01:56.790","Text":"and I cross here,"},{"Start":"01:56.790 ","End":"01:59.010","Text":"this is the point b,"},{"Start":"01:59.010 ","End":"02:02.735","Text":"and the radius is the distance from the center"},{"Start":"02:02.735 ","End":"02:08.705","Text":"to any and every point on the circumference,"},{"Start":"02:08.705 ","End":"02:14.210","Text":"and this distance is r. This would be,"},{"Start":"02:14.210 ","End":"02:16.210","Text":"let\u0027s just mark it a typical point x,"},{"Start":"02:16.210 ","End":"02:18.265","Text":"y though it\u0027s varying."},{"Start":"02:18.265 ","End":"02:20.320","Text":"All such x, y,"},{"Start":"02:20.320 ","End":"02:23.890","Text":"that are of distance r from ab,"},{"Start":"02:23.890 ","End":"02:27.530","Text":"together they comprise the circle."},{"Start":"02:27.530 ","End":"02:30.939","Text":"Let me explain further."},{"Start":"02:30.939 ","End":"02:34.370","Text":"I\u0027m going to give you right away the equation"},{"Start":"02:34.370 ","End":"02:37.565","Text":"of the circle and then we\u0027ll talk about what that means."},{"Start":"02:37.565 ","End":"02:40.888","Text":"Here it is, it\u0027s an equation in x,"},{"Start":"02:40.888 ","End":"02:44.930","Text":"and y involving these 3 numbers a, b,"},{"Start":"02:44.930 ","End":"02:46.970","Text":"and r, which any given instance would"},{"Start":"02:46.970 ","End":"02:50.645","Text":"actually be numbers so we have an equation in x and y."},{"Start":"02:50.645 ","End":"02:55.070","Text":"Now, what I\u0027m going to say here actually applies to"},{"Start":"02:55.070 ","End":"03:02.003","Text":"any equation in x and y. I\u0027m going to try,"},{"Start":"03:02.003 ","End":"03:07.175","Text":"and impress on you the greatness of what is happening here in analytical geometry."},{"Start":"03:07.175 ","End":"03:14.105","Text":"Because this branch of mathematics is a bridge between algebra, and geometry."},{"Start":"03:14.105 ","End":"03:16.920","Text":"If you think about it,"},{"Start":"03:17.660 ","End":"03:22.715","Text":"this equation here has got nothing pictorial about it,"},{"Start":"03:22.715 ","End":"03:26.780","Text":"and this belongs to the world of algebra."},{"Start":"03:26.780 ","End":"03:29.840","Text":"On the other hand, if I look at this picture,"},{"Start":"03:29.840 ","End":"03:31.520","Text":"let\u0027s forget about the axis,"},{"Start":"03:31.520 ","End":"03:34.790","Text":"I see this circle and"},{"Start":"03:34.790 ","End":"03:40.055","Text":"this point actually I want to highlight the center because that\u0027s part of it."},{"Start":"03:40.055 ","End":"03:42.260","Text":"It\u0027s purely geometrical."},{"Start":"03:42.260 ","End":"03:43.760","Text":"So here we have algebra,"},{"Start":"03:43.760 ","End":"03:47.020","Text":"here we have geometry."},{"Start":"03:47.020 ","End":"03:50.685","Text":"The person who invented analytical geometry,"},{"Start":"03:50.685 ","End":"03:52.290","Text":"I\u0027ll write his name down."},{"Start":"03:52.290 ","End":"03:53.650","Text":"His name was,"},{"Start":"03:53.650 ","End":"03:57.110","Text":"a French mathematician named Rene Descartes,"},{"Start":"03:57.110 ","End":"03:59.170","Text":"the philosopher guy who said,"},{"Start":"03:59.170 ","End":"04:01.075","Text":"\u0027\u0027I think therefore I am.\u0027\u0027"},{"Start":"04:01.075 ","End":"04:07.715","Text":"He was the one who built the bridge between algebra, and geometry."},{"Start":"04:07.715 ","End":"04:11.665","Text":"On the world of pictures, and shapes,"},{"Start":"04:11.665 ","End":"04:16.655","Text":"he built a bridge to the world of algebra, and equations."},{"Start":"04:16.655 ","End":"04:19.675","Text":"I\u0027m going to explain the connection here,"},{"Start":"04:19.675 ","End":"04:21.635","Text":"but it applies everywhere."},{"Start":"04:21.635 ","End":"04:24.730","Text":"Everywhere we have an equation and a curve and when"},{"Start":"04:24.730 ","End":"04:27.755","Text":"we say that this equation corresponds to this curve,"},{"Start":"04:27.755 ","End":"04:31.550","Text":"what it means is that if x,"},{"Start":"04:31.550 ","End":"04:33.860","Text":"y satisfies this equation,"},{"Start":"04:33.860 ","End":"04:36.050","Text":"then it lies on this curve."},{"Start":"04:36.050 ","End":"04:37.670","Text":"Assuming we know what"},{"Start":"04:37.670 ","End":"04:45.955","Text":"a Cartesian axis system is in over each point has a pair of numbers and vice versa."},{"Start":"04:45.955 ","End":"04:48.410","Text":"Any point I pick on the circle,"},{"Start":"04:48.410 ","End":"04:50.240","Text":"if I plug it in here,"},{"Start":"04:50.240 ","End":"04:52.370","Text":"it will satisfy the equation."},{"Start":"04:52.370 ","End":"04:56.180","Text":"That is the basic tie-in between the algebra,"},{"Start":"04:56.180 ","End":"05:01.190","Text":"and the geometry between the equation and the geometrical aspect of the curve."},{"Start":"05:01.190 ","End":"05:03.950","Text":"But, let\u0027s stick to the circle this is what we\u0027re"},{"Start":"05:03.950 ","End":"05:07.565","Text":"discussing here and let me give an example."},{"Start":"05:07.565 ","End":"05:10.625","Text":"Let me first clean up a bit."},{"Start":"05:10.625 ","End":"05:18.300","Text":"Let\u0027s say in our particular case at the center is at 3, 5."},{"Start":"05:18.650 ","End":"05:25.810","Text":"Which means that here I have 3 and this b is 5."},{"Start":"05:28.010 ","End":"05:36.630","Text":"Changed my mind, let us be a 7 and we\u0027ll take the radius to be 5."},{"Start":"05:36.630 ","End":"05:43.255","Text":"Just a point on notation, Europeans,"},{"Start":"05:43.255 ","End":"05:52.665","Text":"and others will write a 7 as this with a little crossbar here."},{"Start":"05:52.665 ","End":"05:56.815","Text":"I use this form of a 7 so if you haven\u0027t seen it,"},{"Start":"05:56.815 ","End":"06:00.085","Text":"don\u0027t be wondering what digit this is."},{"Start":"06:00.085 ","End":"06:04.160","Text":"I never mentioned it before but this is time to mention,"},{"Start":"06:04.160 ","End":"06:06.720","Text":"that\u0027s how I write my 7 \u0027s."},{"Start":"06:08.470 ","End":"06:13.430","Text":"Let\u0027s see what this equation becomes in our particular case,"},{"Start":"06:13.430 ","End":"06:20.195","Text":"where this a is 3 so we have x"},{"Start":"06:20.195 ","End":"06:28.580","Text":"minus 3^2 and b is 7 so y minus 7^2,"},{"Start":"06:28.580 ","End":"06:31.025","Text":"and r is 5."},{"Start":"06:31.025 ","End":"06:38.505","Text":"R^2 is 5^2 and we can all do this in our heads is 25,"},{"Start":"06:38.505 ","End":"06:44.950","Text":"and 7 is redundant at this point."},{"Start":"06:45.140 ","End":"06:47.440","Text":"Just to give myself some space,"},{"Start":"06:47.440 ","End":"06:50.260","Text":"I\u0027ll move this bit up."},{"Start":"06:50.260 ","End":"06:55.670","Text":"This is the equation of our particular circle."},{"Start":"06:55.940 ","End":"06:59.590","Text":"I decided that I\u0027m actually not going to bring you"},{"Start":"06:59.590 ","End":"07:04.885","Text":"the proof right now of why this is the equation of a circle."},{"Start":"07:04.885 ","End":"07:09.815","Text":"I\u0027ll save it to the end of the clip because many of you don\u0027t care,"},{"Start":"07:09.815 ","End":"07:14.890","Text":"admit it honestly as to how got to this equation,"},{"Start":"07:14.890 ","End":"07:19.240","Text":"but I have to show you so I\u0027ll save it to the end."},{"Start":"07:19.240 ","End":"07:25.450","Text":"Now, let me again emphasize what is the correlation,"},{"Start":"07:25.450 ","End":"07:30.455","Text":"the connection between this algebra and this geometry."},{"Start":"07:30.455 ","End":"07:34.790","Text":"On the one hand, we have some algebraic equation and on the other hand,"},{"Start":"07:34.790 ","End":"07:42.465","Text":"we have a geometry of a circle with some center and with the radius,"},{"Start":"07:42.465 ","End":"07:44.790","Text":"so-and-so, and let\u0027s see."},{"Start":"07:44.790 ","End":"07:51.184","Text":"What I\u0027d like to do is choose a point and I won\u0027t tell you if it\u0027s on or off the circle."},{"Start":"07:51.184 ","End":"07:52.775","Text":"That\u0027s our game to decide,"},{"Start":"07:52.775 ","End":"07:55.760","Text":"is it or is it not on the circle?"},{"Start":"07:55.760 ","End":"08:05.560","Text":"Let\u0027s take the point 6,11 as our test point."},{"Start":"08:05.560 ","End":"08:07.455","Text":"I don\u0027t know whether it is,"},{"Start":"08:07.455 ","End":"08:13.880","Text":"I could guess but without drawing anything just using the equation I want to know,"},{"Start":"08:13.880 ","End":"08:15.560","Text":"is it on the circle?"},{"Start":"08:15.560 ","End":"08:18.840","Text":"That\u0027s my question. Is it on the circle?"},{"Start":"08:20.690 ","End":"08:25.440","Text":"I\u0027m also going to write now an x another one will do 2 problems already,"},{"Start":"08:25.440 ","End":"08:28.485","Text":"and the question is also for say,"},{"Start":"08:28.485 ","End":"08:30.925","Text":"2,1, is that on a circle?"},{"Start":"08:30.925 ","End":"08:33.050","Text":"Because from an educational point of view,"},{"Start":"08:33.050 ","End":"08:36.020","Text":"I rigged it so that one of them will turn out to be yes,"},{"Start":"08:36.020 ","End":"08:38.705","Text":"and one of them will turn out to be no."},{"Start":"08:38.705 ","End":"08:42.680","Text":"Let\u0027s see how we do this computationally."},{"Start":"08:42.680 ","End":"08:46.730","Text":"When I ask these questions as to whether these points lie"},{"Start":"08:46.730 ","End":"08:50.165","Text":"on the circle or not I want to use this, and not this."},{"Start":"08:50.165 ","End":"08:55.430","Text":"In other words, I don\u0027t want you to start plotting 6,11 and then trying to"},{"Start":"08:55.430 ","End":"09:01.130","Text":"see whether microscope whether it exactly falls on the circle or not."},{"Start":"09:01.130 ","End":"09:05.400","Text":"I mean, to use the algebra that\u0027s what I mean."},{"Start":"09:05.400 ","End":"09:06.870","Text":"Let\u0027s take them one at a time,"},{"Start":"09:06.870 ","End":"09:08.955","Text":"let\u0027s take the 6,11."},{"Start":"09:08.955 ","End":"09:13.200","Text":"What we do is plug in x is 6,"},{"Start":"09:13.200 ","End":"09:14.318","Text":"y is 11,"},{"Start":"09:14.318 ","End":"09:19.440","Text":"and we get 6 minus 3^2"},{"Start":"09:19.440 ","End":"09:25.919","Text":"plus 11"},{"Start":"09:25.919 ","End":"09:31.710","Text":"minus 7^2 equals,"},{"Start":"09:31.710 ","End":"09:37.250","Text":"and I like to write equals question mark because that\u0027s what I\u0027m checking."},{"Start":"09:37.250 ","End":"09:39.635","Text":"I want to get a yes or no answer."},{"Start":"09:39.635 ","End":"09:41.480","Text":"Let\u0027s just see what sets,"},{"Start":"09:41.480 ","End":"09:42.920","Text":"open up the left-hand side,"},{"Start":"09:42.920 ","End":"09:44.285","Text":"and see what we get."},{"Start":"09:44.285 ","End":"09:49.840","Text":"6 minus 3 is 3 so that\u0027s 3^2,"},{"Start":"09:49.840 ","End":"09:53.960","Text":"11 minus 7 is 4 so that\u0027s 4^2."},{"Start":"09:53.960 ","End":"10:00.915","Text":"Basically what we get here is 9 plus 16,"},{"Start":"10:00.915 ","End":"10:04.810","Text":"which is indeed 25 so I say yes,"},{"Start":"10:04.810 ","End":"10:08.540","Text":"this point is on the circle."},{"Start":"10:08.540 ","End":"10:12.255","Text":"Actually, I should put this here, so yes."},{"Start":"10:12.255 ","End":"10:19.485","Text":"I\u0027ll take that as a yes and let\u0027s try the other one, otherwise this first."},{"Start":"10:19.485 ","End":"10:22.620","Text":"This time with x, y being 2,1,"},{"Start":"10:22.620 ","End":"10:27.150","Text":"we will ask whether 2 minus"},{"Start":"10:27.150 ","End":"10:34.110","Text":"3^2 plus 1 minus 7^2 is that equal to 25?"},{"Start":"10:34.110 ","End":"10:40.725","Text":"Let\u0027s see, 2 minus 3 is 1 so this is 1^2 it\u0027s minus 1,"},{"Start":"10:40.725 ","End":"10:43.490","Text":"the minus because we\u0027re squaring it."},{"Start":"10:43.490 ","End":"10:48.740","Text":"Then 1 minus 7 is 6^2."},{"Start":"10:48.850 ","End":"10:55.185","Text":"Anyway, we get 1 plus 37,"},{"Start":"10:55.185 ","End":"11:00.035","Text":"and does this equal to 25?"},{"Start":"11:00.035 ","End":"11:05.110","Text":"Don\u0027t pull out your calculators the answer is obviously no."},{"Start":"11:05.110 ","End":"11:12.285","Text":"I\u0027ll just write that as a no over here."},{"Start":"11:12.285 ","End":"11:14.955","Text":"This is on the circle, this is not."},{"Start":"11:14.955 ","End":"11:18.830","Text":"If you want to just get an extra geometrical verification,"},{"Start":"11:18.830 ","End":"11:22.560","Text":"let\u0027s see, point 2,1."},{"Start":"11:22.560 ","End":"11:25.200","Text":"If this is 3 this is about 2,1."},{"Start":"11:25.200 ","End":"11:27.600","Text":"This would be roughly the point"},{"Start":"11:27.600 ","End":"11:36.530","Text":"2,1 and let\u0027s see 6,11."},{"Start":"11:36.530 ","End":"11:44.315","Text":"Let\u0027s just say this is a and this is b so here we have a and here we have b."},{"Start":"11:44.315 ","End":"11:46.790","Text":"Clearly, a is on the circle,"},{"Start":"11:46.790 ","End":"11:48.275","Text":"b is not on the circle."},{"Start":"11:48.275 ","End":"11:51.920","Text":"I deliberately chose a negative example to say that it"},{"Start":"11:51.920 ","End":"11:55.910","Text":"works both ways is that the point satisfies the equation,"},{"Start":"11:55.910 ","End":"11:57.880","Text":"it is on the circle."},{"Start":"11:57.880 ","End":"12:00.110","Text":"If it\u0027s not on, it doesn\u0027t satisfy,"},{"Start":"12:00.110 ","End":"12:01.370","Text":"it\u0027s not on the circle."},{"Start":"12:01.370 ","End":"12:03.050","Text":"I think I\u0027ve talked enough,"},{"Start":"12:03.050 ","End":"12:05.190","Text":"so I\u0027ll stop here."}],"ID":6624},{"Watched":false,"Name":"Exercises 1-4","Duration":"7m 44s","ChapterTopicVideoID":5110,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5110.jpeg","UploadDate":"2016-02-29T09:22:00.0400000","DurationForVideoObject":"PT7M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.385","Text":"This exercise has 4 parts,"},{"Start":"00:02.385 ","End":"00:06.525","Text":"all involving a particle moving from A to B in the plane."},{"Start":"00:06.525 ","End":"00:09.455","Text":"In each case we\u0027re given the coordinates of A and B."},{"Start":"00:09.455 ","End":"00:12.560","Text":"What we have to do is find 3 things: Delta x,"},{"Start":"00:12.560 ","End":"00:16.710","Text":"Delta y, and the distance from A to B."},{"Start":"00:16.710 ","End":"00:19.980","Text":"Let\u0027s begin with Part 1."},{"Start":"00:19.980 ","End":"00:24.120","Text":"Let\u0027s draw A and B on the coordinate plane below."},{"Start":"00:24.120 ","End":"00:27.765","Text":"Delta x is the difference of the x\u0027s,"},{"Start":"00:27.765 ","End":"00:29.160","Text":"but the order matters."},{"Start":"00:29.160 ","End":"00:30.750","Text":"We\u0027re going from A to B,"},{"Start":"00:30.750 ","End":"00:34.365","Text":"so let\u0027s draw the line"},{"Start":"00:34.365 ","End":"00:39.670","Text":"and put a little arrow here to show that we\u0027re going in this direction."},{"Start":"00:39.860 ","End":"00:45.360","Text":"Delta x is equal"},{"Start":"00:45.360 ","End":"00:51.240","Text":"to the x(B) minus the x(A) minus 1 takeaway minus 3,"},{"Start":"00:51.240 ","End":"00:53.425","Text":"which is plus 2."},{"Start":"00:53.425 ","End":"00:58.140","Text":"For Delta y, we need the differences in the Ys."},{"Start":"00:58.160 ","End":"01:02.070","Text":"Once again, we\u0027re going from A to B."},{"Start":"01:02.070 ","End":"01:05.490","Text":"We take the y(B) minus the y(A),"},{"Start":"01:05.490 ","End":"01:07.245","Text":"which is 0 minus 4,"},{"Start":"01:07.245 ","End":"01:13.510","Text":"so Delta y is minus 4."},{"Start":"01:13.510 ","End":"01:17.500","Text":"The last thing we have to find is the distance from A to B."},{"Start":"01:17.500 ","End":"01:20.630","Text":"Let\u0027s just join A and B."},{"Start":"01:21.010 ","End":"01:26.955","Text":"There is a formula where d,"},{"Start":"01:26.955 ","End":"01:28.196","Text":"this is the distance d,"},{"Start":"01:28.196 ","End":"01:33.020","Text":"the formula is that d is equal to"},{"Start":"01:33.020 ","End":"01:42.070","Text":"the square root of Delta y^2 plus Delta x^2."},{"Start":"01:42.070 ","End":"01:45.410","Text":"This is essentially an immediate application of"},{"Start":"01:45.410 ","End":"01:49.235","Text":"Pythagoras\u0027 theorem in a right-angle triangle."},{"Start":"01:49.235 ","End":"01:50.885","Text":"In our case,"},{"Start":"01:50.885 ","End":"01:55.115","Text":"we get the square root of"},{"Start":"01:55.115 ","End":"02:01.250","Text":"Delta y^2 is minus 4^2."},{"Start":"02:01.250 ","End":"02:04.520","Text":"Delta x^2 is 2^2."},{"Start":"02:04.520 ","End":"02:11.270","Text":"Finally, this comes out to 16 plus 4 square root of 20."},{"Start":"02:11.270 ","End":"02:13.860","Text":"We\u0027re done with Part 1."},{"Start":"02:14.390 ","End":"02:17.460","Text":"Part 2 is very similar to Part 1,"},{"Start":"02:17.460 ","End":"02:20.310","Text":"just different numbers and I\u0027ve pre-drawn"},{"Start":"02:20.310 ","End":"02:25.475","Text":"the coordinates A and B on the coordinate plane,"},{"Start":"02:25.475 ","End":"02:27.460","Text":"There\u0027s A minus 2,"},{"Start":"02:27.460 ","End":"02:29.210","Text":"2 and B minus 4, 2."},{"Start":"02:29.210 ","End":"02:33.540","Text":"I\u0027ve also drawn the triangle."},{"Start":"02:33.540 ","End":"02:37.775","Text":"All we have to do is say what is Delta x,"},{"Start":"02:37.775 ","End":"02:39.980","Text":"Delta y, and the distance."},{"Start":"02:39.980 ","End":"02:45.240","Text":"Delta x is the difference between the Xs,"},{"Start":"02:45.240 ","End":"02:48.540","Text":"but we have to take the x(B) minus the x(A),"},{"Start":"02:48.540 ","End":"02:50.100","Text":"which is minus 4,"},{"Start":"02:50.100 ","End":"02:51.720","Text":"less minus 2,"},{"Start":"02:51.720 ","End":"02:54.690","Text":"so that\u0027s minus 2."},{"Start":"02:54.690 ","End":"03:01.515","Text":"Delta y is the y(B) minus the y(A),"},{"Start":"03:01.515 ","End":"03:03.675","Text":"2 minus minus 2,"},{"Start":"03:03.675 ","End":"03:06.045","Text":"which is 4,"},{"Start":"03:06.045 ","End":"03:09.440","Text":"and Delta x, Delta y."},{"Start":"03:09.440 ","End":"03:13.190","Text":"Finally, we have to find out what the distances from A to B,"},{"Start":"03:13.190 ","End":"03:22.330","Text":"and we use the formula d equals the square root of Delta y^2 plus Delta x^2,"},{"Start":"03:22.330 ","End":"03:28.160","Text":"which is equal to the square root of minus 2^2."},{"Start":"03:28.160 ","End":"03:32.585","Text":"That\u0027s what Delta x, Delta y is for plus 4^2,"},{"Start":"03:32.585 ","End":"03:38.810","Text":"which comes out to be square root of 20."},{"Start":"03:38.810 ","End":"03:41.730","Text":"We\u0027re done with Part 2."},{"Start":"03:41.810 ","End":"03:46.510","Text":"Exercise 3 is similar to the first two,"},{"Start":"03:46.510 ","End":"03:48.500","Text":"but there is a slight difference,"},{"Start":"03:48.500 ","End":"03:54.335","Text":"as we\u0027ll see when we plot the points on the coordinate plane,"},{"Start":"03:54.335 ","End":"03:58.700","Text":"we see that they run a horizontal line."},{"Start":"03:58.700 ","End":"04:01.790","Text":"The A and the B have the same y-coordinate,"},{"Start":"04:01.790 ","End":"04:04.369","Text":"which means we can\u0027t really do a proper triangle here,"},{"Start":"04:04.369 ","End":"04:13.220","Text":"but we can still compute Delta x and Delta y. Delta x is the x(B) minus"},{"Start":"04:13.220 ","End":"04:22.730","Text":"the x which is minus 4.3 minus the x(A) minus minus 3.2,"},{"Start":"04:22.730 ","End":"04:26.760","Text":"the answer comes out to be minus 1.1."},{"Start":"04:27.310 ","End":"04:33.855","Text":"Delta y is equal to the difference in the y-coordinates,"},{"Start":"04:33.855 ","End":"04:35.740","Text":"the y(B) minus the y(A),"},{"Start":"04:35.740 ","End":"04:37.970","Text":"which is minus 2, less minus 2,"},{"Start":"04:37.970 ","End":"04:40.400","Text":"which is 0, which is not surprising because there"},{"Start":"04:40.400 ","End":"04:43.940","Text":"is no vertical distance between A and B."},{"Start":"04:44.080 ","End":"04:51.130","Text":"Finally, we compute the distance d using the same formula."},{"Start":"04:51.130 ","End":"04:59.720","Text":"We had the d was equal to the square root of Delta y^2 plus Delta x^2,"},{"Start":"04:59.720 ","End":"05:09.190","Text":"which equals the square root of 0^2 plus minus 1.1^2."},{"Start":"05:09.190 ","End":"05:10.850","Text":"The zero gives nothing,"},{"Start":"05:10.850 ","End":"05:13.730","Text":"this gives the square root of something squared is just"},{"Start":"05:13.730 ","End":"05:20.285","Text":"that something absolute value, which is 1.1."},{"Start":"05:20.285 ","End":"05:26.205","Text":"We could\u0027ve seen this right away because the horizontal distance Delta x is minus 1.1."},{"Start":"05:26.205 ","End":"05:30.855","Text":"The distance is 1.1, but stick with the formula."},{"Start":"05:30.855 ","End":"05:33.930","Text":"There\u0027s Delta x, there\u0027s Delta y,"},{"Start":"05:33.930 ","End":"05:36.655","Text":"and there\u0027s the distance, we\u0027re done with Part 3."},{"Start":"05:36.655 ","End":"05:40.355","Text":"Exercise 4 is similar to the previous ones."},{"Start":"05:40.355 ","End":"05:44.720","Text":"We have here a square root of 2 but since we\u0027re only approximating,"},{"Start":"05:44.720 ","End":"05:48.800","Text":"let\u0027s write the square root of 2 as 1.4."},{"Start":"05:48.800 ","End":"05:50.315","Text":"That\u0027s close enough."},{"Start":"05:50.315 ","End":"05:54.230","Text":"We\u0027ll sketch the points on the coordinate plane."},{"Start":"05:54.230 ","End":"05:59.010","Text":"I\u0027ve drawn the points as well as the triangle."},{"Start":"05:59.480 ","End":"06:06.020","Text":"Now Delta x. Delta x"},{"Start":"06:06.020 ","End":"06:12.050","Text":"is equal to the difference in the x\u0027s but we have to take B minus A,"},{"Start":"06:12.050 ","End":"06:14.735","Text":"0 minus square root of 2,"},{"Start":"06:14.735 ","End":"06:20.040","Text":"which is minus the square root of 2."},{"Start":"06:21.280 ","End":"06:31.940","Text":"Delta y. The difference in the y\u0027s but B minus A it\u0027s 2.5 minus 3,"},{"Start":"06:31.940 ","End":"06:36.780","Text":"which is minus 0.5."},{"Start":"06:38.010 ","End":"06:44.440","Text":"The distance d, we can compute from the usual formula."},{"Start":"06:44.440 ","End":"06:49.510","Text":"We have that d equals the square root"},{"Start":"06:49.510 ","End":"06:55.765","Text":"of Delta y^2 plus Delta x^2,"},{"Start":"06:55.765 ","End":"07:00.895","Text":"which equals the square root of Delta y^2 is"},{"Start":"07:00.895 ","End":"07:09.990","Text":"0.25 minus the square root"},{"Start":"07:09.990 ","End":"07:13.355","Text":"of 2^2 is just 2,"},{"Start":"07:13.355 ","End":"07:17.390","Text":"say we didn\u0027t need the approximation after all."},{"Start":"07:18.200 ","End":"07:27.545","Text":"That\u0027s equal to the square root of 2.25."},{"Start":"07:27.545 ","End":"07:35.320","Text":"That comes out to be a nice round number, almost 1.5."},{"Start":"07:35.320 ","End":"07:38.909","Text":"We found Delta x, Delta y,"},{"Start":"07:38.909 ","End":"07:44.380","Text":"and d. We\u0027re done with Part 4 and with the whole exercise."}],"ID":5103},{"Watched":false,"Name":"Exercises 5-8","Duration":"3m 3s","ChapterTopicVideoID":5111,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5111.jpeg","UploadDate":"2016-02-29T09:22:17.1930000","DurationForVideoObject":"PT3M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:06.120","Text":"This exercise has four parts and in each we have to describe the sketch,"},{"Start":"00:06.120 ","End":"00:09.195","Text":"the graph of the given equations."},{"Start":"00:09.195 ","End":"00:13.410","Text":"Let\u0027s start with number 5."},{"Start":"00:13.410 ","End":"00:20.535","Text":"I\u0027d like to remind you that if we have the equation,"},{"Start":"00:20.535 ","End":"00:24.600","Text":"x^2 plus y^2 equals r^2,"},{"Start":"00:24.600 ","End":"00:28.664","Text":"that this is a circle centered at the origin with radius,"},{"Start":"00:28.664 ","End":"00:31.995","Text":"r. In this case,"},{"Start":"00:31.995 ","End":"00:38.595","Text":"number 5 is a circle centered at the origin with radius 2 because 4 is 2^2."},{"Start":"00:38.595 ","End":"00:41.710","Text":"This will look something like the following."},{"Start":"00:41.990 ","End":"00:48.585","Text":"Here, a circle centered at the origin with radius 2;"},{"Start":"00:48.585 ","End":"00:51.345","Text":"2, 2, 2, and 2."},{"Start":"00:51.345 ","End":"00:55.210","Text":"That\u0027s part 5."},{"Start":"00:55.910 ","End":"01:00.075","Text":"Part 6 is exactly like part 5,"},{"Start":"01:00.075 ","End":"01:02.970","Text":"but we have a 9 here instead of a 4."},{"Start":"01:02.970 ","End":"01:07.755","Text":"If 9 is going to be our r^2 we\u0027re going to have r=3."},{"Start":"01:07.755 ","End":"01:12.050","Text":"We need a circle with radius 3 centered at the origin."},{"Start":"01:12.050 ","End":"01:15.303","Text":"This will look something like the following."},{"Start":"01:15.303 ","End":"01:16.490","Text":"Here we are,"},{"Start":"01:16.490 ","End":"01:23.710","Text":"a circle centered around the origin and with radius 3 all around."},{"Start":"01:23.710 ","End":"01:30.960","Text":"That\u0027s it for number 6."},{"Start":"01:30.960 ","End":"01:35.155","Text":"Number 7 looks very much like numbers 5."},{"Start":"01:35.155 ","End":"01:36.830","Text":"But here there is an equals,"},{"Start":"01:36.830 ","End":"01:39.170","Text":"and here this is less than or equals."},{"Start":"01:39.170 ","End":"01:47.815","Text":"In the case of x^2 plus y^2 less than or equal to r^2,"},{"Start":"01:47.815 ","End":"01:49.940","Text":"we don\u0027t get the circle."},{"Start":"01:49.940 ","End":"01:54.830","Text":"We get the whole disc which is a circle together with its interior."},{"Start":"01:54.830 ","End":"02:00.805","Text":"In this case, it will look something like this."},{"Start":"02:00.805 ","End":"02:04.370","Text":"Starting with the same sketch as a number 5,"},{"Start":"02:04.370 ","End":"02:06.725","Text":"because of the less than or equal to."},{"Start":"02:06.725 ","End":"02:12.325","Text":"We\u0027ve got to take the whole interior as well."},{"Start":"02:12.325 ","End":"02:19.370","Text":"Something like this, the whole disc."},{"Start":"02:19.370 ","End":"02:21.740","Text":"That\u0027s it for number 7."},{"Start":"02:21.740 ","End":"02:27.305","Text":"Number 8, if we compare it to x^2 plus y^2=r^2,"},{"Start":"02:27.305 ","End":"02:31.310","Text":"then we get r=0 here."},{"Start":"02:31.310 ","End":"02:39.240","Text":"A circle with radius 0 centered at the origin is just a point."},{"Start":"02:39.790 ","End":"02:42.395","Text":"The point 0,0."},{"Start":"02:42.395 ","End":"02:47.210","Text":"Another way to see this is that there\u0027s only one solution to the equation,"},{"Start":"02:47.210 ","End":"02:49.070","Text":"x^2 plus y^2 is 0."},{"Start":"02:49.070 ","End":"02:51.335","Text":"Both x and y have to be 0,"},{"Start":"02:51.335 ","End":"02:53.480","Text":"otherwise we\u0027ll get something positive."},{"Start":"02:53.480 ","End":"02:56.180","Text":"In either case, it\u0027s a single point,"},{"Start":"02:56.180 ","End":"02:58.925","Text":"just the origin 0,0."},{"Start":"02:58.925 ","End":"03:03.030","Text":"We\u0027re done with 8 and with the whole question."}],"ID":5104},{"Watched":false,"Name":"Exercises 9-12","Duration":"6m 38s","ChapterTopicVideoID":5112,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5112.jpeg","UploadDate":"2016-02-29T09:22:57.7130000","DurationForVideoObject":"PT6M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.535","Text":"This exercise consists of 4 parts,"},{"Start":"00:02.535 ","End":"00:04.095","Text":"9, 10,11 and 12."},{"Start":"00:04.095 ","End":"00:08.475","Text":"In each part we\u0027re given a pair of points, A and B."},{"Start":"00:08.475 ","End":"00:10.830","Text":"What we have to do is 3 things."},{"Start":"00:10.830 ","End":"00:12.735","Text":"First, to sketch the points,"},{"Start":"00:12.735 ","End":"00:17.640","Text":"then to find the slope of the line that they determine and finally,"},{"Start":"00:17.640 ","End":"00:21.020","Text":"to determine the perpendicular slope."},{"Start":"00:21.020 ","End":"00:25.575","Text":"Which means the slope of all perpendicular lines to it."},{"Start":"00:25.575 ","End":"00:29.085","Text":"Let\u0027s write some formulae."},{"Start":"00:29.085 ","End":"00:31.605","Text":"If we have a pair of points,"},{"Start":"00:31.605 ","End":"00:34.350","Text":"A and B, let\u0027s say in general,"},{"Start":"00:34.350 ","End":"00:36.780","Text":"A is the point,"},{"Start":"00:36.780 ","End":"00:39.789","Text":"let\u0027s call it (x_1,"},{"Start":"00:39.789 ","End":"00:45.789","Text":"y_1) and let\u0027s say the point B is (x_2,"},{"Start":"00:45.789 ","End":"00:49.639","Text":"y_2) then the slope,"},{"Start":"00:49.639 ","End":"00:59.065","Text":"we have to give it some letter say a. I\u0027ll write down that this is actually the slope,"},{"Start":"00:59.065 ","End":"01:04.470","Text":"is given by the formula (y_2"},{"Start":"01:04.470 ","End":"01:12.180","Text":"- y_1) over (x_2 - x_1)."},{"Start":"01:12.180 ","End":"01:17.410","Text":"The perpendicular slope, let\u0027s call it, say m,"},{"Start":"01:17.410 ","End":"01:25.150","Text":"I\u0027ll just write in brackets that this is the perpendicular slope,"},{"Start":"01:26.270 ","End":"01:31.665","Text":"is given by minus 1 over a,"},{"Start":"01:31.665 ","End":"01:35.150","Text":"the negative reciprocal of the slope."},{"Start":"01:36.570 ","End":"01:42.350","Text":"Let\u0027s begin with number 9."},{"Start":"01:45.030 ","End":"01:50.345","Text":"First thing to do is to plot the points A and B."},{"Start":"01:50.345 ","End":"01:57.110","Text":"Here I\u0027ve drawn A minus 1,3 and B minus 2,0."},{"Start":"01:57.110 ","End":"02:00.020","Text":"Now let\u0027s say that\u0027s the sketching part."},{"Start":"02:00.020 ","End":"02:04.360","Text":"Now the slope which we called a,"},{"Start":"02:04.360 ","End":"02:08.985","Text":"so a is equal to using this formula,"},{"Start":"02:08.985 ","End":"02:15.930","Text":"difference in the y\u0027s over the difference in the x\u0027s and we start B minus A,"},{"Start":"02:15.930 ","End":"02:19.410","Text":"so it\u0027s 0 minus 3,"},{"Start":"02:19.410 ","End":"02:20.730","Text":"this minus this,"},{"Start":"02:20.730 ","End":"02:23.790","Text":"over this minus this,"},{"Start":"02:23.790 ","End":"02:28.500","Text":"which is -2 -(-1),"},{"Start":"02:28.500 ","End":"02:36.470","Text":"so it\u0027s -3 over -1 and this is equal to 3."},{"Start":"02:36.470 ","End":"02:40.265","Text":"The perpendicular slope, which we called m,"},{"Start":"02:40.265 ","End":"02:43.055","Text":"which is a slope of all perpendicular lines to A,"},{"Start":"02:43.055 ","End":"02:51.075","Text":"B is given by the negative reciprocal minus 1 over 3."},{"Start":"02:51.075 ","End":"02:53.700","Text":"That\u0027s it for part 9."},{"Start":"02:53.700 ","End":"02:56.085","Text":"Next is exercise 10,"},{"Start":"02:56.085 ","End":"03:01.695","Text":"similar to before except that the points have changed A and B,"},{"Start":"03:01.695 ","End":"03:06.630","Text":"we begin by plotting them on the coordinate plane."},{"Start":"03:06.950 ","End":"03:10.980","Text":"This is A and this is B,"},{"Start":"03:10.980 ","End":"03:16.355","Text":"and we have to compute the slope based on this formula."},{"Start":"03:16.355 ","End":"03:25.360","Text":"We call the slope a and it\u0027s equal to to the difference of y in Y,"},{"Start":"03:25.360 ","End":"03:34.830","Text":"the y from B minus the y from A minus 1 minus 2 over the x from B minus the x from A,"},{"Start":"03:34.830 ","End":"03:45.040","Text":"2 minus, minus 2 and that comes out to be minus 3 over 4, minus 3/4."},{"Start":"03:45.040 ","End":"03:52.280","Text":"The perpendicular slope, which means the slope of perpendicular lines to A B,"},{"Start":"03:52.280 ","End":"03:54.710","Text":"is given by the other formula m,"},{"Start":"03:54.710 ","End":"03:57.305","Text":"which is 1 over a."},{"Start":"03:57.305 ","End":"03:59.945","Text":"The negative reciprocal of this."},{"Start":"03:59.945 ","End":"04:04.960","Text":"Just change the minus to plus and reverse numerator and denominator,"},{"Start":"04:04.960 ","End":"04:10.635","Text":"so that gives us 4 over 3 and that\u0027s it for number 10."},{"Start":"04:10.635 ","End":"04:12.390","Text":"Number 11 is next."},{"Start":"04:12.390 ","End":"04:17.015","Text":"As before, we begin by plotting A and B in the coordinate plane."},{"Start":"04:17.015 ","End":"04:22.370","Text":"Here\u0027s our A and here\u0027s our B."},{"Start":"04:22.370 ","End":"04:28.440","Text":"Notice that actually the line joining them is horizontal."},{"Start":"04:28.440 ","End":"04:30.919","Text":"If we compute the slope,"},{"Start":"04:30.919 ","End":"04:36.980","Text":"what we called a, using the same formula as before,"},{"Start":"04:36.980 ","End":"04:42.710","Text":"difference of the y\u0027s over the difference in the x\u0027s,"},{"Start":"04:42.710 ","End":"04:49.350","Text":"which is minus 2, minus 1."},{"Start":"04:49.350 ","End":"04:51.670","Text":"The numerator is 0 in any event,"},{"Start":"04:51.670 ","End":"04:53.890","Text":"so the whole thing comes out to be 0."},{"Start":"04:53.890 ","End":"04:58.360","Text":"This makes sense a slope of 0 because the line joining them is horizontal."},{"Start":"04:58.360 ","End":"05:02.105","Text":"As for the perpendicular slope,"},{"Start":"05:02.105 ","End":"05:04.720","Text":"if we just try using the formula,"},{"Start":"05:04.720 ","End":"05:07.719","Text":"we get minus 1 over 0,"},{"Start":"05:07.719 ","End":"05:11.990","Text":"which is basically undefined."},{"Start":"05:13.140 ","End":"05:19.435","Text":"In fact, this also makes sense because the slope of perpendicular lines,"},{"Start":"05:19.435 ","End":"05:24.505","Text":"they\u0027re all vertical lines and vertical lines have an undefined slope."},{"Start":"05:24.505 ","End":"05:27.280","Text":"There\u0027s A, there\u0027s B,"},{"Start":"05:27.280 ","End":"05:30.800","Text":"there\u0027s the slope and the perpendicular slope is undefined."},{"Start":"05:30.800 ","End":"05:35.085","Text":"End of part 11. Finally, number 12."},{"Start":"05:35.085 ","End":"05:38.300","Text":"We\u0027ll begin by plotting the points A and B."},{"Start":"05:38.300 ","End":"05:40.385","Text":"Here they are plotted."},{"Start":"05:40.385 ","End":"05:45.470","Text":"Notice that the line between them is a vertical line."},{"Start":"05:45.470 ","End":"05:49.010","Text":"Now if we try to compute the slope as before,"},{"Start":"05:49.010 ","End":"05:54.140","Text":"we would say that a is equal to difference in the y\u0027s minus"},{"Start":"05:54.140 ","End":"05:59.675","Text":"1 minus 1 over difference in the x\u0027s minus 2 minus,"},{"Start":"05:59.675 ","End":"06:07.430","Text":"minus 2 and we would get minus 2 over 0 and that is undefined."},{"Start":"06:07.430 ","End":"06:12.020","Text":"Now this makes sense because the line joining A and B is a vertical line,"},{"Start":"06:12.020 ","End":"06:14.885","Text":"and a vertical line is not defined."},{"Start":"06:14.885 ","End":"06:16.550","Text":"In fact, when we see an undefined slope,"},{"Start":"06:16.550 ","End":"06:18.565","Text":"it means a vertical line."},{"Start":"06:18.565 ","End":"06:21.120","Text":"Now, to compute m,"},{"Start":"06:21.120 ","End":"06:25.400","Text":"we can\u0027t say minus 1 over undefined but it"},{"Start":"06:25.400 ","End":"06:30.005","Text":"makes sense to say that if this line is vertical,"},{"Start":"06:30.005 ","End":"06:39.640","Text":"perpendicular slopes will be horizontal and horizontal line has a slope of 0, so m=0."}],"ID":5105},{"Watched":false,"Name":"Exercises 13-16","Duration":"2m 54s","ChapterTopicVideoID":5113,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5113.jpeg","UploadDate":"2016-02-29T09:23:14.3800000","DurationForVideoObject":"PT2M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.045","Text":"In these 4 exercises,"},{"Start":"00:03.045 ","End":"00:05.070","Text":"we\u0027re given a point."},{"Start":"00:05.070 ","End":"00:08.130","Text":"We have to find the equation of the vertical line through"},{"Start":"00:08.130 ","End":"00:11.385","Text":"the point and also of the horizontal line."},{"Start":"00:11.385 ","End":"00:14.650","Text":"Let\u0027s begin with the first one."},{"Start":"00:15.080 ","End":"00:21.585","Text":"I\u0027ll plot the point on the coordinate plane here."},{"Start":"00:21.585 ","End":"00:29.310","Text":"Now I\u0027ll draw horizontal and vertical line through this point. Now let\u0027s see."},{"Start":"00:29.310 ","End":"00:31.170","Text":"The vertical line,"},{"Start":"00:31.170 ","End":"00:34.955","Text":"has the same value of x as this point."},{"Start":"00:34.955 ","End":"00:38.510","Text":"All the x values on the vertical line are the same."},{"Start":"00:38.510 ","End":"00:40.640","Text":"This x is minus 1."},{"Start":"00:40.640 ","End":"00:49.410","Text":"So the equation of the vertical line is just x=-1."},{"Start":"00:49.410 ","End":"00:52.880","Text":"Horizontal line, all the values have the same y,"},{"Start":"00:52.880 ","End":"00:56.315","Text":"which would be the same as the 4 over 3 here."},{"Start":"00:56.315 ","End":"01:01.470","Text":"This horizontal line would have the equation y= 4/3."},{"Start":"01:03.580 ","End":"01:08.840","Text":"Now, notice we didn\u0027t have to actually draw the sketch to do this."},{"Start":"01:08.840 ","End":"01:18.440","Text":"I could have straightaway written over here that the vertical line is x=-1."},{"Start":"01:18.440 ","End":"01:26.520","Text":"That the horizontal line was y=4/3."},{"Start":"01:26.520 ","End":"01:29.760","Text":"Let\u0027s say this is the vertical,"},{"Start":"01:29.760 ","End":"01:32.320","Text":"and this is the horizontal."},{"Start":"01:33.140 ","End":"01:37.090","Text":"In actual fact, continuing this way,"},{"Start":"01:37.090 ","End":"01:40.390","Text":"we can solve them without the graph."},{"Start":"01:40.390 ","End":"01:43.645","Text":"I did one just to show you how easy it is."},{"Start":"01:43.645 ","End":"01:46.560","Text":"So the next one, the vertical,"},{"Start":"01:46.560 ","End":"01:52.795","Text":"where x is equal to the value of x here, will be x=0."},{"Start":"01:52.795 ","End":"01:56.255","Text":"That\u0027s the vertical line,"},{"Start":"01:56.255 ","End":"01:58.964","Text":"and the horizontal line"},{"Start":"01:58.964 ","End":"02:09.805","Text":"is y equals minus the square root of 2."},{"Start":"02:09.805 ","End":"02:12.649","Text":"Because on a horizontal line the values of y are the same,"},{"Start":"02:12.649 ","End":"02:14.975","Text":"on a vertical line, the values of x are the same."},{"Start":"02:14.975 ","End":"02:19.190","Text":"Continuing this way, this point,"},{"Start":"02:19.190 ","End":"02:26.090","Text":"we have that the vertical line is x equals square root of 2,"},{"Start":"02:26.090 ","End":"02:27.290","Text":"that\u0027s the vertical,"},{"Start":"02:27.290 ","End":"02:36.410","Text":"and the horizontal is y=-1.3."},{"Start":"02:36.410 ","End":"02:44.595","Text":"Again, here, x=-Pi is our vertical line,"},{"Start":"02:44.595 ","End":"02:49.940","Text":"and y=0 is our horizontal line."},{"Start":"02:49.940 ","End":"02:53.070","Text":"That\u0027s all there is to it."}],"ID":5106},{"Watched":false,"Name":"Exercises 17-20","Duration":"8m 23s","ChapterTopicVideoID":5114,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5114.jpeg","UploadDate":"2016-02-29T09:23:59.3430000","DurationForVideoObject":"PT8M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.470","Text":"In these 4 exercises,"},{"Start":"00:01.470 ","End":"00:04.325","Text":"we\u0027re given a description of a line,"},{"Start":"00:04.325 ","End":"00:07.870","Text":"and we have to write the equation for the given line."},{"Start":"00:08.210 ","End":"00:11.745","Text":"I\u0027d like to remind you that in general,"},{"Start":"00:11.745 ","End":"00:20.820","Text":"the equation of a line is y=ax plus b."},{"Start":"00:20.820 ","End":"00:23.625","Text":"This is true except for vertical lines,"},{"Start":"00:23.625 ","End":"00:26.325","Text":"which we don\u0027t have here."},{"Start":"00:26.325 ","End":"00:33.450","Text":"Y=ax plus b, and a is significant that it\u0027s the slope of the line."},{"Start":"00:33.450 ","End":"00:35.220","Text":"B also has a significance,"},{"Start":"00:35.220 ","End":"00:39.315","Text":"it\u0027s the y-intercept, but I don\u0027t know that we\u0027ll be needing it here."},{"Start":"00:39.315 ","End":"00:44.590","Text":"Now let\u0027s get onto number 17."},{"Start":"00:46.460 ","End":"00:49.520","Text":"I scroll down, and copy the exercise here."},{"Start":"00:49.520 ","End":"00:53.210","Text":"We want the line that passes through minus 1,"},{"Start":"00:53.210 ","End":"00:56.750","Text":"and whose slope is minus 4."},{"Start":"00:56.750 ","End":"01:05.540","Text":"In general, we can write it as y = ax plus b only we know the I already is minus 4,"},{"Start":"01:05.540 ","End":"01:09.215","Text":"so y=minus 4x plus b."},{"Start":"01:09.215 ","End":"01:11.525","Text":"That takes care of the slope condition."},{"Start":"01:11.525 ","End":"01:14.765","Text":"Now we have to make sure it passes through minus 1, 1."},{"Start":"01:14.765 ","End":"01:17.803","Text":"If we substitute x=minus 1,"},{"Start":"01:17.803 ","End":"01:20.720","Text":"and y=minus 1, it should work."},{"Start":"01:20.720 ","End":"01:29.150","Text":"Minus 1 is equal to minus 4 times minus 1 plus b."},{"Start":"01:29.150 ","End":"01:31.820","Text":"If we solve for b, this is minus 1,"},{"Start":"01:31.820 ","End":"01:33.950","Text":"this is plus 4."},{"Start":"01:33.950 ","End":"01:37.445","Text":"If we bring it to the other side, it\u0027s minus 5."},{"Start":"01:37.445 ","End":"01:42.580","Text":"We get that p is equal to minus 5."},{"Start":"01:42.580 ","End":"01:47.585","Text":"Substituting a and b into the equation,"},{"Start":"01:47.585 ","End":"01:49.625","Text":"we get the answer is,"},{"Start":"01:49.625 ","End":"01:57.110","Text":"that y=minus 4x plus 5."},{"Start":"01:57.110 ","End":"02:02.795","Text":"That\u0027s the answer to 17. I wrote plus I should have written minus."},{"Start":"02:02.795 ","End":"02:04.190","Text":"There, that\u0027s fixed it."},{"Start":"02:04.190 ","End":"02:05.884","Text":"Next is number 18,"},{"Start":"02:05.884 ","End":"02:08.750","Text":"very similar to 17, just different numbers."},{"Start":"02:08.750 ","End":"02:12.335","Text":"As before, I\u0027ll copy the information below."},{"Start":"02:12.335 ","End":"02:18.530","Text":"We want the line that passes through 2 minus 2 with slope 1.5."},{"Start":"02:18.530 ","End":"02:23.000","Text":"Once again, we\u0027ll substitute a in here and"},{"Start":"02:23.000 ","End":"02:30.379","Text":"get y=1.5 x plus b."},{"Start":"02:30.379 ","End":"02:32.480","Text":"That takes care of the slope."},{"Start":"02:32.480 ","End":"02:37.310","Text":"Now it has to pass through 2 minus 2 so we substitute this for x, and y."},{"Start":"02:37.310 ","End":"02:46.140","Text":"Minus 2=1.5 times 2 plus b."},{"Start":"02:46.580 ","End":"02:50.460","Text":"Half times 2 is 1 subtracting,"},{"Start":"02:50.460 ","End":"02:51.945","Text":"we get minus 3,"},{"Start":"02:51.945 ","End":"02:56.055","Text":"so b is equal to minus 3."},{"Start":"02:56.055 ","End":"02:59.810","Text":"Finally, we put the values of a and b into the equation,"},{"Start":"02:59.810 ","End":"03:07.603","Text":"and we get y=1.5x,"},{"Start":"03:07.603 ","End":"03:11.210","Text":"and b is minus 3."},{"Start":"03:11.210 ","End":"03:13.910","Text":"That\u0027s the answer for 18."},{"Start":"03:13.910 ","End":"03:18.157","Text":"Now exercise 19 is a bit different from the 2 previous ones."},{"Start":"03:18.157 ","End":"03:20.450","Text":"Here we had a point and a slope,"},{"Start":"03:20.450 ","End":"03:23.510","Text":"but in 19 we\u0027re given 2 points."},{"Start":"03:23.510 ","End":"03:26.450","Text":"We\u0027re going to need another formula."},{"Start":"03:26.450 ","End":"03:30.230","Text":"What we need is a formula that tells us the slope of"},{"Start":"03:30.230 ","End":"03:34.145","Text":"the line passing through 2 given points."},{"Start":"03:34.145 ","End":"03:37.820","Text":"Let\u0027s say that the points are,"},{"Start":"03:37.820 ","End":"03:41.400","Text":"the first point is x1y1,"},{"Start":"03:41.830 ","End":"03:51.630","Text":"and the second point is x2y2."},{"Start":"03:51.630 ","End":"03:53.430","Text":"When we have these 2 points,"},{"Start":"03:53.430 ","End":"03:57.490","Text":"that will give us that the slope, call it a,"},{"Start":"03:57.490 ","End":"04:00.965","Text":"between them is equal to the difference of the y\u0027s,"},{"Start":"04:00.965 ","End":"04:08.210","Text":"y2 minus 1 over the difference in the x\u0027s, x2 minus x1."},{"Start":"04:08.210 ","End":"04:14.190","Text":"This is the formula which will come in handy in number 19 and also in number 20."},{"Start":"04:14.570 ","End":"04:22.350","Text":"In our case, these are the 2 points and I\u0027ll copy them below so I will have more space."},{"Start":"04:22.600 ","End":"04:25.760","Text":"We need the line that passes through 3,"},{"Start":"04:25.760 ","End":"04:28.430","Text":"4 and minus 2, 8."},{"Start":"04:28.430 ","End":"04:30.815","Text":"Using the formula I just gave you,"},{"Start":"04:30.815 ","End":"04:33.710","Text":"we can find what a is,"},{"Start":"04:33.710 ","End":"04:41.795","Text":"which is 8 minus 4 over this minus this of a minus 2,"},{"Start":"04:41.795 ","End":"04:46.580","Text":"minus 3, which is 4 over minus 5,"},{"Start":"04:46.580 ","End":"04:51.170","Text":"or minus 4 over 5."},{"Start":"04:51.170 ","End":"04:55.925","Text":"Now we can proceed as in the exercises before,"},{"Start":"04:55.925 ","End":"04:58.750","Text":"and choose 1 of the points."},{"Start":"04:58.750 ","End":"05:01.995","Text":"Say we\u0027ll go with 3, 4."},{"Start":"05:01.995 ","End":"05:10.400","Text":"The line, is y=minus 4 fifths x,"},{"Start":"05:10.400 ","End":"05:13.325","Text":"which is, this is our a, plus b."},{"Start":"05:13.325 ","End":"05:15.170","Text":"When we substitute our 3,"},{"Start":"05:15.170 ","End":"05:18.140","Text":"4, it should fit,"},{"Start":"05:18.140 ","End":"05:27.170","Text":"4=minus 4 fifths of x,"},{"Start":"05:27.170 ","End":"05:33.090","Text":"which is 3, plus b."},{"Start":"05:33.090 ","End":"05:36.975","Text":"Let\u0027s see. A quick mental computation,"},{"Start":"05:36.975 ","End":"05:38.700","Text":"4 times 3 is 12,"},{"Start":"05:38.700 ","End":"05:44.895","Text":"12 over 5 is 2, and the 2-fifths."},{"Start":"05:44.895 ","End":"05:51.345","Text":"We get that b=4 plus 2 and 2-fifths,"},{"Start":"05:51.345 ","End":"05:56.760","Text":"which is 6 and 2 fifths."},{"Start":"05:56.760 ","End":"06:05.160","Text":"All in all, we get the equation from the y=ax plus b,"},{"Start":"06:05.160 ","End":"06:14.115","Text":"that y = minus 4 fifths x,"},{"Start":"06:14.115 ","End":"06:20.085","Text":"plus 6 and 2 fifths."},{"Start":"06:20.085 ","End":"06:23.290","Text":"That\u0027s it for number 19."},{"Start":"06:23.290 ","End":"06:26.750","Text":"Number 20 is the same idea as number 19,"},{"Start":"06:26.750 ","End":"06:29.465","Text":"just different numbers at different points."},{"Start":"06:29.465 ","End":"06:32.749","Text":"I\u0027ll copy them down below where we have more space."},{"Start":"06:32.749 ","End":"06:36.560","Text":"Here are the points minus 8,0, minus 1,3."},{"Start":"06:36.560 ","End":"06:38.450","Text":"We want the line that goes through them."},{"Start":"06:38.450 ","End":"06:42.275","Text":"As before, we\u0027ll compute the slope first with this formula,"},{"Start":"06:42.275 ","End":"06:45.220","Text":"and we\u0027ll get that the slope a,"},{"Start":"06:45.220 ","End":"06:50.615","Text":"is equal to the difference in the y\u0027s 3 minus"},{"Start":"06:50.615 ","End":"06:57.215","Text":"0 over difference in the x\u0027s minus 1, minus minus 8."},{"Start":"06:57.215 ","End":"07:02.120","Text":"This gives us 3 over 7."},{"Start":"07:02.120 ","End":"07:05.656","Text":"Now we plug the value of a into here,"},{"Start":"07:05.656 ","End":"07:16.260","Text":"and we get the equation is of the form y=3-sevenths, x plus b."},{"Start":"07:16.260 ","End":"07:17.920","Text":"The only thing missing is b,"},{"Start":"07:17.920 ","End":"07:20.510","Text":"but we do know that the line goes through these points."},{"Start":"07:20.510 ","End":"07:23.195","Text":"Let\u0027s pick 1 of them, say minus 80,"},{"Start":"07:23.195 ","End":"07:27.335","Text":"and substitute 0 for y minus 8 for x,"},{"Start":"07:27.335 ","End":"07:37.050","Text":"we get 0=3 sevenths of minus 8 plus b."},{"Start":"07:37.050 ","End":"07:41.490","Text":"Let\u0027s see if we can compute this."},{"Start":"07:41.490 ","End":"07:46.440","Text":"3-sevenths times minus 8 is minus 24 over 7."},{"Start":"07:46.440 ","End":"07:49.380","Text":"B is equal to 24 over 7."},{"Start":"07:49.380 ","End":"07:57.675","Text":"B=24 over 7 or 3, and 3-sevenths."},{"Start":"07:57.675 ","End":"08:00.720","Text":"Now that we have both a and b,"},{"Start":"08:00.720 ","End":"08:04.895","Text":"we can put them in the y=ax plus b equation and get,"},{"Start":"08:04.895 ","End":"08:11.755","Text":"our answer is y=3-sevenths"},{"Start":"08:11.755 ","End":"08:18.920","Text":"of x plus 24 sevenths."},{"Start":"08:18.920 ","End":"08:21.960","Text":"That\u0027s the answer for number 20."}],"ID":5107},{"Watched":false,"Name":"Exercises 21-24","Duration":"2m 50s","ChapterTopicVideoID":5115,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5115.jpeg","UploadDate":"2016-02-29T09:24:14.5730000","DurationForVideoObject":"PT2M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.500","Text":"In these 4 exercises,"},{"Start":"00:01.500 ","End":"00:04.920","Text":"we\u0027re given the description of a line and we"},{"Start":"00:04.920 ","End":"00:08.835","Text":"have to write the equation of the corresponding line."},{"Start":"00:08.835 ","End":"00:11.190","Text":"In number 21 and 22,"},{"Start":"00:11.190 ","End":"00:13.350","Text":"we\u0027re given a slope and an intercept."},{"Start":"00:13.350 ","End":"00:20.680","Text":"I\u0027d like to remind you that when we write a line in the form y equals ax plus b,"},{"Start":"00:22.820 ","End":"00:25.663","Text":"a and b have a particular meaning;"},{"Start":"00:25.663 ","End":"00:34.570","Text":"a is the slope and b is the y-intercept."},{"Start":"00:39.920 ","End":"00:42.945","Text":"This will help us with the first 2."},{"Start":"00:42.945 ","End":"00:47.085","Text":"Let\u0027s go to 21,"},{"Start":"00:47.085 ","End":"00:52.325","Text":"and I\u0027ll just repeat it down below."},{"Start":"00:52.325 ","End":"00:58.580","Text":"What we want is a line with slope minus 2 and y-intercept 6."},{"Start":"00:58.580 ","End":"01:01.550","Text":"Just plug it straight in;"},{"Start":"01:01.550 ","End":"01:03.905","Text":"a is the slope, b is the y-intercept."},{"Start":"01:03.905 ","End":"01:13.640","Text":"The answer is that y is equal to minus 2x plus 6,"},{"Start":"01:13.640 ","End":"01:16.140","Text":"and that\u0027s it for 21."},{"Start":"01:16.520 ","End":"01:19.770","Text":"Let\u0027s move right on to number 22."},{"Start":"01:19.770 ","End":"01:21.170","Text":"It\u0027s so similar."},{"Start":"01:21.170 ","End":"01:23.135","Text":"It\u0027s the same as the above,"},{"Start":"01:23.135 ","End":"01:25.040","Text":"except that here,"},{"Start":"01:25.040 ","End":"01:28.400","Text":"the slope is 1/2 and"},{"Start":"01:28.400 ","End":"01:36.520","Text":"the y-intercept is minus 4."},{"Start":"01:36.520 ","End":"01:42.530","Text":"Once again, we just substitute slope and intercept for a and b and we"},{"Start":"01:42.530 ","End":"01:49.350","Text":"get y equals 1/2x minus 4."},{"Start":"01:49.350 ","End":"01:53.000","Text":"That\u0027s it for 22. In exercise 23,"},{"Start":"01:53.000 ","End":"02:00.860","Text":"we need to remember that slope 0 is an indication of a horizontal line."},{"Start":"02:00.860 ","End":"02:03.575","Text":"Basically, if we rephrase this,"},{"Start":"02:03.575 ","End":"02:08.380","Text":"we want a line that passes through this point and is horizontal."},{"Start":"02:08.380 ","End":"02:12.020","Text":"Horizontal means it has the same y-value everywhere."},{"Start":"02:12.020 ","End":"02:16.010","Text":"This is what we take from minus 4 so the answer is"},{"Start":"02:16.010 ","End":"02:22.695","Text":"simply y equals minus 4 and that\u0027s all there is to it."},{"Start":"02:22.695 ","End":"02:28.895","Text":"In exercise 24, we have to remember that if a line has no slope,"},{"Start":"02:28.895 ","End":"02:31.550","Text":"that means it\u0027s vertical."},{"Start":"02:31.550 ","End":"02:36.139","Text":"What we\u0027re looking for is a vertical line through this point."},{"Start":"02:36.139 ","End":"02:38.270","Text":"Now, on a vertical line,"},{"Start":"02:38.270 ","End":"02:40.190","Text":"the x values remain the same,"},{"Start":"02:40.190 ","End":"02:46.640","Text":"so the x value is got to be 1/4 and the answer is x equals 1/4."},{"Start":"02:46.640 ","End":"02:51.090","Text":"It\u0027s as simple as that. We\u0027re done."}],"ID":5108},{"Watched":false,"Name":"Exercises 25-26","Duration":"3m 23s","ChapterTopicVideoID":5116,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5116.jpeg","UploadDate":"2016-02-29T09:24:32.6000000","DurationForVideoObject":"PT3M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.630","Text":"In Exercise 25, we need to find the equation for the line which has y-intercept 5,"},{"Start":"00:06.630 ","End":"00:10.095","Text":"and x-intercept minus 2."},{"Start":"00:10.095 ","End":"00:17.445","Text":"Let\u0027s remember that the general equation of a line is y equals"},{"Start":"00:17.445 ","End":"00:25.710","Text":"ax plus b and b has a significance that it\u0027s the y-intercept."},{"Start":"00:25.710 ","End":"00:28.995","Text":"A is the slope but we\u0027re not going to use that here."},{"Start":"00:28.995 ","End":"00:31.470","Text":"Now in our case,"},{"Start":"00:31.470 ","End":"00:35.235","Text":"we\u0027re given the y-intercept which is 5, and that\u0027s the b,"},{"Start":"00:35.235 ","End":"00:42.880","Text":"so already we know that the line is of the form y equals ax plus 5."},{"Start":"00:42.880 ","End":"00:46.820","Text":"Now we need some other piece of information to help us find"},{"Start":"00:46.820 ","End":"00:52.170","Text":"a and that is that the x-intercept is minus 2."},{"Start":"00:52.170 ","End":"00:58.535","Text":"Now if the x-intercept is minus 2 which means where the graph crosses the x-axis,"},{"Start":"00:58.535 ","End":"01:00.290","Text":"that\u0027s where y equals 0,"},{"Start":"01:00.290 ","End":"01:03.735","Text":"so we know it passes through the point,"},{"Start":"01:03.735 ","End":"01:07.515","Text":"the x-intercept minus 2 means minus 2,"},{"Start":"01:07.515 ","End":"01:11.760","Text":"0 is on the line,"},{"Start":"01:11.760 ","End":"01:13.575","Text":"so I can substitute,"},{"Start":"01:13.575 ","End":"01:15.435","Text":"I will just repeat that here,"},{"Start":"01:15.435 ","End":"01:18.105","Text":"minus 2, 0 is on the line."},{"Start":"01:18.105 ","End":"01:22.855","Text":"So if I put y equals 0 and x is minus 2, it should work."},{"Start":"01:22.855 ","End":"01:29.865","Text":"0 equals a times minus 2 plus 5,"},{"Start":"01:29.865 ","End":"01:33.990","Text":"so bringing the 2a to the other side equals 5,"},{"Start":"01:33.990 ","End":"01:35.970","Text":"a is 5 over 2,"},{"Start":"01:35.970 ","End":"01:39.860","Text":"or 2.5, a equals,"},{"Start":"01:39.860 ","End":"01:43.550","Text":"let\u0027s write it as decimal 2.5."},{"Start":"01:43.550 ","End":"01:53.540","Text":"Now we have b because b is 5 and we have a which is 2.5 and we can put that in y"},{"Start":"01:53.540 ","End":"01:57.755","Text":"equals ax plus b and get that our answer is y"},{"Start":"01:57.755 ","End":"02:05.250","Text":"equals 2.5x plus 5."},{"Start":"02:05.250 ","End":"02:11.734","Text":"That\u0027s it for number 25 and 26 is very similar to 25, just different numbers."},{"Start":"02:11.734 ","End":"02:17.585","Text":"So I kept this line here which tells us that b is the y-intercept,"},{"Start":"02:17.585 ","End":"02:21.690","Text":"In our case, it\u0027s minus 3,"},{"Start":"02:21.690 ","End":"02:30.290","Text":"so y equals ax minus 3 and to find a,"},{"Start":"02:30.290 ","End":"02:37.605","Text":"we noticed that the x-intercept is 1 which means that the point 1,"},{"Start":"02:37.605 ","End":"02:43.679","Text":"0 is on the line because that\u0027s what it means for x-intercept."},{"Start":"02:43.679 ","End":"02:46.010","Text":"It\u0027s where it crosses the y-axis where y is 0,"},{"Start":"02:46.010 ","End":"02:47.405","Text":"and that\u0027s where x is 1."},{"Start":"02:47.405 ","End":"02:54.205","Text":"Substituting in here, we get that 0 equals a times"},{"Start":"02:54.205 ","End":"03:02.125","Text":"1 minus 3 so a equals 3."},{"Start":"03:02.125 ","End":"03:04.985","Text":"Now we have our a and we have our b,"},{"Start":"03:04.985 ","End":"03:08.625","Text":"which is minus 3,"},{"Start":"03:08.625 ","End":"03:11.885","Text":"and a is plus 3 so putting that into here,"},{"Start":"03:11.885 ","End":"03:16.595","Text":"we get that y equals"},{"Start":"03:16.595 ","End":"03:23.950","Text":"3x minus 3 and we\u0027re done."}],"ID":5109},{"Watched":false,"Name":"Exercise 27","Duration":"3m 8s","ChapterTopicVideoID":5108,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5108.jpeg","UploadDate":"2016-02-29T09:20:54.9470000","DurationForVideoObject":"PT3M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:04.830","Text":"In Exercise 27, we have to find the equation of"},{"Start":"00:04.830 ","End":"00:10.380","Text":"a line that passes through this point and is parallel to this other line."},{"Start":"00:10.380 ","End":"00:13.470","Text":"We have to remember what parallel means."},{"Start":"00:13.470 ","End":"00:16.680","Text":"Two lines are parallel when they have the same slope."},{"Start":"00:16.680 ","End":"00:20.355","Text":"Our first task is to find the slope of this line."},{"Start":"00:20.355 ","End":"00:22.665","Text":"Let\u0027s scroll down a bit."},{"Start":"00:22.665 ","End":"00:25.950","Text":"I\u0027ll write the equation of this line here."},{"Start":"00:25.950 ","End":"00:32.112","Text":"4x+10y equals 30."},{"Start":"00:32.112 ","End":"00:33.710","Text":"To find the slope,"},{"Start":"00:33.710 ","End":"00:36.410","Text":"I should write it as y in terms of x."},{"Start":"00:36.410 ","End":"00:40.560","Text":"If I put the 4x to the other side and divide by 10,"},{"Start":"00:40.560 ","End":"00:50.045","Text":"I will get y=-4/10x+3."},{"Start":"00:50.045 ","End":"00:56.380","Text":"The slope is the coefficient of x."},{"Start":"00:56.380 ","End":"00:58.760","Text":"Our line has to have the same slope."},{"Start":"00:58.760 ","End":"01:02.550","Text":"In other words, we\u0027re given two conditions for our line."},{"Start":"01:04.580 ","End":"01:09.015","Text":"I\u0027ll just cancel top and bottom by 2."},{"Start":"01:09.015 ","End":"01:12.795","Text":"A=-2/5, and that it passes through"},{"Start":"01:12.795 ","End":"01:22.020","Text":"the point 4"},{"Start":"01:22.020 ","End":"01:24.855","Text":", -2."},{"Start":"01:24.855 ","End":"01:31.070","Text":"Remember that the general form of a line is y=ax+b."},{"Start":"01:31.920 ","End":"01:43.390","Text":"Here we already have a which is -2/5, y=-2/5x+b."},{"Start":"01:43.390 ","End":"01:46.900","Text":"The way to find b is to substitute this point,"},{"Start":"01:46.900 ","End":"01:48.955","Text":"which we know is on the line."},{"Start":"01:48.955 ","End":"01:53.450","Text":"We let y be -2 and x should be 4."},{"Start":"01:53.450 ","End":"02:02.380","Text":"So -2=-2/5(x),"},{"Start":"02:02.380 ","End":"02:05.730","Text":"which is 4+b."},{"Start":"02:05.730 ","End":"02:10.295","Text":"Let\u0027s get rid of fractions by multiplying both sides by 5."},{"Start":"02:10.295 ","End":"02:11.180","Text":"We\u0027ll get"},{"Start":"02:11.180 ","End":"02:22.690","Text":"-10=-8+5b."},{"Start":"02:22.690 ","End":"02:23.140","Text":"Let\u0027s see,"},{"Start":"02:23.140 ","End":"02:27.685","Text":"this gives us 5b is -2,"},{"Start":"02:27.685 ","End":"02:37.050","Text":"b=-2/5,"},{"Start":"02:37.050 ","End":"02:40.280","Text":"which is coincidentally the same as a."},{"Start":"02:40.280 ","End":"02:45.905","Text":"Now just substitute it in the y=ax+b."},{"Start":"02:45.905 ","End":"02:48.930","Text":"Here, we\u0027ll put a and b in."},{"Start":"02:48.930 ","End":"02:49.695","Text":"We get"},{"Start":"02:49.695 ","End":"02:50.950","Text":"that"},{"Start":"02:54.500 ","End":"03:04.680","Text":"y=-2/5x-2/5."},{"Start":"03:04.680 ","End":"03:08.020","Text":"Were done for Number 27."}],"ID":5110},{"Watched":false,"Name":"Exercise 28","Duration":"3m 34s","ChapterTopicVideoID":5109,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5109.jpeg","UploadDate":"2016-02-29T09:21:14.4300000","DurationForVideoObject":"PT3M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:05.445","Text":"Number 28 is similar to 27, different numbers."},{"Start":"00:05.445 ","End":"00:07.200","Text":"We have to find the line that passes through"},{"Start":"00:07.200 ","End":"00:10.935","Text":"this point and parallel to another line, this one."},{"Start":"00:10.935 ","End":"00:14.745","Text":"Remember that being parallel means having the same slope."},{"Start":"00:14.745 ","End":"00:18.310","Text":"Let\u0027s first find the slope of this line."},{"Start":"00:19.220 ","End":"00:25.830","Text":"Square root of 2x plus"},{"Start":"00:25.830 ","End":"00:32.475","Text":"5y equals the square root of 5."},{"Start":"00:32.475 ","End":"00:34.570","Text":"Let\u0027s scroll down a bit."},{"Start":"00:34.610 ","End":"00:37.770","Text":"We need to isolate Y."},{"Start":"00:37.770 ","End":"00:48.320","Text":"5y is equal to minus the square root of 2x plus the square root of 5."},{"Start":"00:48.320 ","End":"00:50.225","Text":"Dividing by 5,"},{"Start":"00:50.225 ","End":"00:57.486","Text":"Y equals minus the square root of 2 over 5x,"},{"Start":"00:57.486 ","End":"01:03.390","Text":"plus the square root of 5 over 5."},{"Start":"01:03.390 ","End":"01:10.820","Text":"This number here gives us the slope of the line."},{"Start":"01:10.820 ","End":"01:18.215","Text":"We have the slope a equals minus the square root of 2 over 5."},{"Start":"01:18.215 ","End":"01:21.020","Text":"That\u0027s the slope that we want."},{"Start":"01:21.020 ","End":"01:27.920","Text":"We now have a slope and we have a point which our line passes through,"},{"Start":"01:27.920 ","End":"01:34.550","Text":"which is minus the square root of 2, 4."},{"Start":"01:34.550 ","End":"01:37.940","Text":"We have to find the equation of this line."},{"Start":"01:37.940 ","End":"01:44.545","Text":"Now in general, an equation of a line is of the form y=ax+b."},{"Start":"01:44.545 ","End":"01:47.430","Text":"But in our case, a is this."},{"Start":"01:47.430 ","End":"01:55.160","Text":"We can straight away, say y equals minus the square root of 2 over 5x plus b,"},{"Start":"01:55.160 ","End":"01:58.200","Text":"which we still have to find out."},{"Start":"02:03.260 ","End":"02:08.320","Text":"How we find b is by substituting this point that the line goes through."},{"Start":"02:08.320 ","End":"02:14.090","Text":"We substitute this for x and this for y and we get that 4"},{"Start":"02:14.090 ","End":"02:21.700","Text":"equals minus the square root of 2 over 5 times x,"},{"Start":"02:21.700 ","End":"02:30.775","Text":"which is minus the square root of 2 plus b. Computing this,"},{"Start":"02:30.775 ","End":"02:32.679","Text":"minus and minus cancels,"},{"Start":"02:32.679 ","End":"02:35.935","Text":"square root of 2 times square root of 2 is 2/5."},{"Start":"02:35.935 ","End":"02:42.615","Text":"We get 4 equals 2/5 plus b,"},{"Start":"02:42.615 ","End":"02:52.150","Text":"making b equal to 4 minus 2/5 or 3 and 3/5."},{"Start":"02:55.280 ","End":"03:00.470","Text":"Now we have a and we have b,"},{"Start":"03:00.470 ","End":"03:06.395","Text":"and we just have to write y=ax+b."},{"Start":"03:06.395 ","End":"03:09.500","Text":"Remember y=ax+b in general,"},{"Start":"03:09.500 ","End":"03:12.529","Text":"and we just have to substitute with our numbers,"},{"Start":"03:12.529 ","End":"03:20.800","Text":"which is that y is equal to minus the square root of 2 over"},{"Start":"03:20.800 ","End":"03:30.200","Text":"5x plus b plus 3 and 3/5."},{"Start":"03:30.200 ","End":"03:33.720","Text":"That\u0027s it for number 28."}],"ID":5111},{"Watched":false,"Name":"Exercises 29-30","Duration":"5m 45s","ChapterTopicVideoID":5183,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5183.jpeg","UploadDate":"2016-02-29T14:21:09.0170000","DurationForVideoObject":"PT5M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:03.345","Text":"Now number 29,"},{"Start":"00:03.345 ","End":"00:05.610","Text":"we want the line that passes"},{"Start":"00:05.610 ","End":"00:13.155","Text":"through (2,5) and is perpendicular to this line as opposed to parallel."},{"Start":"00:13.155 ","End":"00:16.860","Text":"Let\u0027s see how we go about this."},{"Start":"00:16.860 ","End":"00:21.450","Text":"The first thing to do is to find the slope of this line here."},{"Start":"00:21.450 ","End":"00:30.870","Text":"Let\u0027s see, 6x- 4y=15 and I\u0027d like to put it in the form y=mx+n"},{"Start":"00:30.870 ","End":"00:35.550","Text":"so let\u0027s bring the y over to the other side"},{"Start":"00:35.550 ","End":"00:41.880","Text":"and 6x-15=4y and let\u0027s switch sides and divide by 4."},{"Start":"00:41.880 ","End":"00:46.480","Text":"We get y=6x/4,6/4"},{"Start":"00:46.790 ","End":"00:54.120","Text":"is 3/2x-15 /4."},{"Start":"00:54.120 ","End":"00:57.480","Text":"The thing is that we now have the slope which is"},{"Start":"00:57.480 ","End":"01:02.490","Text":"3/2 and I\u0027ll highlight it because that\u0027s important."},{"Start":"01:02.490 ","End":"01:06.360","Text":"What I need is not the slope of this line,"},{"Start":"01:06.360 ","End":"01:08.955","Text":"but the slope of the perpendicular line."},{"Start":"01:08.955 ","End":"01:14.640","Text":"There is a formula for getting from slope to a perpendicular slope and it goes like this."},{"Start":"01:14.640 ","End":"01:21.820","Text":"If m_1 is the slope of the original line, in our case,"},{"Start":"01:22.100 ","End":"01:26.200","Text":"3/2, then m_2,"},{"Start":"01:26.200 ","End":"01:28.290","Text":"which is the slope of the perpendicular line,"},{"Start":"01:28.290 ","End":"01:32.445","Text":"is the negative reciprocal -1/m_1."},{"Start":"01:32.445 ","End":"01:41.220","Text":"In practice, what this means is that you flip the fraction and change the sign."},{"Start":"01:41.220 ","End":"01:47.790","Text":"In our case, flipping the fraction is 2/3 and because it\u0027s positive, it\u0027s -2/3."},{"Start":"01:47.790 ","End":"01:54.060","Text":"This is the slope of my perpendicular line and I\u0027ll highlight that."},{"Start":"01:54.060 ","End":"01:57.645","Text":"This is very important because now for the second line,"},{"Start":"01:57.645 ","End":"01:59.010","Text":"the perpendicular one,"},{"Start":"01:59.010 ","End":"02:00.750","Text":"not only do I have a slope,"},{"Start":"02:00.750 ","End":"02:07.155","Text":"but I actually have a point that it passes through, which is (2,5)."},{"Start":"02:07.155 ","End":"02:09.990","Text":"Now, what I have is"},{"Start":"02:09.990 ","End":"02:15.900","Text":"a classic situation of a point and a slope and that\u0027s the standard formula."},{"Start":"02:15.900 ","End":"02:20.265","Text":"What I have is y minus the y of the point,"},{"Start":"02:20.265 ","End":"02:21.765","Text":"which is 5,"},{"Start":"02:21.765 ","End":"02:23.550","Text":"equals the slope,"},{"Start":"02:23.550 ","End":"02:33.905","Text":"which is this minus 2/3 times x minus the x of the point, which is 2."},{"Start":"02:33.905 ","End":"02:37.235","Text":"I was in fact just using the standard formula."},{"Start":"02:37.235 ","End":"02:46.470","Text":"I\u0027ve used it before, y-y1=m(x- x1) in case you suddenly forgot the formula."},{"Start":"02:47.270 ","End":"02:50.835","Text":"This is the answer, we just want to tidy it up a bit."},{"Start":"02:50.835 ","End":"02:54.165","Text":"I\u0027ll put y on this side and let\u0027s see."},{"Start":"02:54.165 ","End":"03:01.890","Text":"Here I have -2/3x and now I have a little side computation to do of what"},{"Start":"03:01.890 ","End":"03:11.820","Text":"is (-2/3)(-2)+5, What does that equal?"},{"Start":"03:11.820 ","End":"03:15.390","Text":"What we have here is minus times minus is plus."},{"Start":"03:15.390 ","End":"03:21.150","Text":"That is 4/3 and 5 I can write as 15/3."},{"Start":"03:21.150 ","End":"03:24.000","Text":"Altogether, what I have here is plus"},{"Start":"03:24.000 ","End":"03:31.950","Text":"19/3 and this is the answer and we are done."},{"Start":"03:31.950 ","End":"03:34.770","Text":"Now we come to number 30,"},{"Start":"03:34.770 ","End":"03:37.485","Text":"which is similar to 29."},{"Start":"03:37.485 ","End":"03:42.068","Text":"This time the line passes through this point and perpendicular to this."},{"Start":"03:42.068 ","End":"03:45.930","Text":"The first thing we do is find the slope of this line,"},{"Start":"03:45.930 ","End":"03:51.630","Text":"13 minus 8y=13, sorry,"},{"Start":"03:51.630 ","End":"03:55.500","Text":"13 x and we want to isolate y"},{"Start":"03:55.500 ","End":"04:01.560","Text":"so what I say is that 8y if I put it on the right-hand side,"},{"Start":"04:01.560 ","End":"04:06.345","Text":"is equal to 13x minus 13. y"},{"Start":"04:06.345 ","End":"04:16.390","Text":"equals 13/8 times x minus 13/8."},{"Start":"04:17.690 ","End":"04:27.900","Text":"The slope of this line is 13/8 but what I need is not this slope,"},{"Start":"04:27.900 ","End":"04:30.825","Text":"but the slope of the perpendicular line."},{"Start":"04:30.825 ","End":"04:33.300","Text":"As in the previous exercise,"},{"Start":"04:33.300 ","End":"04:35.939","Text":"I\u0027ll call this slope m or rather m_1"},{"Start":"04:35.939 ","End":"04:39.555","Text":"and I\u0027ll see what the slope of the perpendicular line is,"},{"Start":"04:39.555 ","End":"04:41.520","Text":"which is accorded m_2,"},{"Start":"04:41.520 ","End":"04:44.260","Text":"which is the negative reciprocal of this."},{"Start":"04:44.260 ","End":"04:50.880","Text":"This will be minus 8/13."},{"Start":"04:50.880 ","End":"04:56.130","Text":"Now we have a slope which is"},{"Start":"04:56.130 ","End":"05:01.545","Text":"this and I have a point that the line passes through, which is this."},{"Start":"05:01.545 ","End":"05:06.120","Text":"Now I use the slope and point equation."},{"Start":"05:06.120 ","End":"05:11.700","Text":"We get y minus the y of the point, which is 1,"},{"Start":"05:11.700 ","End":"05:20.895","Text":"is equal to the slope minus 8/13 times x minus the x of the point, which is 0."},{"Start":"05:20.895 ","End":"05:26.610","Text":"All I need is to simplify now and I get that y equals,"},{"Start":"05:26.610 ","End":"05:28.725","Text":"I bring the 1 over to the other side,"},{"Start":"05:28.725 ","End":"05:34.635","Text":"here I have minus 8/13x."},{"Start":"05:34.635 ","End":"05:45.430","Text":"This is nothing but plus the 1 is this and so this is the answer and we are done."}],"ID":5171},{"Watched":false,"Name":"Exercises 31-34","Duration":"5m 1s","ChapterTopicVideoID":5117,"CourseChapterTopicPlaylistID":56141,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5117.jpeg","UploadDate":"2016-02-29T09:24:59.7630000","DurationForVideoObject":"PT5M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.530 ","End":"00:03.330","Text":"In each of these 4 exercises,"},{"Start":"00:03.330 ","End":"00:05.985","Text":"we\u0027re given the equation of a line."},{"Start":"00:05.985 ","End":"00:09.728","Text":"For each line, we have to compute the lines x and y"},{"Start":"00:09.728 ","End":"00:13.845","Text":"intercepts and use this to sketch the line."},{"Start":"00:13.845 ","End":"00:17.290","Text":"First Number 31."},{"Start":"00:18.110 ","End":"00:20.790","Text":"To find the x and y-intercepts,"},{"Start":"00:20.790 ","End":"00:23.170","Text":"let\u0027s make a little table,"},{"Start":"00:24.260 ","End":"00:27.120","Text":"x, and y."},{"Start":"00:27.120 ","End":"00:33.495","Text":"We\u0027ll put x equals 0 to find the y-intercept and y equals 0 to find the x-intercept."},{"Start":"00:33.495 ","End":"00:37.830","Text":"When x is 0, we get this is 0,"},{"Start":"00:37.830 ","End":"00:39.255","Text":"so 5y is 20,"},{"Start":"00:39.255 ","End":"00:41.640","Text":"so y equals 4,"},{"Start":"00:41.640 ","End":"00:44.325","Text":"so 0,4 is the y-intercept,"},{"Start":"00:44.325 ","End":"00:46.860","Text":"and when y equals 0,"},{"Start":"00:46.860 ","End":"00:48.629","Text":"we get 4x is 20,"},{"Start":"00:48.629 ","End":"00:50.505","Text":"so x is 5."},{"Start":"00:50.505 ","End":"00:53.295","Text":"This gives us 2 points to plot,"},{"Start":"00:53.295 ","End":"00:59.010","Text":"and the y-intercept being 0,4 that would see 1,2,3,4,"},{"Start":"00:59.010 ","End":"01:00.945","Text":"that would be here,"},{"Start":"01:00.945 ","End":"01:04.650","Text":"and the point 5,0,"},{"Start":"01:04.650 ","End":"01:08.490","Text":"1,2,3,4,5 5,0 is here,"},{"Start":"01:08.490 ","End":"01:11.375","Text":"and now we just have to draw a line through them,"},{"Start":"01:11.375 ","End":"01:14.225","Text":"and this gives us the sketch of our graph,"},{"Start":"01:14.225 ","End":"01:17.190","Text":"that\u0027s the Number 31."},{"Start":"01:17.600 ","End":"01:20.700","Text":"Moving on to Number 32,"},{"Start":"01:20.700 ","End":"01:24.844","Text":"have kept a little table, same as before."},{"Start":"01:24.844 ","End":"01:30.004","Text":"We want to know what happens when x is 0 and when y is 0."},{"Start":"01:30.004 ","End":"01:31.910","Text":"Here we put x equals 0,"},{"Start":"01:31.910 ","End":"01:34.745","Text":"we get y equals minus 4,"},{"Start":"01:34.745 ","End":"01:36.755","Text":"and when y is 0,"},{"Start":"01:36.755 ","End":"01:39.415","Text":"we get the 2x is minus 4,"},{"Start":"01:39.415 ","End":"01:42.730","Text":"which means that x is minus 2."},{"Start":"01:42.730 ","End":"01:50.360","Text":"This gives us a couple of points to plot and just scroll a bit,"},{"Start":"01:50.360 ","End":"01:52.385","Text":"give us some more room,"},{"Start":"01:52.385 ","End":"01:55.483","Text":"so 0 minus 4,"},{"Start":"01:55.483 ","End":"02:05.010","Text":"1,2,3,4, that\u0027s minus 4 and minus 2,0."},{"Start":"02:07.840 ","End":"02:12.350","Text":"This of course is the x-intercept and this is the y-intercept."},{"Start":"02:12.350 ","End":"02:17.975","Text":"Now, all we need to do is draw a line through them, and there we are."},{"Start":"02:17.975 ","End":"02:20.970","Text":"Here\u0027s our graph for Number 32."},{"Start":"02:21.790 ","End":"02:27.930","Text":"Let\u0027s move on to Number 33."},{"Start":"02:28.650 ","End":"02:33.840","Text":"In Number 33, using the same technique,"},{"Start":"02:34.670 ","End":"02:38.910","Text":"we get that when x is 0,"},{"Start":"02:38.910 ","End":"02:42.285","Text":"y equals here at 0,"},{"Start":"02:42.285 ","End":"02:45.120","Text":"divide this by minus square root of 3,"},{"Start":"02:45.120 ","End":"02:51.470","Text":"get square root of 12 over minus the square root of 3,"},{"Start":"02:51.470 ","End":"02:56.480","Text":"which is minus the square root of 12 over 3,"},{"Start":"02:56.480 ","End":"02:57.770","Text":"which is 4,"},{"Start":"02:57.770 ","End":"03:00.800","Text":"in other words, minus 2, so when x is 0,"},{"Start":"03:00.800 ","End":"03:05.285","Text":"y is minus 2 and when y is 0,"},{"Start":"03:05.285 ","End":"03:15.180","Text":"we get x is equal to square root of 12 over square root of 4,"},{"Start":"03:15.200 ","End":"03:18.570","Text":"which is square root of 3."},{"Start":"03:18.570 ","End":"03:22.700","Text":"This is approximately equal to 1.7 something."},{"Start":"03:23.040 ","End":"03:26.470","Text":"But I\u0027ll write it here as square root of 3."},{"Start":"03:26.470 ","End":"03:36.100","Text":"But when we sketch it, it gives me an idea that the x-intercept is roughly 1.70,"},{"Start":"03:36.100 ","End":"03:39.084","Text":"so let\u0027s say somewhere here,"},{"Start":"03:39.084 ","End":"03:42.550","Text":"where this intercept is 1.7,"},{"Start":"03:42.550 ","End":"03:46.340","Text":"and the other 1 is 0 minus 2,"},{"Start":"03:46.340 ","End":"03:49.035","Text":"that would be here,"},{"Start":"03:49.035 ","End":"03:51.140","Text":"and that\u0027s a minus 3."},{"Start":"03:51.140 ","End":"03:54.940","Text":"All we need to do is draw a line between them,"},{"Start":"03:54.940 ","End":"04:00.140","Text":"and that\u0027s it for Number 33."},{"Start":"04:00.810 ","End":"04:03.895","Text":"Finally, Number 34,"},{"Start":"04:03.895 ","End":"04:06.790","Text":"same technique when x is 0,"},{"Start":"04:06.790 ","End":"04:10.525","Text":"we get minus 1.5y is minus 3,"},{"Start":"04:10.525 ","End":"04:18.550","Text":"so y is minus 3 over minus 1.5,"},{"Start":"04:18.550 ","End":"04:21.643","Text":"which is to put that in here,"},{"Start":"04:21.643 ","End":"04:25.030","Text":"and when y is 0 for the x-intercept,"},{"Start":"04:25.030 ","End":"04:32.620","Text":"then we get x equals minus 3 because this thing is 0."},{"Start":"04:32.840 ","End":"04:35.130","Text":"Write that in here,"},{"Start":"04:35.130 ","End":"04:37.635","Text":"and now I\u0027ll draw these couple of points,"},{"Start":"04:37.635 ","End":"04:42.737","Text":"x is 0, y equals 2 comes out to here that\u0027s 2,"},{"Start":"04:42.737 ","End":"04:50.895","Text":"minus 3y equals 0 1,2,3 minus 3,"},{"Start":"04:50.895 ","End":"04:53.405","Text":"and now we just have to draw the line."},{"Start":"04:53.405 ","End":"04:57.650","Text":"Here\u0027s a rough sketch of the line, and that\u0027s it."},{"Start":"04:57.650 ","End":"05:00.330","Text":"We\u0027re done with this exercise."}],"ID":5112}],"Thumbnail":null,"ID":56141},{"Name":"Quadratic Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"3m 35s","ChapterTopicVideoID":5131,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5131.jpeg","UploadDate":"2020-09-30T13:18:19.0570000","DurationForVideoObject":"PT3M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"This clip is a brief introduction to the quadratic function."},{"Start":"00:05.085 ","End":"00:08.100","Text":"A quadratic function is a function of"},{"Start":"00:08.100 ","End":"00:11.790","Text":"the form f(x)="},{"Start":"00:11.790 ","End":"00:18.660","Text":"ax^2+bx+c,"},{"Start":"00:18.660 ","End":"00:21.810","Text":"where a, b, and c are some constants."},{"Start":"00:21.810 ","End":"00:26.910","Text":"Now it\u0027s very important that a not be 0."},{"Start":"00:26.910 ","End":"00:31.230","Text":"Because if a is 0 it\u0027s not a quadratic function."},{"Start":"00:31.230 ","End":"00:34.200","Text":"There is no x^2 to it\u0027s linear function,"},{"Start":"00:34.200 ","End":"00:37.005","Text":"so a shouldn\u0027t be 0."},{"Start":"00:37.005 ","End":"00:40.425","Text":"I\u0027ll give you an example to start off with."},{"Start":"00:40.425 ","End":"00:42.190","Text":"How about"},{"Start":"00:43.160 ","End":"00:52.055","Text":"f(x)=4x^2-10x+1."},{"Start":"00:52.055 ","End":"01:00.200","Text":"This is a quadratic function with the constants a=4,"},{"Start":"01:00.200 ","End":"01:07.300","Text":"b=-10, and c=1."},{"Start":"01:07.300 ","End":"01:12.180","Text":"Another example, here I\u0027ll use the y notation,"},{"Start":"01:12.180 ","End":"01:21.560","Text":"y=-x^2+20."},{"Start":"01:21.560 ","End":"01:25.385","Text":"Here, a=-1."},{"Start":"01:25.385 ","End":"01:27.920","Text":"That\u0027s the coefficient of x^2."},{"Start":"01:27.920 ","End":"01:30.815","Text":"b is absent. It\u0027s missing."},{"Start":"01:30.815 ","End":"01:32.935","Text":"When it\u0027s missing it\u0027s 0."},{"Start":"01:32.935 ","End":"01:39.495","Text":"Because you could say +0x here and as for c it\u0027s just 20."},{"Start":"01:39.495 ","End":"01:44.640","Text":"Another example, also in terms"},{"Start":"01:44.640 ","End":"01:51.600","Text":"of take a g(x) for a change different letter,"},{"Start":"01:51.600 ","End":"01:57.040","Text":"g(x) is 50x^2 minus 20x."},{"Start":"02:01.480 ","End":"02:05.855","Text":"Now here in the previous example, we had b=0."},{"Start":"02:05.855 ","End":"02:09.560","Text":"Here c is missing the free co-efficient."},{"Start":"02:09.560 ","End":"02:13.120","Text":"We get a=50,"},{"Start":"02:13.120 ","End":"02:18.965","Text":"b would be minus 20 in this case."},{"Start":"02:18.965 ","End":"02:20.735","Text":"Since there is no other term."},{"Start":"02:20.735 ","End":"02:27.060","Text":"It\u0027s like saying that c=0, plus 0 here."},{"Start":"02:27.950 ","End":"02:30.890","Text":"Let\u0027s take 1 more example."},{"Start":"02:30.890 ","End":"02:32.945","Text":"We\u0027ll take it in terms of y."},{"Start":"02:32.945 ","End":"02:39.005","Text":"Again, y is equal to minus 4x^2."},{"Start":"02:39.005 ","End":"02:41.085","Text":"Just like here,"},{"Start":"02:41.085 ","End":"02:44.085","Text":"b was 0 and here c was 0."},{"Start":"02:44.085 ","End":"02:46.500","Text":"Here both b and c are 0."},{"Start":"02:46.500 ","End":"02:48.030","Text":"We only left with a,"},{"Start":"02:48.030 ","End":"02:55.305","Text":"a is minus 4 b term is missing."},{"Start":"02:55.305 ","End":"02:59.870","Text":"The c term is missing which means that c is 0."},{"Start":"02:59.870 ","End":"03:03.260","Text":"That\u0027s as far as the examples go."},{"Start":"03:03.260 ","End":"03:06.080","Text":"In the rest of the tutorial on"},{"Start":"03:06.080 ","End":"03:11.240","Text":"the quadratic function we\u0027ll learn things like what is the meaning of a,"},{"Start":"03:11.240 ","End":"03:14.645","Text":"b, and c. The graph of the quadratic function"},{"Start":"03:14.645 ","End":"03:18.650","Text":"which happens to be in the shape of a parabola."},{"Start":"03:18.650 ","End":"03:21.695","Text":"I\u0027ll just write the word. But we\u0027ll encounter it later."},{"Start":"03:21.695 ","End":"03:24.920","Text":"We\u0027ll learn about the vertex of the parabola."},{"Start":"03:24.920 ","End":"03:30.097","Text":"We\u0027ll learn about the intersection of the quadratic function with the axis,"},{"Start":"03:30.097 ","End":"03:31.325","Text":"the x and y."},{"Start":"03:31.325 ","End":"03:33.980","Text":"All sorts of other fun stuff like that."},{"Start":"03:33.980 ","End":"03:36.900","Text":"But we\u0027re done for the intro."}],"ID":5113},{"Watched":false,"Name":"About the Constants in the Function","Duration":"9m 40s","ChapterTopicVideoID":5132,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5132.jpeg","UploadDate":"2020-09-30T13:04:22.0270000","DurationForVideoObject":"PT9M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"In this clip, we\u0027ll continue talking about the quadratic function,"},{"Start":"00:05.085 ","End":"00:11.625","Text":"y=ax^2+bx+c, and we\u0027ll focus on the meaning of the constants,"},{"Start":"00:11.625 ","End":"00:18.285","Text":"which are a, b, and c. Let\u0027s begin with the constant a."},{"Start":"00:18.285 ","End":"00:22.710","Text":"Now, the constant a we mentioned cannot be 0."},{"Start":"00:22.710 ","End":"00:25.215","Text":"Otherwise it wouldn\u0027t be a parabola,"},{"Start":"00:25.215 ","End":"00:28.289","Text":"so it\u0027s either positive or negative,"},{"Start":"00:28.289 ","End":"00:31.740","Text":"and it turns out that the positivity or negativity of a"},{"Start":"00:31.740 ","End":"00:35.910","Text":"determines the basic shape of the graph,"},{"Start":"00:35.910 ","End":"00:38.880","Text":"which is either a right side up,"},{"Start":"00:38.880 ","End":"00:42.180","Text":"parabola or an upside down parabola."},{"Start":"00:42.180 ","End":"00:44.680","Text":"I haven\u0027t shown you what a parabola looks like yet,"},{"Start":"00:44.680 ","End":"00:46.119","Text":"I\u0027ll get to it in a moment,"},{"Start":"00:46.119 ","End":"00:48.925","Text":"but I\u0027ll just make note of that,"},{"Start":"00:48.925 ","End":"00:52.900","Text":"first of all just state that the graph is a parabola."},{"Start":"00:52.900 ","End":"00:55.580","Text":"I\u0027ll just write the word parabola."},{"Start":"00:56.550 ","End":"00:59.545","Text":"You\u0027ll soon see 1,"},{"Start":"00:59.545 ","End":"01:05.710","Text":"and if a is bigger than 0,"},{"Start":"01:05.710 ","End":"01:08.420","Text":"then it\u0027s right side up."},{"Start":"01:11.060 ","End":"01:18.295","Text":"Soon, I\u0027ll get to it and I\u0027ll get to the picture in just a moment and if a is negative,"},{"Start":"01:18.295 ","End":"01:23.390","Text":"the parabola will be an upside down parabola."},{"Start":"01:24.710 ","End":"01:28.390","Text":"There are other names and I\u0027ll also get to that."},{"Start":"01:28.390 ","End":"01:31.600","Text":"Let\u0027s start with the a bigger than 0."},{"Start":"01:31.600 ","End":"01:34.540","Text":"If a is bigger than 0,"},{"Start":"01:34.540 ","End":"01:37.119","Text":"then the graph will,"},{"Start":"01:37.119 ","End":"01:38.230","Text":"I\u0027ll try and sketch 1."},{"Start":"01:38.230 ","End":"01:40.015","Text":"First of all, do an axis."},{"Start":"01:40.015 ","End":"01:48.175","Text":"Here we are and what I\u0027ll do is I\u0027ll just freehand sketch some parabolas."},{"Start":"01:48.175 ","End":"01:52.675","Text":"A parabola could look something like this."},{"Start":"01:52.675 ","End":"01:55.015","Text":"It\u0027s supposed to be symmetrical."},{"Start":"01:55.015 ","End":"01:58.930","Text":"It could look something starting from lower down,"},{"Start":"01:58.930 ","End":"02:02.095","Text":"it could start from higher up."},{"Start":"02:02.095 ","End":"02:05.095","Text":"It could be narrower."},{"Start":"02:05.095 ","End":"02:08.210","Text":"It could be wider."},{"Start":"02:08.720 ","End":"02:14.745","Text":"It could be grazing the axis."},{"Start":"02:14.745 ","End":"02:21.015","Text":"That was terrible, a second and it could even be up in the air,"},{"Start":"02:21.015 ","End":"02:23.380","Text":"we have 1 already there."},{"Start":"02:24.600 ","End":"02:27.940","Text":"Some of the things I\u0027d like to say about a parabola is"},{"Start":"02:27.940 ","End":"02:31.000","Text":"that I haven\u0027t drawn them very well,"},{"Start":"02:31.000 ","End":"02:34.885","Text":"but they\u0027re supposed to be symmetrical about the vertical axis."},{"Start":"02:34.885 ","End":"02:38.275","Text":"They are generally called U-shaped."},{"Start":"02:38.275 ","End":"02:41.240","Text":"They hold water, so to speak."},{"Start":"02:42.440 ","End":"02:45.375","Text":"They go on infinitely,"},{"Start":"02:45.375 ","End":"02:47.650","Text":"though I can only draw a finite amount,"},{"Start":"02:47.650 ","End":"02:50.265","Text":"they do carry on all the way to infinity."},{"Start":"02:50.265 ","End":"02:53.003","Text":"They keep going on forever."},{"Start":"02:53.003 ","End":"02:58.850","Text":"The main thing about the a bigger than 0 is that they are all U-shaped,"},{"Start":"02:58.850 ","End":"03:02.195","Text":"that they hold water, you could say."},{"Start":"03:02.195 ","End":"03:06.125","Text":"Now the opposite happens when a is negative."},{"Start":"03:06.125 ","End":"03:15.890","Text":"When a is negative, they\u0027re upside down."},{"Start":"03:15.890 ","End":"03:19.535","Text":"So they will go, I\u0027ll draw an x-axis just to second,"},{"Start":"03:19.535 ","End":"03:25.250","Text":"here\u0027s my axis and the parabolas."},{"Start":"03:25.250 ","End":"03:29.180","Text":"Let me just choose a different color. There we go."},{"Start":"03:29.180 ","End":"03:32.660","Text":"They would be upside down, something like this."},{"Start":"03:32.660 ","End":"03:35.480","Text":"They could be well below the axis."},{"Start":"03:35.480 ","End":"03:38.360","Text":"They could start much higher up."},{"Start":"03:38.360 ","End":"03:41.328","Text":"They could be thinner."},{"Start":"03:41.328 ","End":"03:44.490","Text":"They could be broader."},{"Start":"03:44.920 ","End":"03:48.650","Text":"Again, they can be up in the air or down below."},{"Start":"03:48.650 ","End":"03:55.080","Text":"They don\u0027t have to touch they could even just graze on the axis."},{"Start":"03:55.080 ","End":"03:58.350","Text":"Well, forgive my drawing."},{"Start":"03:58.370 ","End":"04:00.825","Text":"So that\u0027s the general idea."},{"Start":"04:00.825 ","End":"04:03.315","Text":"a bigger than 0,"},{"Start":"04:03.315 ","End":"04:06.770","Text":"a parabola which is the right side up,"},{"Start":"04:06.770 ","End":"04:08.330","Text":"a less than 0,"},{"Start":"04:08.330 ","End":"04:10.190","Text":"upside down parabola,"},{"Start":"04:10.190 ","End":"04:15.315","Text":"does not hold water, propels water."},{"Start":"04:15.315 ","End":"04:16.950","Text":"It\u0027s an upside down cup,"},{"Start":"04:16.950 ","End":"04:18.655","Text":"an upside down U."},{"Start":"04:18.655 ","End":"04:22.625","Text":"There are many ways of saying it."},{"Start":"04:22.625 ","End":"04:25.865","Text":"Notice that a parabola,"},{"Start":"04:25.865 ","End":"04:29.270","Text":"each parabola has a vertex,"},{"Start":"04:29.270 ","End":"04:36.095","Text":"which is the extreme points in the case of the a less than 0,"},{"Start":"04:36.095 ","End":"04:40.100","Text":"each of these would be a vertex."},{"Start":"04:40.630 ","End":"04:45.785","Text":"In the case where we have a right side up parabola,"},{"Start":"04:45.785 ","End":"04:47.180","Text":"the vertex would be, here, here, say here, here, here,"},{"Start":"04:47.180 ","End":"04:56.152","Text":"here, and here."},{"Start":"04:56.152 ","End":"04:59.590","Text":"The thing to notice that in the case"},{"Start":"04:59.590 ","End":"05:04.570","Text":"of the right side up parabola where a is bigger than 0,"},{"Start":"05:04.570 ","End":"05:07.749","Text":"each vertex is a minimum."},{"Start":"05:07.749 ","End":"05:14.810","Text":"So I\u0027ll write vertex is a minimum."},{"Start":"05:16.860 ","End":"05:20.550","Text":"In the case, excuse me,"},{"Start":"05:20.550 ","End":"05:23.236","Text":"of the a less than 0,"},{"Start":"05:23.236 ","End":"05:25.720","Text":"each vertex is a maximum."},{"Start":"05:25.720 ","End":"05:28.090","Text":"It\u0027s the highest point."},{"Start":"05:28.090 ","End":"05:38.450","Text":"Vertex is a maximum."},{"Start":"05:40.050 ","End":"05:46.270","Text":"That\u0027s all I have to say meanwhile about the constant a,"},{"Start":"05:46.270 ","End":"05:50.210","Text":"except possibly to correct a spelling mistake."},{"Start":"05:50.210 ","End":"05:54.640","Text":"Vertex and the green was for the a bigger than"},{"Start":"05:54.640 ","End":"06:01.045","Text":"0 and the purple was for the a less than the 0."},{"Start":"06:01.045 ","End":"06:03.460","Text":"Difficult to sketch freehand,"},{"Start":"06:03.460 ","End":"06:05.600","Text":"but you get the idea."},{"Start":"06:05.790 ","End":"06:09.460","Text":"The next constant I\u0027m going to talk about is actually"},{"Start":"06:09.460 ","End":"06:12.595","Text":"c. I\u0027m going to skip over b meanwhile,"},{"Start":"06:12.595 ","End":"06:16.000","Text":"perhaps later on I\u0027ll have a few words to say about b,"},{"Start":"06:16.000 ","End":"06:19.330","Text":"but its geometrical meaning is not so intuitive."},{"Start":"06:19.330 ","End":"06:25.790","Text":"It\u0027s more difficult to say something about it but later on we will say something anyway."},{"Start":"06:25.790 ","End":"06:28.235","Text":"Next 1 we\u0027ll go to will be c,"},{"Start":"06:28.235 ","End":"06:31.710","Text":"and I\u0027ll just clear everything up from here."},{"Start":"06:32.350 ","End":"06:37.550","Text":"Now let\u0027s talk a bit about the constant c. C has"},{"Start":"06:37.550 ","End":"06:45.050","Text":"a very simple meaning and I brought a parabola with me."},{"Start":"06:45.050 ","End":"06:50.490","Text":"c is the intersection,"},{"Start":"06:50.490 ","End":"06:52.140","Text":"emphasize the c,"},{"Start":"06:52.140 ","End":"06:54.030","Text":"c is the intersection"},{"Start":"06:54.030 ","End":"07:04.690","Text":"of the parabola"},{"Start":"07:04.760 ","End":"07:08.290","Text":"with the y-axis."},{"Start":"07:14.380 ","End":"07:20.600","Text":"This point here where the parabola hits the y-axis, this coordinate,"},{"Start":"07:20.600 ","End":"07:24.650","Text":"the y-coordinate is c. So this point would be 0,"},{"Start":"07:24.650 ","End":"07:27.530","Text":"c. In this case,"},{"Start":"07:27.530 ","End":"07:30.830","Text":"c is negative, it\u0027s below the x-axis."},{"Start":"07:30.830 ","End":"07:35.060","Text":"So c is the intersection of the parabola with the y-axis,"},{"Start":"07:35.060 ","End":"07:38.750","Text":"sometimes it\u0027s called the y-intercept."},{"Start":"07:38.750 ","End":"07:41.030","Text":"When I went to school, that\u0027s what it was called."},{"Start":"07:41.030 ","End":"07:49.670","Text":"So c is the y-intercept."},{"Start":"07:49.670 ","End":"07:52.730","Text":"Now I want to give you a reason why."},{"Start":"07:52.730 ","End":"07:54.455","Text":"The reason is clear."},{"Start":"07:54.455 ","End":"07:57.650","Text":"Because if we substitute,"},{"Start":"07:57.650 ","End":"07:59.880","Text":"I\u0027ll just write that down."},{"Start":"08:00.010 ","End":"08:07.010","Text":"If we substitute, and of course,"},{"Start":"08:07.010 ","End":"08:11.960","Text":"I mean in the equation of the quadratic function,"},{"Start":"08:11.960 ","End":"08:16.464","Text":"if we substitute x=0"},{"Start":"08:16.464 ","End":"08:22.250","Text":", we get y=c."},{"Start":"08:22.250 ","End":"08:24.560","Text":"Isn\u0027t that so, put x=0,"},{"Start":"08:24.560 ","End":"08:27.500","Text":"this is 0, this is 0, and y=c."},{"Start":"08:27.500 ","End":"08:34.630","Text":"That\u0027s the reason why c represents the y-intercept."},{"Start":"08:34.630 ","End":"08:39.515","Text":"Speaking of the y-intercept or the intersection with the y-axis,"},{"Start":"08:39.515 ","End":"08:45.290","Text":"made me wonder, what about the intersection of the parabola with the x-axis?"},{"Start":"08:45.290 ","End":"08:47.224","Text":"Well, in this particular picture,"},{"Start":"08:47.224 ","End":"08:48.560","Text":"it cuts it twice."},{"Start":"08:48.560 ","End":"08:50.240","Text":"There would be 2 points here."},{"Start":"08:50.240 ","End":"08:52.709","Text":"These are also called x-intercepts,"},{"Start":"08:52.709 ","End":"08:56.135","Text":"but how about if I took a different picture?"},{"Start":"08:56.135 ","End":"09:00.089","Text":"Notice that although in this case it cuts twice,"},{"Start":"09:00.089 ","End":"09:05.165","Text":"we could have a parabola that went like this and it wouldn\u0027t cut the x-axis at all."},{"Start":"09:05.165 ","End":"09:10.040","Text":"Or we could have a parabola that just just grazed it."},{"Start":"09:10.040 ","End":"09:14.480","Text":"So it would cut the x-axis only once and besides all that,"},{"Start":"09:14.480 ","End":"09:19.505","Text":"there\u0027s all these upside down parabolas that could cut twice,"},{"Start":"09:19.505 ","End":"09:22.375","Text":"could cut just once,"},{"Start":"09:22.375 ","End":"09:25.085","Text":"or it might not even cut at all."},{"Start":"09:25.085 ","End":"09:30.035","Text":"So all these possibilities makes me think that maybe we should devote in"},{"Start":"09:30.035 ","End":"09:35.510","Text":"a separate section to the intersection of the parabola with the axis both x and y,"},{"Start":"09:35.510 ","End":"09:38.630","Text":"and both for right side up and upside down parabola."},{"Start":"09:38.630 ","End":"09:41.160","Text":"So that will be up next.1"}],"ID":5114},{"Watched":false,"Name":"Intersection with the Axes","Duration":"15m 58s","ChapterTopicVideoID":5133,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5133.jpeg","UploadDate":"2020-09-30T13:13:40.5130000","DurationForVideoObject":"PT15M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.970","Text":"Continuing with the quadratic function, y=ax^2+bx+c."},{"Start":"00:05.970 ","End":"00:10.665","Text":"We\u0027ll talk now about the intersection with the axes."},{"Start":"00:10.665 ","End":"00:16.890","Text":"Well, actually we\u0027ve already talked about the intersection with the y-axis."},{"Start":"00:16.890 ","End":"00:20.805","Text":"So as far as the y-axis is concerned,"},{"Start":"00:20.805 ","End":"00:29.190","Text":"we know the answer and we know that the intersection is (0,C)."},{"Start":"00:29.190 ","End":"00:33.820","Text":"We did this just a little while ago."},{"Start":"00:34.370 ","End":"00:37.890","Text":"Really quick sketch."},{"Start":"00:37.890 ","End":"00:40.870","Text":"These are the axes."},{"Start":"00:40.870 ","End":"00:44.465","Text":"This is the parabola,"},{"Start":"00:44.465 ","End":"00:48.987","Text":"and this point here is the intersection with the y-axis,"},{"Start":"00:48.987 ","End":"00:53.270","Text":"y=0, don\u0027t get the 2 mixed up and this is the x."},{"Start":"00:53.270 ","End":"00:57.080","Text":"We already talked about this very quickly in our heads."},{"Start":"00:57.080 ","End":"01:00.350","Text":"The reason was that at 1 of the y-axis, x=0."},{"Start":"01:00.350 ","End":"01:02.470","Text":"If you put x=0 in here,"},{"Start":"01:02.470 ","End":"01:05.215","Text":"this is 0, this is 0, y=C."},{"Start":"01:05.215 ","End":"01:08.000","Text":"That\u0027s very straightforward and simple."},{"Start":"01:08.000 ","End":"01:12.320","Text":"Things are going to be a little bit more difficult with the x-axis."},{"Start":"01:12.320 ","End":"01:15.170","Text":"So let\u0027s see what happens."},{"Start":"01:15.170 ","End":"01:19.955","Text":"In the case of the axis in general, (0,C),"},{"Start":"01:19.955 ","End":"01:26.335","Text":"now the x-axis intersection with,"},{"Start":"01:26.335 ","End":"01:33.925","Text":"what we do is we put at y=0 this time because on the x-axis, y=0,"},{"Start":"01:33.925 ","End":"01:43.090","Text":"so we get 0=ax^2+bx+c."},{"Start":"01:44.600 ","End":"01:55.835","Text":"Now, this is the famous quadratic equation, ax^2+bx+c=0."},{"Start":"01:55.835 ","End":"01:59.845","Text":"Remember, as always that a is not equal to 0."},{"Start":"01:59.845 ","End":"02:07.140","Text":"Essentially, to find the x-intercepts,"},{"Start":"02:07.140 ","End":"02:08.870","Text":"and there might be more than 1,"},{"Start":"02:08.870 ","End":"02:10.820","Text":"as you can see in this example,"},{"Start":"02:10.820 ","End":"02:13.220","Text":"we just solve the quadratic equation."},{"Start":"02:13.220 ","End":"02:14.735","Text":"Let\u0027s take an example."},{"Start":"02:14.735 ","End":"02:17.160","Text":"Suppose I have"},{"Start":"02:18.190 ","End":"02:25.638","Text":"y=x^2-4x+3,"},{"Start":"02:25.638 ","End":"02:29.150","Text":"how do we solve quadratic equations?"},{"Start":"02:29.150 ","End":"02:31.190","Text":"There\u0027s a few techniques,"},{"Start":"02:31.190 ","End":"02:35.360","Text":"but the famous 1 is the formula."},{"Start":"02:35.360 ","End":"02:38.990","Text":"Let me write the quadratic formula."},{"Start":"02:38.990 ","End":"02:40.820","Text":"It\u0027s so important to think,"},{"Start":"02:40.820 ","End":"02:43.085","Text":"you know what, I think, I\u0027ll put it in red."},{"Start":"02:43.085 ","End":"02:45.110","Text":"When we have this equation here,"},{"Start":"02:45.110 ","End":"02:55.115","Text":"we say that x=-b+/- square root"},{"Start":"02:55.115 ","End":"03:00.068","Text":"of b^2-4ac,"},{"Start":"03:00.068 ","End":"03:02.910","Text":"this whole thing, over 2a."},{"Start":"03:05.240 ","End":"03:09.140","Text":"Notice that there\u0027s a plus or minus here,"},{"Start":"03:09.140 ","End":"03:13.715","Text":"which means that we take 1 time the plus 1 times the minus,"},{"Start":"03:13.715 ","End":"03:17.045","Text":"and we get 2 solutions."},{"Start":"03:17.045 ","End":"03:20.360","Text":"Usually, 1 of them is called the 1 with the plus,"},{"Start":"03:20.360 ","End":"03:23.345","Text":"let\u0027s say it\u0027s called x_1."},{"Start":"03:23.345 ","End":"03:26.345","Text":"The 1 with the minus,"},{"Start":"03:26.345 ","End":"03:28.930","Text":"we call it x_2."},{"Start":"03:28.930 ","End":"03:32.910","Text":"But there up to 2 possible values of x."},{"Start":"03:32.910 ","End":"03:36.670","Text":"Now, let\u0027s apply this to our case."},{"Start":"03:37.820 ","End":"03:44.696","Text":"What we have is that,"},{"Start":"03:44.696 ","End":"03:46.849","Text":"what I\u0027m solving is not the function,"},{"Start":"03:46.849 ","End":"03:50.195","Text":"but the equation where this is equal to 0."},{"Start":"03:50.195 ","End":"03:52.144","Text":"To find the intercepts,"},{"Start":"03:52.144 ","End":"03:55.925","Text":"and solve the equation, this thing equal 0."},{"Start":"03:55.925 ","End":"03:59.570","Text":"The y was just there for the function."},{"Start":"03:59.570 ","End":"04:03.935","Text":"That\u0027s the function, and we\u0027re looking for the intercepts."},{"Start":"04:03.935 ","End":"04:08.210","Text":"We have here the a=1,"},{"Start":"04:08.210 ","End":"04:16.875","Text":"b=-4, and c=3."},{"Start":"04:16.875 ","End":"04:20.360","Text":"Then a mere substitution,"},{"Start":"04:20.360 ","End":"04:25.760","Text":"we get that x=-b."},{"Start":"04:25.760 ","End":"04:27.175","Text":"So that\u0027s minus,"},{"Start":"04:27.175 ","End":"04:33.440","Text":"-(-4)+/- square"},{"Start":"04:33.440 ","End":"04:54.992","Text":"root of(-4)^2-4(1)(3)"},{"Start":"04:54.992 ","End":"05:03.610","Text":", the whole thing is over 2a, which is 2(1)."},{"Start":"05:06.280 ","End":"05:09.095","Text":"I\u0027ll continue on the same line."},{"Start":"05:09.095 ","End":"05:16.520","Text":"This is equal to -(-4=4+/- the square root."},{"Start":"05:16.520 ","End":"05:18.950","Text":"I think we can do this in our heads,"},{"Start":"05:18.950 ","End":"05:24.370","Text":"4(1)(3)=12,"},{"Start":"05:26.120 ","End":"05:30.360","Text":"(-4)^2=16,16-12=4 and all this is over 2."},{"Start":"05:30.360 ","End":"05:35.975","Text":"I chose this example so it would have a nice square root, make life easier."},{"Start":"05:35.975 ","End":"05:39.125","Text":"The square root of 4=2."},{"Start":"05:39.125 ","End":"05:41.660","Text":"So what we basically get is,"},{"Start":"05:41.660 ","End":"05:45.575","Text":"I\u0027ll put a little like a fork."},{"Start":"05:45.575 ","End":"05:47.603","Text":"The 2 possibilities,"},{"Start":"05:47.603 ","End":"05:49.265","Text":"1 time I\u0027ll take the plus,"},{"Start":"05:49.265 ","End":"05:52.215","Text":"and 1 time I\u0027ll take the minus."},{"Start":"05:52.215 ","End":"05:53.850","Text":"If I take the plus,"},{"Start":"05:53.850 ","End":"05:56.730","Text":"I get the square root of 4=2 of course."},{"Start":"05:56.730 ","End":"05:59.610","Text":"So I get (4+2)/2,"},{"Start":"05:59.610 ","End":"06:03.010","Text":"which is 6/2, which is 3."},{"Start":"06:03.010 ","End":"06:05.915","Text":"And that\u0027s my x_1, if you like."},{"Start":"06:05.915 ","End":"06:08.150","Text":"Then the other case, I take the minus 4,"},{"Start":"06:08.150 ","End":"06:12.230","Text":"minus 2 over 2, so it\u0027s 2/2=1."},{"Start":"06:12.230 ","End":"06:14.555","Text":"That will be my x2."},{"Start":"06:14.555 ","End":"06:19.410","Text":"And it doesn\u0027t matter really what order 1 and 3 are the solutions."},{"Start":"06:19.410 ","End":"06:28.770","Text":"I could say that x=1,3."},{"Start":"06:28.770 ","End":"06:31.590","Text":"Note that a=1,"},{"Start":"06:31.590 ","End":"06:33.455","Text":"and 1 is bigger than 0."},{"Start":"06:33.455 ","End":"06:36.649","Text":"This picture, though I drew it in general,"},{"Start":"06:36.649 ","End":"06:42.904","Text":"applies for parabolas where a is bigger than 0,"},{"Start":"06:42.904 ","End":"06:44.555","Text":"which it\u0027s facing up."},{"Start":"06:44.555 ","End":"06:46.100","Text":"We\u0027ve got a lot of information already."},{"Start":"06:46.100 ","End":"06:48.980","Text":"We\u0027ve got 3 intercepts and we\u0027ve got it facing the right way."},{"Start":"06:48.980 ","End":"06:53.590","Text":"We actually got 2 intersections with the x-axis."},{"Start":"06:53.590 ","End":"06:56.705","Text":"Remember, we had 3 basic paradigms,"},{"Start":"06:56.705 ","End":"07:02.555","Text":"at least for the cases where a was positive and the parabola was facing up,"},{"Start":"07:02.555 ","End":"07:10.135","Text":"we had the possibility that the parabola cut the axis in 2 places."},{"Start":"07:10.135 ","End":"07:18.479","Text":"We had a case where the parabola just graze touched."},{"Start":"07:18.479 ","End":"07:22.015","Text":"The x-axis was a tangent, if you like."},{"Start":"07:22.015 ","End":"07:26.575","Text":"We had just 1 intersection."},{"Start":"07:26.575 ","End":"07:31.015","Text":"In fact, we also have the case where"},{"Start":"07:31.015 ","End":"07:37.190","Text":"the parabola was totally up in the air, something like this."},{"Start":"07:38.070 ","End":"07:43.240","Text":"Extend it a bit. Completely missed the x-axis."},{"Start":"07:43.240 ","End":"07:48.700","Text":"I\u0027d like to understand what makes these 3 different cases happen."},{"Start":"07:48.700 ","End":"07:51.925","Text":"What causes it to be this, this, or this?"},{"Start":"07:51.925 ","End":"07:58.390","Text":"Well, let\u0027s look here in our example and notice that"},{"Start":"07:58.390 ","End":"08:04.975","Text":"we got 2 solutions because of this fork in the cases."},{"Start":"08:04.975 ","End":"08:09.490","Text":"We had a plus and a minus which produced 2."},{"Start":"08:09.490 ","End":"08:13.180","Text":"The reason we\u0027ve got a plus and a minus is because of the formula."},{"Start":"08:13.180 ","End":"08:15.340","Text":"We have plus or minus the square root,"},{"Start":"08:15.340 ","End":"08:17.500","Text":"and under the square root sign,"},{"Start":"08:17.500 ","End":"08:19.945","Text":"we happen to have 4,"},{"Start":"08:19.945 ","End":"08:25.600","Text":"which was fortunate because 4 is a positive number."},{"Start":"08:25.600 ","End":"08:28.420","Text":"Therefore it has a square root and if it has a square root,"},{"Start":"08:28.420 ","End":"08:33.370","Text":"it actually has 2 square roots plus and minus of the other."},{"Start":"08:33.370 ","End":"08:36.955","Text":"That\u0027s what gave us, this divergence,"},{"Start":"08:36.955 ","End":"08:38.499","Text":"gave us the 2 solutions,"},{"Start":"08:38.499 ","End":"08:43.100","Text":"is because what was under the square root was positive."},{"Start":"08:45.870 ","End":"08:53.620","Text":"Let\u0027s just imagine that we had a minus here."},{"Start":"08:53.620 ","End":"08:56.335","Text":"I\u0027m not fiddling the arithmetic,"},{"Start":"08:56.335 ","End":"09:00.595","Text":"we\u0027re just supposing we had a minus here."},{"Start":"09:00.595 ","End":"09:03.700","Text":"How would it have turned out differently?"},{"Start":"09:03.700 ","End":"09:06.160","Text":"Would have got to here and said, okay,"},{"Start":"09:06.160 ","End":"09:12.400","Text":"x is something plus or minus the square root of minus 4 something, something."},{"Start":"09:12.400 ","End":"09:15.310","Text":"Then we said, what\u0027s the square root of minus 4?"},{"Start":"09:15.310 ","End":"09:18.745","Text":"Then we realize, oh, negative numbers don\u0027t have square roots."},{"Start":"09:18.745 ","End":"09:21.070","Text":"So we wouldn\u0027t have any solution,"},{"Start":"09:21.070 ","End":"09:23.245","Text":"which means there\u0027s no solution for x."},{"Start":"09:23.245 ","End":"09:25.075","Text":"If there is no solutions for x,"},{"Start":"09:25.075 ","End":"09:26.890","Text":"it means there is no x-intercept."},{"Start":"09:26.890 ","End":"09:29.410","Text":"It means we\u0027d be in this case,"},{"Start":"09:29.410 ","End":"09:33.910","Text":"no solution, does not cut the axis."},{"Start":"09:33.910 ","End":"09:36.115","Text":"That\u0027s what makes the difference."},{"Start":"09:36.115 ","End":"09:38.755","Text":"Now let\u0027s take a third scenario."},{"Start":"09:38.755 ","End":"09:43.010","Text":"Supposing that instead of this,"},{"Start":"09:43.230 ","End":"09:45.970","Text":"we had a 0 here."},{"Start":"09:45.970 ","End":"09:47.950","Text":"Then what would have happened?"},{"Start":"09:47.950 ","End":"09:50.320","Text":"Square root of 0, no problem."},{"Start":"09:50.320 ","End":"09:53.395","Text":"Positive or negative they have the square roots."},{"Start":"09:53.395 ","End":"09:56.875","Text":"Sorry, positive or 0 only the negative doesn\u0027t."},{"Start":"09:56.875 ","End":"10:02.230","Text":"We would\u0027ve said 4 plus or minus 0 over 2,"},{"Start":"10:02.230 ","End":"10:04.840","Text":"but plus or minus 0 is the same thing."},{"Start":"10:04.840 ","End":"10:07.705","Text":"So essentially you would have got the same answer here and here."},{"Start":"10:07.705 ","End":"10:09.580","Text":"In our case we would have had 2 and 2,"},{"Start":"10:09.580 ","End":"10:12.865","Text":"which is not really 2 separate answers."},{"Start":"10:12.865 ","End":"10:14.905","Text":"The answer would have just been 2."},{"Start":"10:14.905 ","End":"10:17.575","Text":"Would have had only 1 answer,"},{"Start":"10:17.575 ","End":"10:23.575","Text":"and we would\u0027ve been in the middle case with exactly 1 intercept,"},{"Start":"10:23.575 ","End":"10:26.305","Text":"which is this 1 over here."},{"Start":"10:26.305 ","End":"10:31.480","Text":"That\u0027s essentially what gives rise to the 3 different cases."},{"Start":"10:31.480 ","End":"10:37.850","Text":"What\u0027s under the square root can be positive, negative, or 0."},{"Start":"10:38.040 ","End":"10:42.940","Text":"We see that what\u0027s most important in determining which kind of"},{"Start":"10:42.940 ","End":"10:46.960","Text":"parabola we get is what\u0027s under the square root sign."},{"Start":"10:46.960 ","End":"10:48.220","Text":"We tried 4,"},{"Start":"10:48.220 ","End":"10:51.100","Text":"we tried minus 4, we tried 0."},{"Start":"10:51.100 ","End":"10:55.465","Text":"Let me just mark what\u0027s under the square root sign."},{"Start":"10:55.465 ","End":"10:59.215","Text":"In general, this b^2 minus 4ac,"},{"Start":"10:59.215 ","End":"11:03.610","Text":"highlight it in yellow and in our particular case that came out to be 4 though we tried"},{"Start":"11:03.610 ","End":"11:09.205","Text":"out also hypothetically both minus 4 and 0."},{"Start":"11:09.205 ","End":"11:15.415","Text":"This actually has a name and it\u0027s called the discriminant."},{"Start":"11:15.415 ","End":"11:19.670","Text":"Let me write that down."},{"Start":"11:20.730 ","End":"11:25.015","Text":"Where should I write it? Let\u0027s say right above here."},{"Start":"11:25.015 ","End":"11:32.695","Text":"That Delta is the b^2 minus 4ac."},{"Start":"11:32.695 ","End":"11:35.650","Text":"It\u0027s what\u0027s under the square root sign without the square root."},{"Start":"11:35.650 ","End":"11:38.000","Text":"That\u0027s Delta."},{"Start":"11:40.950 ","End":"11:43.255","Text":"Use a different color."},{"Start":"11:43.255 ","End":"11:47.390","Text":"That\u0027s called the discriminant."},{"Start":"11:51.930 ","End":"11:58.285","Text":"The discriminant of the quadratic function or of the quadratic equation."},{"Start":"11:58.285 ","End":"12:01.825","Text":"Discriminant Delta b^2 minus 4ac."},{"Start":"12:01.825 ","End":"12:05.540","Text":"In our case, Delta was equal to 4."},{"Start":"12:07.590 ","End":"12:11.245","Text":"Then we got this picture from Delta equals 4."},{"Start":"12:11.245 ","End":"12:18.310","Text":"Or in general, this is the kind of graph you would get where Delta is bigger than 0."},{"Start":"12:18.310 ","End":"12:25.450","Text":"This is the kind of thing we got only 3 solution to the quadratic equation,"},{"Start":"12:25.450 ","End":"12:28.420","Text":"which corresponds to this 1 cutting place."},{"Start":"12:28.420 ","End":"12:31.795","Text":"This was what happened when Delta was equal to 0,"},{"Start":"12:31.795 ","End":"12:34.344","Text":"and when Delta was less than 0,"},{"Start":"12:34.344 ","End":"12:36.100","Text":"we got no solutions at all."},{"Start":"12:36.100 ","End":"12:39.415","Text":"The parabola did not cut the x-axis."},{"Start":"12:39.415 ","End":"12:42.955","Text":"You could say it\u0027s kind of floating above."},{"Start":"12:42.955 ","End":"12:47.934","Text":"Of course, this is not the full picture because as you might remember,"},{"Start":"12:47.934 ","End":"12:52.345","Text":"there are also parabolas that face down,"},{"Start":"12:52.345 ","End":"12:58.615","Text":"that spill water, look like mountains and not valleys, whatever."},{"Start":"12:58.615 ","End":"13:05.170","Text":"Here we\u0027ll take the cases where a is less than 0."},{"Start":"13:05.170 ","End":"13:07.240","Text":"Remember, a can\u0027t be 0,"},{"Start":"13:07.240 ","End":"13:09.715","Text":"it\u0027s either greater than or less than."},{"Start":"13:09.715 ","End":"13:12.010","Text":"In this particular case,"},{"Start":"13:12.010 ","End":"13:17.470","Text":"we can get that if Delta is greater than 0,"},{"Start":"13:17.470 ","End":"13:19.030","Text":"there\u0027ll be 2 solutions,"},{"Start":"13:19.030 ","End":"13:27.370","Text":"but 2 solutions and it\u0027s facing down would give us something like this."},{"Start":"13:27.370 ","End":"13:30.530","Text":"Here\u0027s the 2 solutions."},{"Start":"13:31.050 ","End":"13:34.525","Text":"If we only had 1 solution,"},{"Start":"13:34.525 ","End":"13:36.684","Text":"which would happen if Delta was 0,"},{"Start":"13:36.684 ","End":"13:41.365","Text":"then it would look something like this."},{"Start":"13:41.365 ","End":"13:44.845","Text":"Okay. Not the greatest drawing just touches once,"},{"Start":"13:44.845 ","End":"13:50.800","Text":"or it could be 1 of those floating parabolas that doesn\u0027t intersect at all,"},{"Start":"13:50.800 ","End":"13:55.150","Text":"and it would look something like this, no intersection."},{"Start":"13:55.150 ","End":"13:58.070","Text":"I\u0027ll just label the others."},{"Start":"13:58.680 ","End":"14:01.900","Text":"This 1 would be, again,"},{"Start":"14:01.900 ","End":"14:03.670","Text":"Delta bigger than 0."},{"Start":"14:03.670 ","End":"14:05.395","Text":"Delta is the discriminant,"},{"Start":"14:05.395 ","End":"14:09.880","Text":"and here Delta is exactly equal to 0,"},{"Start":"14:09.880 ","End":"14:13.520","Text":"and here Delta is negative."},{"Start":"14:13.950 ","End":"14:16.480","Text":"These are the 6 basic cases."},{"Start":"14:16.480 ","End":"14:21.850","Text":"If you look at it classified according to a being positive,"},{"Start":"14:21.850 ","End":"14:24.640","Text":"then they all face up and then you get the cut twice,"},{"Start":"14:24.640 ","End":"14:27.985","Text":"once or no times at all according to the Delta."},{"Start":"14:27.985 ","End":"14:29.995","Text":"If a is less than 0,"},{"Start":"14:29.995 ","End":"14:31.930","Text":"then it\u0027s always facing down."},{"Start":"14:31.930 ","End":"14:34.585","Text":"But it could cut the x-axis either twice,"},{"Start":"14:34.585 ","End":"14:38.235","Text":"once or nonce again,"},{"Start":"14:38.235 ","End":"14:43.740","Text":"just according to the discriminant being positive, 2 solutions."},{"Start":"14:43.740 ","End":"14:47.855","Text":"0 1 solution or negative no solutions."},{"Start":"14:47.855 ","End":"14:52.130","Text":"That\u0027s basically the classification and all"},{"Start":"14:52.130 ","End":"14:56.340","Text":"I want to say at this point about a and Delta."},{"Start":"14:56.340 ","End":"14:59.690","Text":"Now I\u0027d like to give you a preview of what I\u0027m going to talk about next."},{"Start":"14:59.690 ","End":"15:03.730","Text":"Let\u0027s get back to our particular parabola,"},{"Start":"15:03.730 ","End":"15:10.770","Text":"the 1 we drew up here and notice that we\u0027ve got a lot of the important features in."},{"Start":"15:10.770 ","End":"15:14.020","Text":"We know which way up it\u0027s facing,"},{"Start":"15:14.020 ","End":"15:15.610","Text":"we got that from a."},{"Start":"15:15.610 ","End":"15:20.980","Text":"We know how many x-intercepts it has."},{"Start":"15:20.980 ","End":"15:23.005","Text":"In this case 2."},{"Start":"15:23.005 ","End":"15:28.990","Text":"We can sketch the y-intercept, the 2 x-intercepts."},{"Start":"15:28.990 ","End":"15:32.915","Text":"General shape, whether it\u0027s facing up or facing down."},{"Start":"15:32.915 ","End":"15:35.660","Text":"The most obvious thing that\u0027s missing,"},{"Start":"15:35.660 ","End":"15:42.544","Text":"if I was to say what\u0027s another important feature of this parabola,"},{"Start":"15:42.544 ","End":"15:47.985","Text":"what I would think is obviously missing would be the vertex,"},{"Start":"15:47.985 ","End":"15:51.755","Text":"the coordinates of this particular special point."},{"Start":"15:51.755 ","End":"15:56.285","Text":"In fact, that\u0027s what we\u0027re going to talk about in the next section."},{"Start":"15:56.285 ","End":"15:59.180","Text":"The vertex of the parabola."}],"ID":5115},{"Watched":false,"Name":"The Apex of the Parabola","Duration":"8m 28s","ChapterTopicVideoID":5134,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5134.jpeg","UploadDate":"2020-09-30T13:15:57.3900000","DurationForVideoObject":"PT8M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:03.900","Text":"Continuing with the quadratic function,"},{"Start":"00:03.900 ","End":"00:08.445","Text":"we\u0027re going to now talk more about the vertex of the parabola."},{"Start":"00:08.445 ","End":"00:16.125","Text":"The parabola being the equation of the quadratic function y=ax^2+bx+c."},{"Start":"00:16.125 ","End":"00:24.000","Text":"Now I\u0027d like to point out the very first thing is that every parabola has a vertex."},{"Start":"00:24.000 ","End":"00:26.610","Text":"I\u0027ll quickly illustrate that."},{"Start":"00:26.610 ","End":"00:31.005","Text":"Have an axis, have an axis."},{"Start":"00:31.005 ","End":"00:35.290","Text":"Not so good. Try again."},{"Start":"00:36.620 ","End":"00:39.730","Text":"A little bit better."},{"Start":"00:40.330 ","End":"00:44.420","Text":"Here\u0027s a parabola."},{"Start":"00:44.420 ","End":"00:47.495","Text":"It has a vertex here."},{"Start":"00:47.495 ","End":"00:50.045","Text":"This is an upward-facing parabola."},{"Start":"00:50.045 ","End":"00:56.195","Text":"Downward-facing parabola also has a vertex here."},{"Start":"00:56.195 ","End":"00:59.180","Text":"The difference is that with the upward-facing,"},{"Start":"00:59.180 ","End":"01:01.910","Text":"the vertex is actually a minimum."},{"Start":"01:01.910 ","End":"01:06.080","Text":"Which you\u0027ll learn more about when you do calculus."},{"Start":"01:06.080 ","End":"01:09.395","Text":"But its the lowest point is what I\u0027m trying to say."},{"Start":"01:09.395 ","End":"01:12.750","Text":"If it\u0027s a downward-facing parabola,"},{"Start":"01:12.750 ","End":"01:15.890","Text":"then the vertex is the maximum or the highest point,"},{"Start":"01:15.890 ","End":"01:19.400","Text":"the peak of the mountain, bottom of the valley."},{"Start":"01:19.400 ","End":"01:24.600","Text":"Let\u0027s call, I\u0027ll write that down because it\u0027s important."},{"Start":"01:28.460 ","End":"01:35.080","Text":"Every parabola has a vertex."},{"Start":"01:41.230 ","End":"01:43.490","Text":"Let\u0027s give it a name,"},{"Start":"01:43.490 ","End":"01:45.500","Text":"a pair of coordinates."},{"Start":"01:45.500 ","End":"01:48.549","Text":"Let\u0027s call it,"},{"Start":"01:48.549 ","End":"01:54.350","Text":"we\u0027ll give it a name like (x,"},{"Start":"01:54.350 ","End":"01:57.050","Text":"y) but will indicate that it belongs to the vertex."},{"Start":"01:57.050 ","End":"02:01.745","Text":"I\u0027ll put a little subscript v here, (x_v, y_v)."},{"Start":"02:01.745 ","End":"02:05.190","Text":"I want to point out that some books,"},{"Start":"02:05.920 ","End":"02:11.015","Text":"some people, some call it,"},{"Start":"02:11.015 ","End":"02:15.485","Text":"I\u0027ve seen this in the literature (h, k)."},{"Start":"02:15.485 ","End":"02:17.360","Text":"If you encounter that,"},{"Start":"02:17.360 ","End":"02:20.430","Text":"then you can use (h, k)."},{"Start":"02:20.440 ","End":"02:23.880","Text":"Now it has a formula."},{"Start":"02:25.790 ","End":"02:30.700","Text":"Sorry, it has a formula."},{"Start":"02:33.250 ","End":"02:42.270","Text":"The formula is, I\u0027ll give you separately for the x_v, sorry."},{"Start":"02:43.970 ","End":"02:52.120","Text":"Yes, the x of the vertex is minus b over 2a."},{"Start":"02:53.660 ","End":"02:58.850","Text":"What are b and a? These are the b and a that appear in the formula for"},{"Start":"02:58.850 ","End":"03:03.935","Text":"the quadratic function and y of the vertex."},{"Start":"03:03.935 ","End":"03:07.460","Text":"It has a specific equation,"},{"Start":"03:07.460 ","End":"03:11.490","Text":"but I\u0027m not going to give it to you because it\u0027s harder to remember."},{"Start":"03:12.100 ","End":"03:18.315","Text":"After all, if we have the x of a vertex,"},{"Start":"03:18.315 ","End":"03:21.395","Text":"and all we have to do is substitute in the function to get the y."},{"Start":"03:21.395 ","End":"03:23.030","Text":"We don\u0027t need an explicit formula."},{"Start":"03:23.030 ","End":"03:31.180","Text":"Let me write it as f(x) vertex and its simplest to explain with an example."},{"Start":"03:31.180 ","End":"03:34.120","Text":"Let me give you an example."},{"Start":"03:34.520 ","End":"03:42.990","Text":"I think I want to"},{"Start":"03:42.990 ","End":"03:52.875","Text":"highlight the x_v and y_v."},{"Start":"03:52.875 ","End":"03:55.990","Text":"I\u0027m sorry, I highlighted that."},{"Start":"03:57.890 ","End":"04:00.330","Text":"As an example,"},{"Start":"04:00.330 ","End":"04:09.430","Text":"we\u0027ll take the function y=x^2-4x+3."},{"Start":"04:10.640 ","End":"04:15.290","Text":"That\u0027s a parabola and in order to use the formula,"},{"Start":"04:15.290 ","End":"04:17.660","Text":"we should say what a, b, and c are."},{"Start":"04:17.660 ","End":"04:25.460","Text":"Here we have a is equal to the coefficient of x^2 is 1,"},{"Start":"04:25.460 ","End":"04:29.815","Text":"b equals coefficient of x-4,"},{"Start":"04:29.815 ","End":"04:33.980","Text":"and c equals the 3 coefficient is 3."},{"Start":"04:33.980 ","End":"04:44.135","Text":"I can say that my x of the vertex is equal to minus b over 2a from here,"},{"Start":"04:44.135 ","End":"04:53.100","Text":"which is minus b is minus 4 over 2 and a is 1."},{"Start":"04:53.100 ","End":"04:54.855","Text":"Altogether 4 over 2,"},{"Start":"04:54.855 ","End":"04:56.745","Text":"which is equal to 2."},{"Start":"04:56.745 ","End":"05:01.015","Text":"I can highlight that and say that in my case."},{"Start":"05:01.015 ","End":"05:03.705","Text":"I don\u0027t know what\u0027s with this marker."},{"Start":"05:03.705 ","End":"05:06.820","Text":"Anyway, x_v=2."},{"Start":"05:08.770 ","End":"05:12.965","Text":"These are the vertices also give them a little bit of blue here,"},{"Start":"05:12.965 ","End":"05:16.860","Text":"or at least that\u0027s the x, that\u0027s the vertex."},{"Start":"05:17.770 ","End":"05:24.450","Text":"Now, all we have to do is compute the y of the vertex and the y of the vertex we"},{"Start":"05:24.450 ","End":"05:31.155","Text":"get by substituting so y of the vertex."},{"Start":"05:31.155 ","End":"05:36.740","Text":"Let\u0027s just get rid of this excess blue."},{"Start":"05:36.740 ","End":"05:45.725","Text":"The y of the vertex is just f(x) of the vertex is f(2)."},{"Start":"05:45.725 ","End":"05:51.460","Text":"I think we could do this in our heads, that\u0027s f(2)."},{"Start":"05:52.190 ","End":"05:59.148","Text":"Well, this is f(x) so I should have said y is equal to,"},{"Start":"05:59.148 ","End":"06:02.040","Text":"when I say y I usually mean f(x) and vice versa."},{"Start":"06:02.040 ","End":"06:04.725","Text":"Okay, so f(2) is 2^2."},{"Start":"06:04.725 ","End":"06:12.030","Text":"I\u0027ll write it anyway 2^2-4(2)+3."},{"Start":"06:12.030 ","End":"06:17.550","Text":"Which is equal to 4-8+3."},{"Start":"06:17.550 ","End":"06:20.355","Text":"That looks like -1."},{"Start":"06:20.355 ","End":"06:22.714","Text":"Now we have the y vertex,"},{"Start":"06:22.714 ","End":"06:29.840","Text":"and I\u0027ll highlight that too. Very good."},{"Start":"06:29.840 ","End":"06:40.845","Text":"The vertex, let\u0027s write it separately, is (2,-1)."},{"Start":"06:40.845 ","End":"06:43.370","Text":"Now, this example is not randomly chosen."},{"Start":"06:43.370 ","End":"06:48.510","Text":"This is just a continuation of the example we had previously."},{"Start":"06:48.920 ","End":"06:53.900","Text":"I\u0027ll just refresh the picture we had before."},{"Start":"06:53.900 ","End":"06:56.570","Text":"Before we had something like this."},{"Start":"06:56.570 ","End":"07:03.245","Text":"We had y-axis and x-axis."},{"Start":"07:03.245 ","End":"07:05.945","Text":"We had all the intercepts."},{"Start":"07:05.945 ","End":"07:12.605","Text":"If you recall, we had that the y-intercept was 3."},{"Start":"07:12.605 ","End":"07:19.540","Text":"We had that the x-intercepts were 1 and 3."},{"Start":"07:19.540 ","End":"07:24.470","Text":"We knew that the parabola was facing upwards."},{"Start":"07:24.470 ","End":"07:31.800","Text":"Let\u0027s see my brave attempt to draw a parabola there."},{"Start":"07:31.900 ","End":"07:35.375","Text":"Let\u0027s see if we can get it to go through here."},{"Start":"07:35.375 ","End":"07:41.270","Text":"Not too bad. I\u0027ll leave it at that."},{"Start":"07:41.270 ","End":"07:46.294","Text":"Now we have an extra point of interest,"},{"Start":"07:46.294 ","End":"07:47.690","Text":"which is the vertex,"},{"Start":"07:47.690 ","End":"07:52.490","Text":"which was, as we said from here (2,-1)."},{"Start":"07:52.490 ","End":"07:55.955","Text":"Now that\u0027s really looking like a lot of information."},{"Start":"07:55.955 ","End":"08:01.100","Text":"Yeah. Again, we have the y-intercept,"},{"Start":"08:01.100 ","End":"08:04.130","Text":"2 x-intercepts, though we might not have to,"},{"Start":"08:04.130 ","End":"08:08.425","Text":"we might only have 1 or even none at all and the vertex."},{"Start":"08:08.425 ","End":"08:11.210","Text":"That already gives us quite a lot to start off drawing with."},{"Start":"08:11.210 ","End":"08:13.445","Text":"We also know that it\u0027s facing upwards."},{"Start":"08:13.445 ","End":"08:15.710","Text":"Again because the a was bigger than 0."},{"Start":"08:15.710 ","End":"08:17.120","Text":"I\u0027m not happy with this drawing."},{"Start":"08:17.120 ","End":"08:19.535","Text":"I\u0027m going to redraw it just a second."},{"Start":"08:19.535 ","End":"08:21.935","Text":"That\u0027s a bit better."},{"Start":"08:21.935 ","End":"08:29.220","Text":"Now, next, I\u0027d like to make some comments about the vertex."}],"ID":5116},{"Watched":false,"Name":"Exercise 1","Duration":"6m 59s","ChapterTopicVideoID":5125,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5125.jpeg","UploadDate":"2016-02-29T13:02:25.8300000","DurationForVideoObject":"PT6M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.140 ","End":"00:04.485","Text":"In this exercise, we have to graph the given parabola,"},{"Start":"00:04.485 ","End":"00:09.045","Text":"which includes labeling the vertex, the axis,"},{"Start":"00:09.045 ","End":"00:10.680","Text":"or the axis of symmetry,"},{"Start":"00:10.680 ","End":"00:15.670","Text":"and the intercepts, meaning x and y-intercepts."},{"Start":"00:16.250 ","End":"00:19.094","Text":"We\u0027ll do this in steps."},{"Start":"00:19.094 ","End":"00:24.659","Text":"First step is to see if the parabola is upward or downward facing."},{"Start":"00:24.659 ","End":"00:28.920","Text":"This we see immediately from the co-efficient of x^2,"},{"Start":"00:28.920 ","End":"00:31.605","Text":"which is minus 1, and if it\u0027s negative,"},{"Start":"00:31.605 ","End":"00:35.950","Text":"we know that the parabola is downward facing."},{"Start":"00:37.450 ","End":"00:39.560","Text":"I\u0027ll write that down."},{"Start":"00:39.560 ","End":"00:43.500","Text":"It\u0027s downward facing, meaning roughly something like this."},{"Start":"00:43.570 ","End":"00:48.125","Text":"Next step we\u0027ll do the intercepts."},{"Start":"00:48.125 ","End":"00:52.410","Text":"Let\u0027s take the x-intercept first."},{"Start":"00:54.070 ","End":"00:59.180","Text":"For the x-intercept, we need to let y=0,"},{"Start":"00:59.180 ","End":"01:03.061","Text":"and that gives us a quadratic equation,"},{"Start":"01:03.061 ","End":"01:11.825","Text":"minus x^2 plus 2x plus 3=0."},{"Start":"01:11.825 ","End":"01:14.280","Text":"Let\u0027s put in y=0."},{"Start":"01:14.280 ","End":"01:18.280","Text":"Now, we already know how to solve quadratic equations."},{"Start":"01:18.280 ","End":"01:28.635","Text":"I\u0027ll just tell you the 2 solutions for x. X can either be equal to minus 1 or to 3."},{"Start":"01:28.635 ","End":"01:32.520","Text":"Those are the 2 solutions to the quadratic equation,"},{"Start":"01:32.520 ","End":"01:39.750","Text":"which means that there are actually 2 x-intercepts."},{"Start":"01:39.750 ","End":"01:46.230","Text":"The x-intercepts"},{"Start":"01:46.230 ","End":"01:51.015","Text":"are the points minus 1,"},{"Start":"01:51.015 ","End":"01:58.905","Text":"0 and 3, 0."},{"Start":"01:58.905 ","End":"02:02.100","Text":"I can draw these on the graph already,"},{"Start":"02:02.100 ","End":"02:03.360","Text":"so minus 1,"},{"Start":"02:03.360 ","End":"02:05.805","Text":"0 would be here,"},{"Start":"02:05.805 ","End":"02:07.560","Text":"so this is minus 1,"},{"Start":"02:07.560 ","End":"02:10.930","Text":"and 1, 2, 3."},{"Start":"02:12.050 ","End":"02:16.615","Text":"3, 0 would be this point here."},{"Start":"02:16.615 ","End":"02:21.185","Text":"Now the y-intercept, let\u0027s call that Step 3."},{"Start":"02:21.185 ","End":"02:24.740","Text":"The y-intercept, you just let x=0."},{"Start":"02:24.740 ","End":"02:30.860","Text":"If x is 0, we get y=3."},{"Start":"02:30.860 ","End":"02:33.170","Text":"We can immediately say that"},{"Start":"02:33.170 ","End":"02:39.485","Text":"the y-intercept"},{"Start":"02:39.485 ","End":"02:43.890","Text":"is 0,"},{"Start":"02:43.890 ","End":"02:48.780","Text":"3 and 1,"},{"Start":"02:48.780 ","End":"02:51.785","Text":"2, 3, this is 3 here."},{"Start":"02:51.785 ","End":"02:56.070","Text":"That would be the y-intercept."},{"Start":"02:58.340 ","End":"03:04.685","Text":"Next, call it Step number 4 is the vertex."},{"Start":"03:04.685 ","End":"03:09.815","Text":"Well, the vertex has an x and a y."},{"Start":"03:09.815 ","End":"03:16.820","Text":"The formula is that the x of the vertex is given by minus b/2a,"},{"Start":"03:16.820 ","End":"03:18.740","Text":"where a, b,"},{"Start":"03:18.740 ","End":"03:21.215","Text":"and c are in the quadratic equation."},{"Start":"03:21.215 ","End":"03:25.005","Text":"Minus b/2a would be minus 2,"},{"Start":"03:25.005 ","End":"03:26.580","Text":"because our b is 2,"},{"Start":"03:26.580 ","End":"03:31.885","Text":"over 2a, which is twice minus 1."},{"Start":"03:31.885 ","End":"03:35.095","Text":"This comes out to be exactly 1."},{"Start":"03:35.095 ","End":"03:38.915","Text":"To find the y of the vertex,"},{"Start":"03:38.915 ","End":"03:42.920","Text":"we just substitute x=1 in the original equation."},{"Start":"03:42.920 ","End":"03:44.240","Text":"What if x is 1?"},{"Start":"03:44.240 ","End":"03:46.550","Text":"We get minus 1 plus 2 plus 3,"},{"Start":"03:46.550 ","End":"03:49.020","Text":"so y is 4."},{"Start":"03:50.630 ","End":"03:55.750","Text":"That means that the vertex"},{"Start":"03:58.940 ","End":"04:02.235","Text":"is given by 1,"},{"Start":"04:02.235 ","End":"04:06.600","Text":"4, and we can also draw that here,"},{"Start":"04:06.600 ","End":"04:10.890","Text":"let\u0027s say this is 1 and let\u0027s say this is 4,"},{"Start":"04:10.890 ","End":"04:17.500","Text":"so 1, 4 would be this point here, that\u0027s the vertex."},{"Start":"04:21.260 ","End":"04:30.340","Text":"Next step, call it 5, is the axis."},{"Start":"04:31.730 ","End":"04:37.470","Text":"The axis is the vertical line that goes through the vertex."},{"Start":"04:37.470 ","End":"04:40.320","Text":"Since it\u0027s a vertical line it\u0027s defined by x,"},{"Start":"04:40.320 ","End":"04:43.540","Text":"so the axis would be the line x=1."},{"Start":"04:48.560 ","End":"04:52.390","Text":"It will look like this."},{"Start":"04:52.900 ","End":"04:56.855","Text":"This is also called the axis of symmetry."},{"Start":"04:56.855 ","End":"05:00.835","Text":"The final thing we have to do is,"},{"Start":"05:00.835 ","End":"05:03.810","Text":"let\u0027s call it Step 6,"},{"Start":"05:03.810 ","End":"05:09.160","Text":"is to do a quick rough sketch."},{"Start":"05:09.470 ","End":"05:13.240","Text":"This would look something like this."},{"Start":"05:14.840 ","End":"05:19.065","Text":"That\u0027s it for Part A."},{"Start":"05:19.065 ","End":"05:21.995","Text":"Now, we come to Part B,"},{"Start":"05:21.995 ","End":"05:25.025","Text":"which to solve the inequality,"},{"Start":"05:25.025 ","End":"05:32.010","Text":"which is the same quadratic expression as in Part A,"},{"Start":"05:32.010 ","End":"05:42.275","Text":"it\u0027s just the same formula just this time as an inequality."},{"Start":"05:42.275 ","End":"05:48.479","Text":"Well, we\u0027ve done most of the work because the sketch really gives us everything."},{"Start":"05:49.260 ","End":"05:54.770","Text":"What we have to do is say which part of this graph is less than 0,"},{"Start":"05:54.770 ","End":"05:57.610","Text":"means below the x-axis?"},{"Start":"05:57.610 ","End":"06:02.449","Text":"We can sketch the parts that are below."},{"Start":"06:02.690 ","End":"06:12.720","Text":"That would be this part"},{"Start":"06:12.720 ","End":"06:16.245","Text":"here and this part here,"},{"Start":"06:16.245 ","End":"06:21.070","Text":"but not including these 2 points."},{"Start":"06:21.810 ","End":"06:27.850","Text":"We see, I\u0027ve highlighted the part of the graph that\u0027s below the x-axis."},{"Start":"06:27.850 ","End":"06:33.150","Text":"This corresponds to x bigger than 3,"},{"Start":"06:33.150 ","End":"06:35.355","Text":"but not including, remember the 3,"},{"Start":"06:35.355 ","End":"06:41.260","Text":"or x less than minus 1."},{"Start":"06:41.260 ","End":"06:46.760","Text":"Once we\u0027ve done the first part of the exercise and sketch,"},{"Start":"06:46.760 ","End":"06:50.400","Text":"the graph it\u0027s very easy to solve the inequality."},{"Start":"06:51.260 ","End":"06:54.750","Text":"X less than minus 1 or x bigger than 3,"},{"Start":"06:54.750 ","End":"06:58.570","Text":"that\u0027s the solution for Part B. We\u0027re done."}],"ID":5117},{"Watched":false,"Name":"Exercise 2","Duration":"5m 59s","ChapterTopicVideoID":5126,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5126.jpeg","UploadDate":"2016-02-29T13:02:59.4530000","DurationForVideoObject":"PT5M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:01.580 ","End":"00:04.500","Text":"In part a of this exercise,"},{"Start":"00:04.500 ","End":"00:08.775","Text":"we have to graph the parabola from its equation,"},{"Start":"00:08.775 ","End":"00:11.175","Text":"and we have to label the vertex,"},{"Start":"00:11.175 ","End":"00:12.975","Text":"the axis of symmetry,"},{"Start":"00:12.975 ","End":"00:14.680","Text":"and the intercepts,"},{"Start":"00:14.680 ","End":"00:17.460","Text":"where it cuts the x and y-axis."},{"Start":"00:17.460 ","End":"00:19.680","Text":"The first thing to do, however,"},{"Start":"00:19.680 ","End":"00:24.435","Text":"is to decide whether the parabola is upward or downward-facing."},{"Start":"00:24.435 ","End":"00:27.780","Text":"We can see by the co-efficient of x squared 2,"},{"Start":"00:27.780 ","End":"00:33.435","Text":"which is positive that the parabola is upward facing."},{"Start":"00:33.435 ","End":"00:35.310","Text":"I\u0027ll write it as upward,"},{"Start":"00:35.310 ","End":"00:39.460","Text":"and I\u0027ll show you what this means like this and not the other way around."},{"Start":"00:39.460 ","End":"00:43.945","Text":"Next, let\u0027s find the x-intercept."},{"Start":"00:43.945 ","End":"00:47.015","Text":"We\u0027ll call that part 2."},{"Start":"00:47.015 ","End":"00:52.175","Text":"For the x-intercept, it means where it cuts the x-axis,"},{"Start":"00:52.175 ","End":"00:53.555","Text":"that\u0027s where y is 0."},{"Start":"00:53.555 ","End":"00:56.075","Text":"I just substitute y equals 0 here,"},{"Start":"00:56.075 ","End":"01:05.915","Text":"and it gives us a quadratic equation, 2x squared plus 3x minus 5 equals 0."},{"Start":"01:05.915 ","End":"01:09.169","Text":"I\u0027m assuming you know how to solve quadratic equations."},{"Start":"01:09.169 ","End":"01:11.150","Text":"I won\u0027t go into all the details."},{"Start":"01:11.150 ","End":"01:12.730","Text":"There are 2 solutions,"},{"Start":"01:12.730 ","End":"01:21.710","Text":"x can be either be equal minus 2.5 or 1."},{"Start":"01:21.710 ","End":"01:25.560","Text":"That means that the x-intercepts"},{"Start":"01:29.390 ","End":"01:35.115","Text":"in the plural are minus 2.5,"},{"Start":"01:35.115 ","End":"01:39.855","Text":"0 and 1, 0."},{"Start":"01:39.855 ","End":"01:41.920","Text":"The y is 0."},{"Start":"01:41.960 ","End":"01:44.880","Text":"If I sketch these, 1,"},{"Start":"01:44.880 ","End":"01:50.400","Text":"0 comes out somewhere here and minus 2.5,"},{"Start":"01:50.400 ","End":"01:53.310","Text":"0 would come out here,"},{"Start":"01:53.310 ","End":"01:56.535","Text":"label this minus 2.5."},{"Start":"01:56.535 ","End":"01:59.715","Text":"That\u0027s done with the x-intercept."},{"Start":"01:59.715 ","End":"02:04.180","Text":"Part 3, we\u0027ll do the y-intercept."},{"Start":"02:06.770 ","End":"02:10.940","Text":"This is much easier because the y-intercept is where"},{"Start":"02:10.940 ","End":"02:14.920","Text":"the curve hits the y-axis and that\u0027s where x is 0."},{"Start":"02:14.920 ","End":"02:17.525","Text":"We just put x equals 0 in this equation,"},{"Start":"02:17.525 ","End":"02:21.450","Text":"and we see immediately that y equals minus 5."},{"Start":"02:21.450 ","End":"02:25.350","Text":"If y is minus 5,"},{"Start":"02:25.940 ","End":"02:32.470","Text":"that indicates that we want the point 0, minus 5."},{"Start":"02:32.470 ","End":"02:37.230","Text":"Now, I\u0027ll sketch these on the coordinate plane."},{"Start":"02:37.810 ","End":"02:41.135","Text":"There\u0027s our y-intercept."},{"Start":"02:41.135 ","End":"02:43.860","Text":"Had to scroll down a bit."},{"Start":"02:45.470 ","End":"02:54.240","Text":"Now, the x of"},{"Start":"02:54.240 ","End":"02:58.815","Text":"the vertex is given by minus b over 2a,"},{"Start":"02:58.815 ","End":"03:02.520","Text":"in this case, minus 3"},{"Start":"03:02.520 ","End":"03:09.945","Text":"over 2 times 2,"},{"Start":"03:09.945 ","End":"03:17.310","Text":"which is minus 3/4."},{"Start":"03:17.310 ","End":"03:20.470","Text":"The y of the vertex"},{"Start":"03:22.010 ","End":"03:28.880","Text":"is determined by substituting the x into the original equation."},{"Start":"03:28.880 ","End":"03:33.800","Text":"If I put x equals minus 3/4 here,"},{"Start":"03:33.800 ","End":"03:38.790","Text":"a quick computation will show you that y is equal to"},{"Start":"03:38.790 ","End":"03:46.170","Text":"minus 6 and 1/8."},{"Start":"03:46.170 ","End":"03:49.585","Text":"I\u0027ll need to scroll down a bit to sketch that."},{"Start":"03:49.585 ","End":"03:52.710","Text":"This is the vertex."},{"Start":"03:52.710 ","End":"03:59.730","Text":"A vertical line through this will give us our axis, so number 5."},{"Start":"03:59.930 ","End":"04:06.000","Text":"The axis is just the vertical line given"},{"Start":"04:06.000 ","End":"04:11.940","Text":"by x equals minus 3/4 and if I sketch,"},{"Start":"04:11.940 ","End":"04:17.740","Text":"it will look like this roughly."},{"Start":"04:23.450 ","End":"04:31.350","Text":"The final part is to draw the sketch of the whole thing of the graph."},{"Start":"04:33.640 ","End":"04:37.465","Text":"I\u0027ll join a line through these points."},{"Start":"04:37.465 ","End":"04:40.540","Text":"We get something like this."},{"Start":"04:41.960 ","End":"04:46.600","Text":"We\u0027re done with part a."},{"Start":"04:47.750 ","End":"04:52.975","Text":"For part b, most of the work has been done in the sketch."},{"Start":"04:52.975 ","End":"04:58.200","Text":"We have to find where this quadratic expression is less than or equal to 0."},{"Start":"04:58.200 ","End":"05:00.470","Text":"Let\u0027s look at the sketch."},{"Start":"05:00.470 ","End":"05:03.295","Text":"Just scrolling down again,"},{"Start":"05:03.295 ","End":"05:08.740","Text":"we have to see which parts of the graph is less than or equal to 0."},{"Start":"05:08.740 ","End":"05:13.550","Text":"That means on the axis or below it."},{"Start":"05:13.590 ","End":"05:24.045","Text":"Let\u0027s see this part here which I\u0027m highlighting up to and including the points here."},{"Start":"05:24.045 ","End":"05:27.830","Text":"If I draw this on the axes,"},{"Start":"05:27.830 ","End":"05:36.315","Text":"that will come out to be from here to here and everything in the middle."},{"Start":"05:36.315 ","End":"05:41.930","Text":"This is the part, and we can see that its equation would"},{"Start":"05:41.930 ","End":"05:48.350","Text":"be minus 2.5 less than or equal to x,"},{"Start":"05:48.350 ","End":"05:50.915","Text":"less than or equal to 1."},{"Start":"05:50.915 ","End":"05:58.710","Text":"This is the solution to part b, and we\u0027re done."}],"ID":5118},{"Watched":false,"Name":"Exercise 3","Duration":"5m 24s","ChapterTopicVideoID":5127,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5127.jpeg","UploadDate":"2016-02-29T13:03:29.5170000","DurationForVideoObject":"PT5M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:03.660","Text":"In Part a of this exercise,"},{"Start":"00:03.660 ","End":"00:10.130","Text":"we have to graph the parabola with the given equation and to label the vertex,"},{"Start":"00:10.130 ","End":"00:14.770","Text":"the axis is symmetry on the x and y-intercepts."},{"Start":"00:15.820 ","End":"00:23.164","Text":"Let\u0027s start off first with deciding whether the parabola is upward or downward-facing."},{"Start":"00:23.164 ","End":"00:25.805","Text":"We examine the coefficient of x^2,"},{"Start":"00:25.805 ","End":"00:27.110","Text":"we see that it\u0027s 1."},{"Start":"00:27.110 ","End":"00:36.510","Text":"That\u0027s a positive number so the parabola is upward facing, something like this."},{"Start":"00:36.510 ","End":"00:40.080","Text":"Next step, x-intercept."},{"Start":"00:40.080 ","End":"00:41.430","Text":"For the x-intercept,"},{"Start":"00:41.430 ","End":"00:45.435","Text":"we have to let y equals 0 because we\u0027re on the x-axis,"},{"Start":"00:45.435 ","End":"00:50.140","Text":"and so we get a quadratic equation, x squared plus x plus 1 is equal to 0."},{"Start":"00:54.560 ","End":"00:59.790","Text":"Now when we start solving it, by the formula,"},{"Start":"00:59.790 ","End":"01:09.370","Text":"minus 1 plus or minus square root of b squared minus 4ac comes out to be 1 minus 4."},{"Start":"01:09.940 ","End":"01:12.410","Text":"We don\u0027t even have to go on anymore."},{"Start":"01:12.410 ","End":"01:14.915","Text":"We see that what\u0027s under the square root is negative,"},{"Start":"01:14.915 ","End":"01:17.885","Text":"so there are no solutions."},{"Start":"01:17.885 ","End":"01:24.750","Text":"There is no x-intercept,"},{"Start":"01:26.480 ","End":"01:32.610","Text":"which means that the graph doesn\u0027t cut the x-axis at all."},{"Start":"01:32.950 ","End":"01:36.870","Text":"Let\u0027s see about the y-intercept."},{"Start":"01:40.130 ","End":"01:47.439","Text":"For the y-intercept,"},{"Start":"01:47.439 ","End":"01:52.350","Text":"we let x equals 0 because where it cuts the y-axis,"},{"Start":"01:52.350 ","End":"01:56.970","Text":"x equals 0 and we just substitute and we get y equals 1."},{"Start":"01:57.890 ","End":"02:03.110","Text":"It\u0027s equal to 0,1 because when we substitute x equals 0,"},{"Start":"02:03.110 ","End":"02:04.850","Text":"we get y equals 1,"},{"Start":"02:04.850 ","End":"02:10.475","Text":"and this we can plot on the graph, x equals 0, y equals 1."},{"Start":"02:10.475 ","End":"02:13.195","Text":"Somewhere here."},{"Start":"02:13.195 ","End":"02:18.970","Text":"No x-intercept and only 1 y-intercept."},{"Start":"02:18.980 ","End":"02:22.335","Text":"Next, the vertex."},{"Start":"02:22.335 ","End":"02:24.540","Text":"Let\u0027s call that Part 4."},{"Start":"02:24.540 ","End":"02:27.315","Text":"The vertex has an x and a y."},{"Start":"02:27.315 ","End":"02:30.330","Text":"Let\u0027s see the x of the vertex."},{"Start":"02:30.330 ","End":"02:33.465","Text":"There\u0027s a formula, minus b over 2a."},{"Start":"02:33.465 ","End":"02:40.610","Text":"In our case, we get minus 1/2 and the y of the vertex,"},{"Start":"02:40.610 ","End":"02:45.115","Text":"we get by substituting this value in here."},{"Start":"02:45.115 ","End":"02:47.430","Text":"I won\u0027t do all the computation,"},{"Start":"02:47.430 ","End":"02:48.690","Text":"I\u0027ll just give you the answer."},{"Start":"02:48.690 ","End":"02:51.645","Text":"It comes out to be 3/4."},{"Start":"02:51.645 ","End":"03:01.583","Text":"The vertex is minus 1/2,"},{"Start":"03:01.583 ","End":"03:05.295","Text":"3/4, which comes out to be,"},{"Start":"03:05.295 ","End":"03:10.395","Text":"minus 1/2 is here and 3/4 is somewhere here,"},{"Start":"03:10.395 ","End":"03:15.220","Text":"so it\u0027s somewhere around here."},{"Start":"03:17.360 ","End":"03:24.545","Text":"Let me just indicate that this is minus 1/2,"},{"Start":"03:24.545 ","End":"03:27.240","Text":"and that this is 3/4."},{"Start":"03:28.630 ","End":"03:31.475","Text":"Once we have the vertex,"},{"Start":"03:31.475 ","End":"03:35.970","Text":"we immediately can get the axis of symmetry."},{"Start":"03:38.920 ","End":"03:47.300","Text":"The equation of the axis of symmetry is x equals minus 1/2."},{"Start":"03:47.300 ","End":"03:55.885","Text":"The axis is given by the equation x equals minus 1/2,"},{"Start":"03:55.885 ","End":"04:01.260","Text":"which is a vertical line going through the vertex, something like this."},{"Start":"04:02.440 ","End":"04:08.670","Text":"Finally, we need a sketch."},{"Start":"04:09.220 ","End":"04:11.900","Text":"We don\u0027t have very many points."},{"Start":"04:11.900 ","End":"04:15.035","Text":"We have the y-intercept and the vertex."},{"Start":"04:15.035 ","End":"04:18.365","Text":"That would be 1 of them is here, and 1 of them is here."},{"Start":"04:18.365 ","End":"04:20.960","Text":"We have an axis, not really enough to go on,"},{"Start":"04:20.960 ","End":"04:23.770","Text":"but we only needed something rough."},{"Start":"04:23.770 ","End":"04:30.525","Text":"How about something like this."},{"Start":"04:30.525 ","End":"04:34.810","Text":"That will do. That\u0027s done for Part a."},{"Start":"04:35.360 ","End":"04:41.540","Text":"Now Part b is not going to be difficult because we\u0027ve"},{"Start":"04:41.540 ","End":"04:48.035","Text":"already done most of the work when we drew the graph of x squared plus x plus 1."},{"Start":"04:48.035 ","End":"04:52.280","Text":"We just are asking which part of the graph is negative, in other words,"},{"Start":"04:52.280 ","End":"04:55.640","Text":"below the x-axis and if we look at the graph,"},{"Start":"04:55.640 ","End":"04:59.290","Text":"we see there is no part below the x-axis."},{"Start":"04:59.290 ","End":"05:04.250","Text":"So we could either say that in Part b,"},{"Start":"05:04.670 ","End":"05:09.840","Text":"there is no such x or we can"},{"Start":"05:09.840 ","End":"05:17.870","Text":"write empty set for the answer."},{"Start":"05:17.870 ","End":"05:23.670","Text":"Either way, it means that there is no such x, and we\u0027re done."}],"ID":5119},{"Watched":false,"Name":"Exercise 4","Duration":"5m 57s","ChapterTopicVideoID":5128,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5128.jpeg","UploadDate":"2016-02-29T13:04:03.2330000","DurationForVideoObject":"PT5M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.650 ","End":"00:03.825","Text":"In part A of this exercise,"},{"Start":"00:03.825 ","End":"00:07.455","Text":"we have to sketch the parabola with this equation."},{"Start":"00:07.455 ","End":"00:09.930","Text":"We have to label the vertex,"},{"Start":"00:09.930 ","End":"00:11.535","Text":"the axis of symmetry,"},{"Start":"00:11.535 ","End":"00:14.620","Text":"and the x and y-intercepts."},{"Start":"00:15.380 ","End":"00:21.570","Text":"The first step in sketching is to know which way up the parabola is."},{"Start":"00:21.570 ","End":"00:24.630","Text":"We look at the coefficient of x^2."},{"Start":"00:24.630 ","End":"00:27.060","Text":"We see it\u0027s minus 1 which is negative."},{"Start":"00:27.060 ","End":"00:31.840","Text":"We know that the parabola is a downward facing 1."},{"Start":"00:32.230 ","End":"00:36.440","Text":"Which means has a general shape like this."},{"Start":"00:37.230 ","End":"00:40.610","Text":"Now, let\u0027s go to the intercept."},{"Start":"00:40.610 ","End":"00:44.550","Text":"Let\u0027s start off with the x-intercept."},{"Start":"00:46.420 ","End":"00:54.680","Text":"The x-intercept, or the may be more than 1 or may not be any."},{"Start":"00:54.680 ","End":"00:57.960","Text":"We let y equals 0."},{"Start":"00:57.960 ","End":"01:00.540","Text":"That\u0027s where the graph hits the x-axis."},{"Start":"01:00.540 ","End":"01:05.860","Text":"Then we get a quadratic equation minus x^2 plus 4x minus 5 equals to 0."},{"Start":"01:10.960 ","End":"01:18.440","Text":"We start to solve it using the quadratic formula and say that x is minus 4 plus or"},{"Start":"01:18.440 ","End":"01:25.865","Text":"minus the square root of b^2 which is 16 minus 4ac,"},{"Start":"01:25.865 ","End":"01:33.890","Text":"which is minus 4 times minus 1 times minus 5."},{"Start":"01:33.890 ","End":"01:38.975","Text":"Already we see that what\u0027s under the square root sign is 16 minus 20."},{"Start":"01:38.975 ","End":"01:42.845","Text":"This whole thing is minus 4."},{"Start":"01:42.845 ","End":"01:47.080","Text":"It\u0027s a negative number, it doesn\u0027t have any square roots."},{"Start":"01:47.080 ","End":"01:49.970","Text":"Since there\u0027s no solution to this exercise,"},{"Start":"01:49.970 ","End":"01:52.760","Text":"basically, there are no x-intercepts."},{"Start":"01:52.760 ","End":"01:54.830","Text":"When I write x intercepts none,"},{"Start":"01:54.830 ","End":"01:58.130","Text":"the graph does not cut the x axis."},{"Start":"01:58.130 ","End":"02:01.070","Text":"Well, let\u0027s try the y-intercept."},{"Start":"02:01.070 ","End":"02:03.200","Text":"Y-intercept is easier."},{"Start":"02:03.200 ","End":"02:04.969","Text":"That\u0027s where it cuts the y-axis."},{"Start":"02:04.969 ","End":"02:06.710","Text":"In other words, x is 0."},{"Start":"02:06.710 ","End":"02:13.460","Text":"Just substitute x equals 0 here and get y equals minus 5."},{"Start":"02:13.460 ","End":"02:22.230","Text":"In other words, 0 minus 5 is the y intercept."},{"Start":"02:27.230 ","End":"02:31.380","Text":"I\u0027ll plot that. I have to scroll down a bit."},{"Start":"02:31.550 ","End":"02:35.010","Text":"There\u0027s our y-intercept."},{"Start":"02:35.010 ","End":"02:39.100","Text":"The next thing is to find the vertex."},{"Start":"02:40.010 ","End":"02:45.610","Text":"Let\u0027s call that part 4, the vertex."},{"Start":"02:49.220 ","End":"02:54.250","Text":"Start off with the x of the vertex."},{"Start":"02:56.450 ","End":"02:59.970","Text":"Let\u0027s write x_v, the x of the vertex."},{"Start":"02:59.970 ","End":"03:03.840","Text":"This is given by the general formula minus b over 2a,"},{"Start":"03:03.840 ","End":"03:05.835","Text":"which, in our case,"},{"Start":"03:05.835 ","End":"03:08.400","Text":"minus b is minus 4,"},{"Start":"03:08.400 ","End":"03:12.150","Text":"and 2a is minus 2,"},{"Start":"03:12.150 ","End":"03:17.680","Text":"twice minus 1 and that gives us 2."},{"Start":"03:22.760 ","End":"03:30.830","Text":"Scroll a bit. The y of the vertex is obtained by substituting 2 in the original equation."},{"Start":"03:30.830 ","End":"03:32.300","Text":"We can see it here."},{"Start":"03:32.300 ","End":"03:34.655","Text":"If we put x equals 2,"},{"Start":"03:34.655 ","End":"03:38.780","Text":"we get minus 4 plus 8."},{"Start":"03:38.780 ","End":"03:41.690","Text":"We\u0027re up to plus 4 minus 5,"},{"Start":"03:41.690 ","End":"03:44.460","Text":"which is minus 1."},{"Start":"03:44.500 ","End":"03:47.225","Text":"In other words, the vertex,"},{"Start":"03:47.225 ","End":"03:48.425","Text":"if I write it here,"},{"Start":"03:48.425 ","End":"03:53.945","Text":"will be 2, minus 1."},{"Start":"03:53.945 ","End":"03:56.370","Text":"I can sketch that."},{"Start":"03:56.540 ","End":"03:59.760","Text":"This is 2, this is minus 1,"},{"Start":"03:59.760 ","End":"04:03.760","Text":"our point of the vertex will be here."},{"Start":"04:06.030 ","End":"04:09.265","Text":"Once we have the vertex,"},{"Start":"04:09.265 ","End":"04:12.400","Text":"we can easily find the axis."},{"Start":"04:12.400 ","End":"04:16.600","Text":"The axis of symmetry is a vertical line through the vertex."},{"Start":"04:16.600 ","End":"04:20.740","Text":"Since the vertex has a coordinate x equals 2,"},{"Start":"04:20.740 ","End":"04:26.080","Text":"the vertical line for the axis will be given by the equation x equals 2."},{"Start":"04:26.080 ","End":"04:27.598","Text":"If we sketch it,"},{"Start":"04:27.598 ","End":"04:31.810","Text":"it will look something like this here."},{"Start":"04:32.090 ","End":"04:37.070","Text":"All we\u0027re left with now is to draw the sketch."},{"Start":"04:38.520 ","End":"04:41.650","Text":"Although we don\u0027t really have very many points,"},{"Start":"04:41.650 ","End":"04:45.385","Text":"we have the vertex and we have the y-intercept."},{"Start":"04:45.385 ","End":"04:48.100","Text":"But we just need this roughly."},{"Start":"04:48.100 ","End":"04:53.300","Text":"How about something like this?"},{"Start":"04:53.300 ","End":"04:57.450","Text":"That\u0027s done with the sketch."},{"Start":"04:58.120 ","End":"05:01.325","Text":"For part b, we have to solve"},{"Start":"05:01.325 ","End":"05:07.115","Text":"the inequality minus x squared plus 4x minus 5 is less than 0."},{"Start":"05:07.115 ","End":"05:12.010","Text":"But we\u0027ve already sketched the graph of y equals this thing."},{"Start":"05:12.010 ","End":"05:13.700","Text":"That\u0027s down here."},{"Start":"05:13.700 ","End":"05:17.435","Text":"Let me just scroll down a little bit."},{"Start":"05:17.435 ","End":"05:23.825","Text":"We have to find where this thing is less than 0."},{"Start":"05:23.825 ","End":"05:29.240","Text":"Less than 0 means below the x-axis."},{"Start":"05:29.240 ","End":"05:36.725","Text":"The whole graph is below the x-axis because it\u0027s downward facing and its vertex is here."},{"Start":"05:36.725 ","End":"05:39.425","Text":"The whole thing is below the x-axis."},{"Start":"05:39.425 ","End":"05:47.110","Text":"The answer for part b would be all x."},{"Start":"05:47.120 ","End":"05:53.540","Text":"For every x, the quadratic expression will be negative."},{"Start":"05:53.540 ","End":"05:55.950","Text":"That\u0027s all there is to it."}],"ID":5120},{"Watched":false,"Name":"Exercise 5","Duration":"8m 55s","ChapterTopicVideoID":5129,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5129.jpeg","UploadDate":"2016-02-29T13:04:53.2670000","DurationForVideoObject":"PT8M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:01.460 ","End":"00:07.109","Text":"In part a, we have to graph a parabola and a line."},{"Start":"00:07.109 ","End":"00:13.030","Text":"We\u0027ll start with the parabola and use the same techniques as we used earlier."},{"Start":"00:13.520 ","End":"00:19.560","Text":"First thing we will do is say whether it\u0027s facing upwards or downwards."},{"Start":"00:19.560 ","End":"00:23.654","Text":"Look at the coefficient of x^2 is 1, which is positive."},{"Start":"00:23.654 ","End":"00:27.675","Text":"The parabola is upward facing."},{"Start":"00:27.675 ","End":"00:30.375","Text":"Something like this."},{"Start":"00:30.375 ","End":"00:34.900","Text":"Next step was finding the intercepts."},{"Start":"00:35.900 ","End":"00:43.500","Text":"For the x-intercept,"},{"Start":"00:43.500 ","End":"00:50.610","Text":"we let y= 0 and essentially solve the quadratic equation,"},{"Start":"01:00.310 ","End":"01:04.789","Text":"x^2+3x+2=0. I assume you know how to solve quadratic equations,"},{"Start":"01:04.789 ","End":"01:20.870","Text":"so I\u0027ll just spoil the surprise and give you the answers that x=-2"},{"Start":"01:20.870 ","End":"01:25.230","Text":", or x=-1."},{"Start":"01:25.780 ","End":"01:28.970","Text":"That\u0027s for the x-intercept."},{"Start":"01:28.970 ","End":"01:36.740","Text":"For the y-intercept, we let x=0 and we get y=2."},{"Start":"01:36.740 ","End":"01:43.590","Text":"We get that (0,2) is the y-intercept."},{"Start":"01:48.340 ","End":"01:52.490","Text":"If I plot the intercepts,"},{"Start":"01:52.490 ","End":"02:01.040","Text":"I\u0027ll get (0,2) that\u0027s going to be the y-intercept."},{"Start":"02:01.040 ","End":"02:04.532","Text":"Well, this gives us an intercept of"},{"Start":"02:04.532 ","End":"02:13.250","Text":"(-1,0), or (-2,0)."},{"Start":"02:13.250 ","End":"02:14.719","Text":"These are the 2 intercepts,"},{"Start":"02:14.719 ","End":"02:21.930","Text":"so (-1,0) is here and (-2,0) is here."},{"Start":"02:24.070 ","End":"02:29.375","Text":"Next thing is the vertex."},{"Start":"02:29.375 ","End":"02:36.050","Text":"Now there is a bit of a shortcut we can use in finding the x of the vertex."},{"Start":"02:36.050 ","End":"02:40.460","Text":"It\u0027s when we have two x-intercepts,"},{"Start":"02:40.460 ","End":"02:45.230","Text":"like here we have -1 and -2, at this point,"},{"Start":"02:45.230 ","End":"02:47.420","Text":"let me just also add the 2 here,"},{"Start":"02:47.420 ","End":"02:57.425","Text":"then the vertex lies halfway between the two x-intercepts."},{"Start":"02:57.425 ","End":"03:03.440","Text":"The x_v is the average of -2 and -1,"},{"Start":"03:03.440 ","End":"03:08.195","Text":"which is -1.5 or minus 1/2."},{"Start":"03:08.195 ","End":"03:18.785","Text":"If we substitute this value into the original equation, x^2+3x+2,"},{"Start":"03:18.785 ","End":"03:28.610","Text":"then we get -0.25"},{"Start":"03:28.610 ","End":"03:31.650","Text":"if we\u0027ve already started with decimals."},{"Start":"03:32.050 ","End":"03:38.045","Text":"Let me scroll a bit."},{"Start":"03:38.045 ","End":"03:44.810","Text":"The vertex will be the x of"},{"Start":"03:44.810 ","End":"03:54.005","Text":"the vertex -1.5 and -0.25 for the y,"},{"Start":"03:54.005 ","End":"04:00.140","Text":"which will be, this is -1.5 and this is about 1/4 of the way down."},{"Start":"04:00.140 ","End":"04:05.180","Text":"Say here, we get these 3 points kind of close together,"},{"Start":"04:05.180 ","End":"04:07.830","Text":"together with this one."},{"Start":"04:08.560 ","End":"04:13.931","Text":"Now I will try and sketch by drawing a line through these 1,"},{"Start":"04:13.931 ","End":"04:16.280","Text":"2, 3, 4 points that we have."},{"Start":"04:16.280 ","End":"04:18.870","Text":"We\u0027ll get something like this,"},{"Start":"04:19.480 ","End":"04:23.015","Text":"something like this for the parabola."},{"Start":"04:23.015 ","End":"04:26.369","Text":"Now let\u0027s look at the line."},{"Start":"04:26.470 ","End":"04:30.275","Text":"Again, we can do the intercepts."},{"Start":"04:30.275 ","End":"04:37.745","Text":"For the line, when x is 0, y is 5."},{"Start":"04:37.745 ","End":"04:41.750","Text":"We know that when x is 0,"},{"Start":"04:41.750 ","End":"04:42.830","Text":"y is 5,"},{"Start":"04:42.830 ","End":"04:44.525","Text":"that\u0027s gonna be on the graph."},{"Start":"04:44.525 ","End":"04:47.120","Text":"And if y is 0,"},{"Start":"04:47.120 ","End":"04:49.910","Text":"then 5 plus x is 0,"},{"Start":"04:49.910 ","End":"04:52.250","Text":"so x becomes -5."},{"Start":"04:52.250 ","End":"04:56.130","Text":"So we get the point (-5,0)."},{"Start":"04:57.220 ","End":"04:59.300","Text":"If I plot those,"},{"Start":"04:59.300 ","End":"05:01.637","Text":"let\u0027s see (0,5) 1,"},{"Start":"05:01.637 ","End":"05:02.990","Text":"2, 3, 4, 5."},{"Start":"05:02.990 ","End":"05:04.280","Text":"It would be here,"},{"Start":"05:04.280 ","End":"05:10.695","Text":"-5 here and (-5,0),"},{"Start":"05:10.695 ","End":"05:12.590","Text":"let\u0027s see, 1, 2, 3, 4, 5."},{"Start":"05:12.590 ","End":"05:15.770","Text":"I\u0027m not going to label it everything\u0027s crowded here,"},{"Start":"05:15.770 ","End":"05:19.945","Text":"but this is the other point on the line."},{"Start":"05:19.945 ","End":"05:24.440","Text":"If I sketch a straight line through these two points,"},{"Start":"05:24.440 ","End":"05:26.975","Text":"I should get something like this."},{"Start":"05:26.975 ","End":"05:29.840","Text":"Here we are green one."},{"Start":"05:29.840 ","End":"05:32.160","Text":"Parabola is in red."},{"Start":"05:32.440 ","End":"05:37.620","Text":"Now turning to part B."},{"Start":"05:41.260 ","End":"05:44.970","Text":"Wrong color, never mind."},{"Start":"05:46.560 ","End":"05:54.940","Text":"We have to solve the inequality that the parabola is bigger than the line."},{"Start":"05:54.940 ","End":"05:56.620","Text":"That\u0027s what basically it says."},{"Start":"05:56.620 ","End":"06:01.810","Text":"Where is the parabola above the line?"},{"Start":"06:01.810 ","End":"06:09.340","Text":"Well, we can see that something happens outside these two points."},{"Start":"06:09.340 ","End":"06:12.200","Text":"These are the two points of intersection."},{"Start":"06:12.450 ","End":"06:19.240","Text":"Each of these has a corresponding x value."},{"Start":"06:19.240 ","End":"06:21.455","Text":"This one might be,"},{"Start":"06:21.455 ","End":"06:23.390","Text":"I\u0027m not sure exactly where it is,"},{"Start":"06:23.390 ","End":"06:27.540","Text":"but it might be say here."},{"Start":"06:27.550 ","End":"06:30.750","Text":"Let\u0027s see below,"},{"Start":"06:30.790 ","End":"06:35.255","Text":"the x values here and here."},{"Start":"06:35.255 ","End":"06:39.335","Text":"When x is between these two, which is,"},{"Start":"06:39.335 ","End":"06:43.610","Text":"the red is below the green and outside these two,"},{"Start":"06:43.610 ","End":"06:47.600","Text":"then the red is above the green,"},{"Start":"06:47.600 ","End":"06:49.250","Text":"and that\u0027s what we want."},{"Start":"06:49.250 ","End":"06:55.418","Text":"Now it has to be strictly bigger than what we would like"},{"Start":"06:55.418 ","End":"07:02.540","Text":"is this part here not including and to the right."},{"Start":"07:02.540 ","End":"07:09.870","Text":"Also this point here and to the left of it."},{"Start":"07:10.450 ","End":"07:16.490","Text":"Now the way to compute these points would be to"},{"Start":"07:16.490 ","End":"07:22.700","Text":"actually make the parabola equal to the line."},{"Start":"07:22.700 ","End":"07:26.615","Text":"Let\u0027s see what would happen if we did that."},{"Start":"07:26.615 ","End":"07:37.770","Text":"We\u0027d get that x^2+3x+2=5+x."},{"Start":"07:38.110 ","End":"07:42.080","Text":"Bringing it to the other side,"},{"Start":"07:42.080 ","End":"07:51.380","Text":"we get that x^2+2x-3=0."},{"Start":"07:51.380 ","End":"07:53.180","Text":"If we solve this,"},{"Start":"07:53.180 ","End":"07:54.590","Text":"I\u0027ll just tell you the answers."},{"Start":"07:54.590 ","End":"07:58.320","Text":"X will either equal 1 or -3."},{"Start":"07:58.990 ","End":"08:03.200","Text":"In other words, this is the point 1,"},{"Start":"08:03.200 ","End":"08:06.410","Text":"this is the point -3."},{"Start":"08:06.410 ","End":"08:11.420","Text":"I\u0027ll just use another color to highlight."},{"Start":"08:11.420 ","End":"08:20.975","Text":"In other words, the part where the red is above the green,"},{"Start":"08:20.975 ","End":"08:24.305","Text":"where the parabola is bigger than the line will either be x"},{"Start":"08:24.305 ","End":"08:29.190","Text":"bigger than 1 or this part here."},{"Start":"08:29.230 ","End":"08:36.155","Text":"This would be x less than -3."},{"Start":"08:36.155 ","End":"08:40.625","Text":"In other words, the solution is right down here."},{"Start":"08:40.625 ","End":"08:48.485","Text":"For part b is that x is less than -3,"},{"Start":"08:48.485 ","End":"08:54.720","Text":"or x bigger than 1 and we\u0027re done."}],"ID":5121},{"Watched":false,"Name":"Exercise 6","Duration":"9m 6s","ChapterTopicVideoID":5130,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5130.jpeg","UploadDate":"2016-02-29T13:05:45.2070000","DurationForVideoObject":"PT9M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.830 ","End":"00:03.330","Text":"In part A,"},{"Start":"00:03.330 ","End":"00:05.490","Text":"we\u0027re given 2 equations,"},{"Start":"00:05.490 ","End":"00:11.100","Text":"one of a parabola and one of a line and we need to sketch the graphs."},{"Start":"00:11.100 ","End":"00:17.280","Text":"Let\u0027s start with the parabola and we go through the usual stages like we did before."},{"Start":"00:17.280 ","End":"00:23.324","Text":"First one is that we say whether it\u0027s upward facing or downward facing."},{"Start":"00:23.324 ","End":"00:28.830","Text":"We look at the coefficient of x^2 see that it\u0027s minus 1 which is negative."},{"Start":"00:28.830 ","End":"00:34.630","Text":"We know that this is a downward facing parabola."},{"Start":"00:34.630 ","End":"00:37.010","Text":"Something like this."},{"Start":"00:37.010 ","End":"00:40.100","Text":"That\u0027s what I mean by downward."},{"Start":"00:40.100 ","End":"00:43.585","Text":"Let me look at the intercepts."},{"Start":"00:43.585 ","End":"00:46.290","Text":"Let\u0027s start with the X-intercept,"},{"Start":"00:46.290 ","End":"00:50.590","Text":"Y-intercept or they might not be any."},{"Start":"00:53.780 ","End":"00:55.830","Text":"For the X-intercepts,"},{"Start":"00:55.830 ","End":"00:59.510","Text":"we need to let y=0 because that\u0027s where we are on the axis."},{"Start":"00:59.510 ","End":"01:04.490","Text":"If we let y=0 we get the equation that this is equal to"},{"Start":"01:04.490 ","End":"01:14.430","Text":"0 minus x^2 minus x plus 6 = 0."},{"Start":"01:14.430 ","End":"01:17.060","Text":"I\u0027m going to assume you know how to solve"},{"Start":"01:17.060 ","End":"01:20.270","Text":"quadratic equations and I\u0027ll just give you the answer."},{"Start":"01:20.270 ","End":"01:27.210","Text":"That x can either equal minus 3 or 2 and"},{"Start":"01:27.210 ","End":"01:33.945","Text":"that makes the X-intercepts if we add the y part so we have minus 3,"},{"Start":"01:33.945 ","End":"01:39.530","Text":"0 or 2,"},{"Start":"01:39.530 ","End":"01:42.575","Text":"0 and if I draw these here,"},{"Start":"01:42.575 ","End":"01:44.030","Text":"I\u0027ll see 1,"},{"Start":"01:44.030 ","End":"01:47.730","Text":"2 and 0 would be here and minus 1,"},{"Start":"01:47.730 ","End":"01:49.110","Text":"minus 2, minus 3,"},{"Start":"01:49.110 ","End":"01:51.135","Text":"0 would be here."},{"Start":"01:51.135 ","End":"01:53.565","Text":"Just labeled that this is 2,"},{"Start":"01:53.565 ","End":"01:56.890","Text":"this is minus 3."},{"Start":"01:57.890 ","End":"02:05.505","Text":"The next thing is the Y-intercept. Let\u0027s see."},{"Start":"02:05.505 ","End":"02:11.770","Text":"I\u0027ll call that stage 3 the Y-intercept."},{"Start":"02:13.030 ","End":"02:19.130","Text":"There\u0027s always a Y-intercept it\u0027s exactly where x = 0 and if we put x = 0"},{"Start":"02:19.130 ","End":"02:27.295","Text":"in the equation which is basically this one."},{"Start":"02:27.295 ","End":"02:32.715","Text":"Scroll the other bid up and just put x equals 0 here we get y equals 6."},{"Start":"02:32.715 ","End":"02:36.710","Text":"I\u0027ll just write it down as 0,"},{"Start":"02:36.710 ","End":"02:40.072","Text":"6 and sketch that here as 0,"},{"Start":"02:40.072 ","End":"02:41.637","Text":"1, 2, 3, 4,"},{"Start":"02:41.637 ","End":"02:43.185","Text":"5 up a bit more."},{"Start":"02:43.185 ","End":"02:45.240","Text":"Let\u0027s say here."},{"Start":"02:45.240 ","End":"02:51.480","Text":"This is 0, 6."},{"Start":"02:51.480 ","End":"02:58.890","Text":"Those are the intercepts and we usually also draw the vertex of the parabola."},{"Start":"02:58.890 ","End":"03:05.525","Text":"For the vertex, we use the shortcut that the x of the vertex,"},{"Start":"03:05.525 ","End":"03:09.890","Text":"the x_v, is halfway between the 2 roots of the equation."},{"Start":"03:09.890 ","End":"03:11.870","Text":"If I take the point halfway,"},{"Start":"03:11.870 ","End":"03:14.060","Text":"if I add these and divide by 2,"},{"Start":"03:14.060 ","End":"03:19.035","Text":"I get minus 1.5."},{"Start":"03:19.035 ","End":"03:24.200","Text":"It\u0027s minus 1.5 and then the y of the vertex I get by"},{"Start":"03:24.200 ","End":"03:31.950","Text":"substituting minus 1/2 into the equation."},{"Start":"03:31.950 ","End":"03:35.420","Text":"If I\u0027m not going to detail"},{"Start":"03:35.420 ","End":"03:39.905","Text":"calculation the answer comes out to be comes out to be 6 and 1/4,"},{"Start":"03:39.905 ","End":"03:44.060","Text":"which is just a bit higher than this somewhere"},{"Start":"03:44.060 ","End":"03:53.040","Text":"here and we have enough points to make a sketch."},{"Start":"03:54.710 ","End":"03:58.175","Text":"That\u0027s the graph of the parabola in red."},{"Start":"03:58.175 ","End":"04:00.815","Text":"Now let\u0027s go for the line."},{"Start":"04:00.815 ","End":"04:03.905","Text":"We can use the system of the intercepts."},{"Start":"04:03.905 ","End":"04:05.225","Text":"That will be quick."},{"Start":"04:05.225 ","End":"04:12.750","Text":"When x is 0, y = 3 so we have the 0,"},{"Start":"04:12.750 ","End":"04:17.520","Text":"3 and if y is 0,"},{"Start":"04:17.520 ","End":"04:25.470","Text":"then we get that minus 3x plus 3 is 0, x is 1."},{"Start":"04:25.470 ","End":"04:29.070","Text":"When y is 0, x is 1."},{"Start":"04:29.070 ","End":"04:35.625","Text":"That gives us a couple of extra points on the line."},{"Start":"04:35.625 ","End":"04:43.590","Text":"Let say 1,0 is here"},{"Start":"04:43.590 ","End":"04:49.680","Text":"and 0,3 is here,"},{"Start":"04:49.680 ","End":"04:56.460","Text":"3 that\u0027s 1."},{"Start":"04:56.460 ","End":"04:58.575","Text":"We join these 2 points."},{"Start":"04:58.575 ","End":"05:01.060","Text":"Let\u0027s do it in green."},{"Start":"05:03.290 ","End":"05:11.425","Text":"That\u0027s the line in green and I extended the parabola a bit."},{"Start":"05:11.425 ","End":"05:17.760","Text":"The green is the minus 3x plus 3,"},{"Start":"05:17.760 ","End":"05:21.820","Text":"the red is the minus x squared minus x plus 6."},{"Start":"05:22.010 ","End":"05:27.440","Text":"That\u0027s basically it for part a."},{"Start":"05:27.440 ","End":"05:30.245","Text":"The line is in green,"},{"Start":"05:30.245 ","End":"05:33.950","Text":"the parabola which I extended from its original in"},{"Start":"05:33.950 ","End":"05:37.880","Text":"red and we\u0027re ready to move on to part b."},{"Start":"05:37.880 ","End":"05:41.180","Text":"I\u0027ve indicated with the red and the green which is the parabola which is"},{"Start":"05:41.180 ","End":"05:45.725","Text":"the line and then b we have to solve the inequality,"},{"Start":"05:45.725 ","End":"05:50.460","Text":"where is the parabola below the line?"},{"Start":"05:51.130 ","End":"05:54.035","Text":"To help solve the inequality,"},{"Start":"05:54.035 ","End":"05:55.730","Text":"let\u0027s first solve the equality."},{"Start":"05:55.730 ","End":"05:59.970","Text":"In other words, where did the green and red meet?"},{"Start":"06:01.450 ","End":"06:09.195","Text":"What I\u0027m asking is when do the condition minus x squared minus x"},{"Start":"06:09.195 ","End":"06:16.140","Text":"plus 6 equals minus 3x plus 3."},{"Start":"06:16.140 ","End":"06:17.875","Text":"What does this hold?"},{"Start":"06:17.875 ","End":"06:24.965","Text":"If I bring everything to the right-hand side and make it equal to 0 I\u0027ll get plus"},{"Start":"06:24.965 ","End":"06:34.230","Text":"x^2 plus x minus 3x which is minus 2x plus 3 minus 6 which is minus 3 = 0."},{"Start":"06:34.230 ","End":"06:43.355","Text":"Again, a quadratic equation and assuming you know how to solve this quadratic equation,"},{"Start":"06:43.355 ","End":"06:44.990","Text":"I\u0027ll just write down the answer."},{"Start":"06:44.990 ","End":"06:54.565","Text":"That x can either equal minus 1 or x can equal 3."},{"Start":"06:54.565 ","End":"07:00.440","Text":"What this gives us basically is that these 2 cut just"},{"Start":"07:00.440 ","End":"07:08.899","Text":"above minus 1 and they meet below."},{"Start":"07:08.899 ","End":"07:11.250","Text":"I need to scroll down."},{"Start":"07:15.350 ","End":"07:23.920","Text":"This is the 3 somewhere around here."},{"Start":"07:26.170 ","End":"07:35.569","Text":"Now it\u0027s fairly clear that we can\u0027t see everything at once the whole graph but the green"},{"Start":"07:35.569 ","End":"07:40.785","Text":"is above the red"},{"Start":"07:40.785 ","End":"07:47.720","Text":"outside from this point leftward and from this point rightward."},{"Start":"07:47.720 ","End":"07:57.300","Text":"In other words, if I draw it in a different color with x is from this point on all"},{"Start":"07:57.300 ","End":"08:06.655","Text":"along here and so on but not including the point because it\u0027s a strict inequality."},{"Start":"08:06.655 ","End":"08:12.780","Text":"We\u0027ll see that when x is bigger than 3 the green is"},{"Start":"08:12.780 ","End":"08:20.265","Text":"above the red and also not including this point,"},{"Start":"08:20.265 ","End":"08:29.865","Text":"but leftwards of it from x less than minus 1 also the grid is above the red."},{"Start":"08:29.865 ","End":"08:35.280","Text":"For part b, we could write the answer as x."},{"Start":"08:35.280 ","End":"08:36.915","Text":"Part b, that is,"},{"Start":"08:36.915 ","End":"08:46.580","Text":"x is less than minus 1 or x is bigger than 3."},{"Start":"08:46.580 ","End":"08:51.860","Text":"Just again to show you the top part here."},{"Start":"08:51.860 ","End":"08:55.670","Text":"This is where the green goes above the red here and"},{"Start":"08:55.670 ","End":"09:01.730","Text":"below this is where the green goes above the red."},{"Start":"09:01.730 ","End":"09:06.180","Text":"We\u0027re done with this exercise."}],"ID":5122},{"Watched":false,"Name":"Exercise 7","Duration":"7m 46s","ChapterTopicVideoID":8100,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8100.jpeg","UploadDate":"2016-12-08T04:35:37.5570000","DurationForVideoObject":"PT7M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.505","Text":"In this exercise, we have a word problem,"},{"Start":"00:02.505 ","End":"00:06.570","Text":"where Debby throws a ball straight up into the air and we\u0027re given"},{"Start":"00:06.570 ","End":"00:12.570","Text":"the equation of h in terms of t as follows,"},{"Start":"00:12.570 ","End":"00:16.140","Text":"where h is the height of the ball above the ground,"},{"Start":"00:16.140 ","End":"00:20.280","Text":"and t is the number of seconds after she throws it,"},{"Start":"00:20.280 ","End":"00:22.440","Text":"and we\u0027re working in feet."},{"Start":"00:22.440 ","End":"00:23.970","Text":"Then the 3 questions,"},{"Start":"00:23.970 ","End":"00:26.355","Text":"how high is the bowl after 2 seconds?"},{"Start":"00:26.355 ","End":"00:29.070","Text":"What\u0027s the greatest height the ball reaches?"},{"Start":"00:29.070 ","End":"00:31.020","Text":"When does it hit the ground?"},{"Start":"00:31.020 ","End":"00:34.150","Text":"Let\u0027s start with a."},{"Start":"00:34.150 ","End":"00:38.455","Text":"Now, we\u0027re given the height as anytime t,"},{"Start":"00:38.455 ","End":"00:44.570","Text":"so after 2 seconds we just have to let t equals 2 in the equation,"},{"Start":"00:44.570 ","End":"00:52.060","Text":"and that gives us that h of 2 is minus 16 times 2^2,"},{"Start":"00:52.060 ","End":"00:54.440","Text":"just substituting in here,"},{"Start":"00:54.440 ","End":"00:59.980","Text":"plus 48 times 2 plus 108."},{"Start":"00:59.980 ","End":"01:04.965","Text":"This equals 2^2 is 4 times 16 is 64,"},{"Start":"01:04.965 ","End":"01:10.470","Text":"48 times 2 is 96 plus 108."},{"Start":"01:10.470 ","End":"01:14.100","Text":"This comes out to, let\u0027s see,"},{"Start":"01:14.100 ","End":"01:17.280","Text":"96 and 108 is 204,"},{"Start":"01:17.280 ","End":"01:19.245","Text":"204 less 64,"},{"Start":"01:19.245 ","End":"01:23.160","Text":"I make it a 140 feet."},{"Start":"01:23.160 ","End":"01:30.785","Text":"Part b, now the maximum height,"},{"Start":"01:30.785 ","End":"01:38.820","Text":"I\u0027m going to assume that you know that when we have a function of, in this case it\u0027s t,"},{"Start":"01:38.860 ","End":"01:45.875","Text":"we have at^2 plus bt plus c,"},{"Start":"01:45.875 ","End":"01:56.450","Text":"that the maximum or minimum of the parabola occurs when t is equal to minus b/2a."},{"Start":"01:56.450 ","End":"01:59.405","Text":"Sometimes it\u0027s not t it\u0027s x,"},{"Start":"01:59.405 ","End":"02:03.265","Text":"whatever it is, minus b/2a."},{"Start":"02:03.265 ","End":"02:06.965","Text":"In our case, looking here,"},{"Start":"02:06.965 ","End":"02:13.060","Text":"we have that a equals minus 16,"},{"Start":"02:13.060 ","End":"02:17.670","Text":"b is 48, and c,"},{"Start":"02:17.670 ","End":"02:20.580","Text":"that we don\u0027t need it, is 108."},{"Start":"02:20.580 ","End":"02:25.500","Text":"What we have is minus b/2a in our case is"},{"Start":"02:25.500 ","End":"02:33.765","Text":"minus 48/2a is minus 32,"},{"Start":"02:33.765 ","End":"02:40.960","Text":"and this comes out to be divide top and bottom by minus 16."},{"Start":"02:40.960 ","End":"02:43.220","Text":"We\u0027ve got 3/2."},{"Start":"02:43.460 ","End":"02:48.155","Text":"If you prefer decimal, that\u0027s 1.5."},{"Start":"02:48.155 ","End":"02:52.795","Text":"Now we know that the time is 1.5,"},{"Start":"02:52.795 ","End":"02:54.310","Text":"but that\u0027s not what we\u0027re asked,"},{"Start":"02:54.310 ","End":"02:59.740","Text":"we\u0027re not asked when is the maximum height, but what is the maximum height?"},{"Start":"02:59.740 ","End":"03:04.160","Text":"We need to substitute t equals 1,1/2,"},{"Start":"03:05.220 ","End":"03:08.575","Text":"I\u0027ll go with 3/2."},{"Start":"03:08.575 ","End":"03:11.800","Text":"We have h of"},{"Start":"03:11.800 ","End":"03:19.800","Text":"3/2 equals minus 16, 3/2^2."},{"Start":"03:19.800 ","End":"03:21.270","Text":"Just like here is that there\u0027s"},{"Start":"03:21.270 ","End":"03:31.230","Text":"a 3/2 instead of 2 plus 48 times 3/2 plus 108."},{"Start":"03:31.230 ","End":"03:37.140","Text":"This is equal to 3/2^2 is 9/4,"},{"Start":"03:37.140 ","End":"03:40.395","Text":"4 into 16 goes 4 times."},{"Start":"03:40.395 ","End":"03:46.575","Text":"This gives us minus 4 times 9, minus 36."},{"Start":"03:46.575 ","End":"03:55.000","Text":"48 times 3/2 if I divide by 2 is 24 times 3 is 72 plus 108,"},{"Start":"03:56.060 ","End":"03:59.505","Text":"72 and 108 is 180,"},{"Start":"03:59.505 ","End":"04:07.395","Text":"180 minus 36 is 144."},{"Start":"04:07.395 ","End":"04:10.310","Text":"Good job it came out bigger than this because"},{"Start":"04:10.310 ","End":"04:14.170","Text":"the maximum has to be bigger than a particular time."},{"Start":"04:14.170 ","End":"04:17.205","Text":"This is the answer to part b,"},{"Start":"04:17.205 ","End":"04:21.765","Text":"might as well highlight, and a it\u0027s this and b it\u0027s this."},{"Start":"04:21.765 ","End":"04:23.940","Text":"Strictly speaking, I should add the units,"},{"Start":"04:23.940 ","End":"04:27.075","Text":"this is feet, this is feet."},{"Start":"04:27.075 ","End":"04:31.570","Text":"Then when does the ball hit the ground?"},{"Start":"04:31.570 ","End":"04:37.790","Text":"Well, the condition for the ball to hit the ground is for the height to be 0."},{"Start":"04:37.790 ","End":"04:39.380","Text":"If I set the height to be"},{"Start":"04:39.380 ","End":"04:47.900","Text":"0 minus 16t^2 plus 48t"},{"Start":"04:47.900 ","End":"04:55.100","Text":"plus 108 is equal to 0."},{"Start":"04:55.100 ","End":"04:59.840","Text":"Let\u0027s see. I like to simplify where possible."},{"Start":"04:59.840 ","End":"05:02.060","Text":"I see everything divides by 4."},{"Start":"05:02.060 ","End":"05:05.380","Text":"In fact, let me divide by minus 4."},{"Start":"05:05.380 ","End":"05:13.755","Text":"Then we\u0027ll get plus 4t^2 minus 12t minus,"},{"Start":"05:13.755 ","End":"05:19.340","Text":"divide this by 4 and it\u0027s 27 equals 0."},{"Start":"05:19.340 ","End":"05:21.830","Text":"Now we have a quadratic equation,"},{"Start":"05:21.830 ","End":"05:28.345","Text":"and using the formula we get that t equals,"},{"Start":"05:28.345 ","End":"05:33.020","Text":"I\u0027m assuming you know the formula by now, minus b,"},{"Start":"05:33.020 ","End":"05:38.030","Text":"which is 12 plus or minus the square root of"},{"Start":"05:38.030 ","End":"05:44.670","Text":"b^2 is a 144 minus 4ac,"},{"Start":"05:44.670 ","End":"05:46.070","Text":"because there\u0027s a minus here,"},{"Start":"05:46.070 ","End":"05:50.960","Text":"I\u0027ll write plus 4 times 4 times 27,"},{"Start":"05:50.960 ","End":"05:54.515","Text":"should have been a minus 27 and a minus here."},{"Start":"05:54.515 ","End":"06:00.440","Text":"Then all this over 2a, 2a is 8."},{"Start":"06:00.440 ","End":"06:03.320","Text":"Let me do a side calculation here."},{"Start":"06:03.320 ","End":"06:09.470","Text":"4 times 4 times 27 is 16 times 27,"},{"Start":"06:09.470 ","End":"06:13.195","Text":"which is 432,"},{"Start":"06:13.195 ","End":"06:22.900","Text":"144 plus 432 is 576."}],"ID":8216},{"Watched":false,"Name":"Intro to Economic Functions","Duration":"10m 40s","ChapterTopicVideoID":8198,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8198.jpeg","UploadDate":"2020-09-30T13:17:48.3500000","DurationForVideoObject":"PT10M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this clip, I\u0027ll be introducing some of"},{"Start":"00:02.460 ","End":"00:10.155","Text":"the basic and important functions used in Economics Functions in the mathematical sense."},{"Start":"00:10.155 ","End":"00:15.435","Text":"The first 1 will be the Demand Function."},{"Start":"00:15.435 ","End":"00:22.515","Text":"It will be written as usually capital P as a function of capital Q."},{"Start":"00:22.515 ","End":"00:25.035","Text":"Now, what are P and Q?"},{"Start":"00:25.035 ","End":"00:33.380","Text":"P is the price and Q is the quantity,"},{"Start":"00:33.380 ","End":"00:36.800","Text":"meaning the number of items that are sold."},{"Start":"00:36.800 ","End":"00:42.785","Text":"Q items are sold at a price of P each."},{"Start":"00:42.785 ","End":"00:47.420","Text":"Typically, this is a decreasing function"},{"Start":"00:47.420 ","End":"00:50.690","Text":"because as the quantity you want to sell goes up,"},{"Start":"00:50.690 ","End":"00:52.205","Text":"the price goes down."},{"Start":"00:52.205 ","End":"00:53.570","Text":"Or if you look at it the other way,"},{"Start":"00:53.570 ","End":"00:54.680","Text":"the lower the price,"},{"Start":"00:54.680 ","End":"00:56.540","Text":"the greater the quantity sold."},{"Start":"00:56.540 ","End":"00:58.325","Text":"Let\u0027s give an example."},{"Start":"00:58.325 ","End":"01:06.980","Text":"The example is P(Q)=-2Q plus 40."},{"Start":"01:06.980 ","End":"01:10.115","Text":"In this example, if Q=1,"},{"Start":"01:10.115 ","End":"01:13.279","Text":"in other words, 1 item sold,"},{"Start":"01:13.279 ","End":"01:20.240","Text":"then the price will equal minus 2 times 1 plus 40 is 38."},{"Start":"01:20.240 ","End":"01:24.815","Text":"Let\u0027s say it\u0027s in dollars, so $38."},{"Start":"01:24.815 ","End":"01:27.365","Text":"If Q is 2,"},{"Start":"01:27.365 ","End":"01:30.005","Text":"we\u0027d expect the answer to be less, let\u0027s see,"},{"Start":"01:30.005 ","End":"01:38.640","Text":"minus 2 times 2 is minus 4 plus 40 is $36, and so on."},{"Start":"01:38.750 ","End":"01:41.370","Text":"You substitute the value of Q,"},{"Start":"01:41.370 ","End":"01:47.280","Text":"you get value of P. That was Example 1,"},{"Start":"01:47.280 ","End":"01:50.185","Text":"I\u0027ll underline this too."},{"Start":"01:50.185 ","End":"01:58.460","Text":"Now I want to give another example of a special kind of a Demand Function."},{"Start":"01:58.460 ","End":"02:06.740","Text":"Sometimes a Demand Function can be constant or fixed, it never changes."},{"Start":"02:06.740 ","End":"02:13.355","Text":"An example would be P(Q)=5."},{"Start":"02:13.355 ","End":"02:16.370","Text":"Whatever Q is, P is 5."},{"Start":"02:16.370 ","End":"02:17.780","Text":"It could be many reasons for this."},{"Start":"02:17.780 ","End":"02:20.714","Text":"It could be a government-controlled price or it could be"},{"Start":"02:20.714 ","End":"02:26.840","Text":"the manufacturer suggests that this box of cereal should be sold for $5."},{"Start":"02:26.840 ","End":"02:30.815","Text":"If the quantity sold was 7 units,"},{"Start":"02:30.815 ","End":"02:33.565","Text":"the price per unit is 5."},{"Start":"02:33.565 ","End":"02:36.725","Text":"If you sold 11 units,"},{"Start":"02:36.725 ","End":"02:39.430","Text":"still the price per unit is 5."},{"Start":"02:39.430 ","End":"02:44.015","Text":"I think you get the idea it\u0027s a constant Demand Function."},{"Start":"02:44.015 ","End":"02:48.590","Text":"Now an important note, and that is,"},{"Start":"02:48.590 ","End":"02:54.500","Text":"that not always is the Demand Function given as P in terms of Q."},{"Start":"02:54.500 ","End":"02:59.675","Text":"In some books or some professors would do it the other way round,"},{"Start":"02:59.675 ","End":"03:05.990","Text":"and you would have Q as a function of P. This would"},{"Start":"03:05.990 ","End":"03:12.275","Text":"give you the quantity sold as a function of the price per unit,"},{"Start":"03:12.275 ","End":"03:14.375","Text":"and again, it would be decreasing."},{"Start":"03:14.375 ","End":"03:18.185","Text":"The lower the price, the more you expect to sell."},{"Start":"03:18.185 ","End":"03:23.060","Text":"I\u0027ll just say this is sometimes I don\u0027t know how frequently that occurs,"},{"Start":"03:23.060 ","End":"03:24.365","Text":"but you should be aware of it."},{"Start":"03:24.365 ","End":"03:27.965","Text":"For example, you might be given that"},{"Start":"03:27.965 ","End":"03:37.040","Text":"Q=100 minus P over 7,"},{"Start":"03:37.040 ","End":"03:39.350","Text":"let\u0027s say P is the price in dollars."},{"Start":"03:39.350 ","End":"03:44.405","Text":"If P equals 2 the price $2 each,"},{"Start":"03:44.405 ","End":"03:46.670","Text":"the quantity sold would be,"},{"Start":"03:46.670 ","End":"03:51.600","Text":"Q would=100 minus 2 over 7,"},{"Start":"03:51.600 ","End":"03:55.485","Text":"98 over 7 would be 14."},{"Start":"03:55.485 ","End":"03:58.460","Text":"Yeah, you should be aware that it could be the other way."},{"Start":"03:58.460 ","End":"04:02.990","Text":"On another note, this applies not just to Demand Functions,"},{"Start":"04:02.990 ","End":"04:08.000","Text":"but to pretty much most of the functions in this clip,"},{"Start":"04:08.000 ","End":"04:13.715","Text":"is that often instead of the letter Q because it\u0027s the independent variable,"},{"Start":"04:13.715 ","End":"04:18.710","Text":"it\u0027s often replaced by x to make it more mathematical."},{"Start":"04:18.710 ","End":"04:20.930","Text":"In exercises or elsewhere,"},{"Start":"04:20.930 ","End":"04:23.580","Text":"you would see, for example,"},{"Start":"04:23.580 ","End":"04:26.760","Text":"P as a function of x."},{"Start":"04:26.760 ","End":"04:33.170","Text":"Not to be alarm, not to get used to just the letter Q could be x or any other letter."},{"Start":"04:33.170 ","End":"04:37.045","Text":"Let\u0027s move on to the next function."},{"Start":"04:37.045 ","End":"04:45.130","Text":"Now, the next function we\u0027ll consider will be the Total Revenue Function."},{"Start":"04:45.130 ","End":"04:53.105","Text":"Total Revenue, and it\u0027s abbreviated usually TR and it\u0027s a function of Q,"},{"Start":"04:53.105 ","End":"04:55.745","Text":"sometimes just R for revenue,"},{"Start":"04:55.745 ","End":"04:57.985","Text":"we will call it TR."},{"Start":"04:57.985 ","End":"05:01.790","Text":"The Total Revenue, another way of saying it is the total receipts,"},{"Start":"05:01.790 ","End":"05:04.100","Text":"R also stands for receipts."},{"Start":"05:04.100 ","End":"05:08.510","Text":"Let\u0027s say for sure that this is receipts,"},{"Start":"05:08.510 ","End":"05:16.175","Text":"now what this would be is the price per unit times the number of units sold."},{"Start":"05:16.175 ","End":"05:19.595","Text":"It\u0027s going to be equal to the price,"},{"Start":"05:19.595 ","End":"05:23.165","Text":"I\u0027m leaving a space here deliberately,"},{"Start":"05:23.165 ","End":"05:26.570","Text":"times the number of items sold,"},{"Start":"05:26.570 ","End":"05:28.310","Text":"which is the quantity."},{"Start":"05:28.310 ","End":"05:36.635","Text":"Now, the price is actually a function of Q."},{"Start":"05:36.635 ","End":"05:42.330","Text":"In fact, this is just equal to the Demand Function."},{"Start":"05:42.890 ","End":"05:48.475","Text":"Q is the quantity,"},{"Start":"05:48.475 ","End":"05:51.725","Text":"the number of items sold,"},{"Start":"05:51.725 ","End":"05:53.300","Text":"price times quantity,"},{"Start":"05:53.300 ","End":"05:56.120","Text":"the total receipts, and we call that the Total Revenue."},{"Start":"05:56.120 ","End":"05:57.904","Text":"Let\u0027s move on to the next 1."},{"Start":"05:57.904 ","End":"06:02.495","Text":"The next 1 is going to be the Total Cost Function, copied from here."},{"Start":"06:02.495 ","End":"06:06.500","Text":"Now just erase revenue and erase the R,"},{"Start":"06:06.500 ","End":"06:10.650","Text":"and Total Cost Function,"},{"Start":"06:10.840 ","End":"06:14.310","Text":"TC(Q) also depends on the quantity."},{"Start":"06:14.310 ","End":"06:16.470","Text":"Sometimes instead of TC,"},{"Start":"06:16.470 ","End":"06:18.560","Text":"we write just C for cost,"},{"Start":"06:18.560 ","End":"06:21.835","Text":"but we\u0027ll write TC, total cost."},{"Start":"06:21.835 ","End":"06:28.280","Text":"The total cost, basically it\u0027s all the costs that you"},{"Start":"06:28.280 ","End":"06:35.610","Text":"incurred in producing these Q items."},{"Start":"06:35.610 ","End":"06:38.850","Text":"Another given example. Now what was this,"},{"Start":"06:38.850 ","End":"06:44.640","Text":"we had Example 3 before I think so this makes this 1 Number 4."},{"Start":"06:45.520 ","End":"06:56.425","Text":"We\u0027ll take TC(Q)=Q^2 plus Q plus 1."},{"Start":"06:56.425 ","End":"06:57.605","Text":"It could be anything,"},{"Start":"06:57.605 ","End":"06:58.850","Text":"and this is an example."},{"Start":"06:58.850 ","End":"07:02.030","Text":"In this case, if we were told that the number of"},{"Start":"07:02.030 ","End":"07:06.095","Text":"items to be produced or that were produced is 4,"},{"Start":"07:06.095 ","End":"07:10.895","Text":"then we would get the total cost of producing those 4 items, TC(4),"},{"Start":"07:10.895 ","End":"07:13.460","Text":"and just plug it in here,"},{"Start":"07:13.460 ","End":"07:16.845","Text":"4^2 plus 4 plus 1,"},{"Start":"07:16.845 ","End":"07:18.632","Text":"that comes out to 21,"},{"Start":"07:18.632 ","End":"07:22.060","Text":"and presumably this is in dollars."},{"Start":"07:22.100 ","End":"07:25.575","Text":"Let\u0027s move on to the next function."},{"Start":"07:25.575 ","End":"07:27.990","Text":"It\u0027s going to be the Average Cost Function,"},{"Start":"07:27.990 ","End":"07:29.700","Text":"I\u0027ll just copy this."},{"Start":"07:29.700 ","End":"07:33.768","Text":"Let\u0027s see, I\u0027ll erase total and I\u0027ll erase the T,"},{"Start":"07:33.768 ","End":"07:36.370","Text":"I\u0027m going to write here,"},{"Start":"07:36.370 ","End":"07:44.010","Text":"the average and make this AC(Q)."},{"Start":"07:44.010 ","End":"07:51.970","Text":"What it is is the average cost of producing not the whole Q units,"},{"Start":"07:51.970 ","End":"07:55.120","Text":"but just 1 unit."},{"Start":"07:55.120 ","End":"07:56.980","Text":"There\u0027s a formula for this."},{"Start":"07:56.980 ","End":"08:01.030","Text":"If we know the total cost,"},{"Start":"08:01.030 ","End":"08:03.170","Text":"so that would be"},{"Start":"08:04.430 ","End":"08:11.705","Text":"TC(Q) and we divide it by the number of units which is Q,"},{"Start":"08:11.705 ","End":"08:16.030","Text":"then that will give us the average for each unit."},{"Start":"08:16.030 ","End":"08:21.790","Text":"Of course, we can\u0027t do the average cost if the quantity was 0,"},{"Start":"08:21.790 ","End":"08:27.939","Text":"so let me note that it has to be Q bigger than 0."},{"Start":"08:27.939 ","End":"08:30.069","Text":"I should have mentioned earlier,"},{"Start":"08:30.069 ","End":"08:32.615","Text":"it almost goes without saying in economics that"},{"Start":"08:32.615 ","End":"08:36.670","Text":"the quantity is always bigger or equal to 0."},{"Start":"08:36.670 ","End":"08:38.950","Text":"You can produce 0 items,"},{"Start":"08:38.950 ","End":"08:41.710","Text":"but you can\u0027t produce a negative number of items."},{"Start":"08:41.710 ","End":"08:43.450","Text":"If you\u0027re going to do an average,"},{"Start":"08:43.450 ","End":"08:45.640","Text":"you can\u0027t do an average of 0 items,"},{"Start":"08:45.640 ","End":"08:48.130","Text":"it has to be at least 1."},{"Start":"08:48.530 ","End":"08:52.820","Text":"The next economics function I want to"},{"Start":"08:52.820 ","End":"08:58.315","Text":"introduce is a very important 1, the Profit Function."},{"Start":"08:58.315 ","End":"08:59.935","Text":"What are we going to call it?"},{"Start":"08:59.935 ","End":"09:02.835","Text":"We\u0027re going to call it P, no,"},{"Start":"09:02.835 ","End":"09:08.765","Text":"we can\u0027t because P is already taken for price for the Demand Function."},{"Start":"09:08.765 ","End":"09:13.460","Text":"Instead of P, we\u0027ll use the Greek letter Pi."},{"Start":"09:13.460 ","End":"09:18.510","Text":"It\u0027s going to be Pi also depends on Q,"},{"Start":"09:18.510 ","End":"09:25.445","Text":"and we can define the profit in terms of the revenue and the cost functions."},{"Start":"09:25.445 ","End":"09:29.270","Text":"The profit is what we get if we take the revenue,"},{"Start":"09:29.270 ","End":"09:31.550","Text":"which is how much we took in,"},{"Start":"09:31.550 ","End":"09:35.811","Text":"in receipts, and we subtract from that costs,"},{"Start":"09:35.811 ","End":"09:38.479","Text":"what\u0027s left over is the profit."},{"Start":"09:38.479 ","End":"09:41.520","Text":"This is equal to TR,"},{"Start":"09:41.520 ","End":"09:45.900","Text":"the income minus the expenses,"},{"Start":"09:45.900 ","End":"09:54.530","Text":"the cost, the Total Revenue minus Total Costs is going to be the profit."},{"Start":"09:54.530 ","End":"09:57.035","Text":"Let me just label these."},{"Start":"09:57.035 ","End":"10:04.370","Text":"This is the profit and this is the total receipts"},{"Start":"10:04.370 ","End":"10:14.460","Text":"so revenue and we subtract from that total costs."},{"Start":"10:14.460 ","End":"10:19.370","Text":"Note that it could come out negative or"},{"Start":"10:19.370 ","End":"10:25.355","Text":"0 if the revenue is bigger than the cost, then profit."},{"Start":"10:25.355 ","End":"10:27.320","Text":"If revenue is less than cost,"},{"Start":"10:27.320 ","End":"10:29.030","Text":"it\u0027s a negative profit."},{"Start":"10:29.030 ","End":"10:31.310","Text":"You get cool, a negative profit or loss."},{"Start":"10:31.310 ","End":"10:34.075","Text":"You can just leave it as a negative number,"},{"Start":"10:34.075 ","End":"10:37.120","Text":"so that\u0027s something to bear in mind."},{"Start":"10:37.120 ","End":"10:40.280","Text":"That\u0027s the end of this clip."}],"ID":8346},{"Watched":false,"Name":"Exercise 8","Duration":"4m 45s","ChapterTopicVideoID":8101,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8101.jpeg","UploadDate":"2016-12-08T04:36:13.6870000","DurationForVideoObject":"PT4M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"Here we have a word problem."},{"Start":"00:01.830 ","End":"00:07.650","Text":"The company that sells items and makes a profit according to this formula in terms of x,"},{"Start":"00:07.650 ","End":"00:09.690","Text":"which is the number of items itself."},{"Start":"00:09.690 ","End":"00:17.730","Text":"We have to assume looking ahead that a loss is a negative profit."},{"Start":"00:17.730 ","End":"00:23.205","Text":"We have three parts; how many items were sold if the company lost 1,000 dollars?"},{"Start":"00:23.205 ","End":"00:28.680","Text":"In Part b; how many items do you have to sell to get the maximum profit?"},{"Start":"00:28.680 ","End":"00:32.250","Text":"In Part c; what is the maximum profit?"},{"Start":"00:32.250 ","End":"00:35.175","Text":"We\u0027ll start with part a."},{"Start":"00:35.175 ","End":"00:37.910","Text":"As I said, a loss is a negative profit,"},{"Start":"00:37.910 ","End":"00:44.455","Text":"so we\u0027ll have that p(x)=-1,000,"},{"Start":"00:44.455 ","End":"00:47.655","Text":"assuming all numbers are in dollars."},{"Start":"00:47.655 ","End":"00:57.470","Text":"What we get is that 30x-x^2=-1,000."},{"Start":"00:57.470 ","End":"00:59.570","Text":"Then if I put everything on the one side,"},{"Start":"00:59.570 ","End":"01:00.950","Text":"let\u0027s say on the right-hand side,"},{"Start":"01:00.950 ","End":"01:10.520","Text":"I\u0027ll get x^2-30x-1,000=0,"},{"Start":"01:10.520 ","End":"01:13.170","Text":"and we have to find x."},{"Start":"01:13.330 ","End":"01:15.920","Text":"We\u0027ll solve this quadratic equation with"},{"Start":"01:15.920 ","End":"01:18.400","Text":"the formula which I\u0027m assuming you know by heart."},{"Start":"01:18.400 ","End":"01:21.458","Text":"We have then x=-b which is 30+- the square root of b^2,"},{"Start":"01:21.458 ","End":"01:21.461","Text":"and that makes it √900, minus 4ac,"},{"Start":"01:21.461 ","End":"01:21.468","Text":"but there\u0027s already a minus so it\u0027s going to be +4*1*1,000/2a."},{"Start":"01:21.468 ","End":"01:22.470","Text":"What"},{"Start":"01:44.800 ","End":"01:46.850","Text":"we have here"},{"Start":"01:46.850 ","End":"01:51.860","Text":"under the square root sign is the square root of"},{"Start":"01:51.860 ","End":"02:01.400","Text":"4000+900 is 4900=70,"},{"Start":"02:01.400 ","End":"02:10.200","Text":"so our two solutions become 30+70/2 and 30-70/2."},{"Start":"02:12.280 ","End":"02:22.635","Text":"Now, this one becomes negative and the number of items sold can only be positive,"},{"Start":"02:22.635 ","End":"02:25.365","Text":"so this one can be ruled out."},{"Start":"02:25.365 ","End":"02:31.930","Text":"We have this, so 30+70 is 100, 100/2 is 50."},{"Start":"02:31.930 ","End":"02:37.460","Text":"The number of items sold is 50 units,"},{"Start":"02:37.460 ","End":"02:41.490","Text":"so that\u0027s the answer to Part a."},{"Start":"02:42.070 ","End":"02:45.590","Text":"In Part b, I\u0027m assuming you\u0027ve studied"},{"Start":"02:45.590 ","End":"02:49.580","Text":"the properties of the parabola or quadratic function,"},{"Start":"02:49.580 ","End":"02:57.330","Text":"that the minimum or maximum the apex occurs at -b/2a;"},{"Start":"02:57.770 ","End":"03:01.810","Text":"the independent variable x."},{"Start":"03:03.590 ","End":"03:12.090","Text":"Well, I should have written it using this equation;"},{"Start":"03:12.090 ","End":"03:15.510","Text":"a is 1,"},{"Start":"03:15.510 ","End":"03:19.080","Text":"b is -30, c is -1,000."},{"Start":"03:19.080 ","End":"03:19.740","Text":"I used it in"},{"Start":"03:19.740 ","End":"03:20.830","Text":"the"},{"Start":"03:31.970 ","End":"03:32.981","Text":"formula."},{"Start":"03:32.981 ","End":"03:40.350","Text":"-b/2a=30/2=15. The answer is that 15 units give the maximum profit,"},{"Start":"03:40.610 ","End":"03:44.865","Text":"and I\u0027ll highlight that, items."},{"Start":"03:44.865 ","End":"03:46.485","Text":"Maybe I\u0027ll even write that."},{"Start":"03:46.485 ","End":"03:51.380","Text":"That is items, it\u0027s not dollars or anything else."},{"Start":"03:51.380 ","End":"03:57.860","Text":"Items. Now, in Part c we want to know what is the maximum profit."},{"Start":"03:57.860 ","End":"04:03.905","Text":"We just take this 15 and plug it into the formula for profit."},{"Start":"04:03.905 ","End":"04:13.710","Text":"What we want is p(15)=30*15-15^2,"},{"Start":"04:13.970 ","End":"04:17.560","Text":"and let\u0027s see what that gives;"},{"Start":"04:22.940 ","End":"04:27.960","Text":"30*15 is 450,"},{"Start":"04:27.960 ","End":"04:31.640","Text":"15^2 is 225, and that gives 225."},{"Start":"04:31.640 ","End":"04:39.260","Text":"The answer for the maximum profit would be 225."},{"Start":"04:39.260 ","End":"04:43.830","Text":"I\u0027ll put a unit dollars and I\u0027ll highlight it."},{"Start":"04:43.830 ","End":"04:46.180","Text":"We\u0027re done."}],"ID":8217},{"Watched":false,"Name":"Exercise 9","Duration":"6m 58s","ChapterTopicVideoID":8192,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8192.jpeg","UploadDate":"2017-01-01T13:55:07.1630000","DurationForVideoObject":"PT6M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, we\u0027re given the demand function of"},{"Start":"00:03.570 ","End":"00:09.180","Text":"a certain product it means that x is the quantity or"},{"Start":"00:09.180 ","End":"00:13.290","Text":"the number of units sold and p is the price that depends on"},{"Start":"00:13.290 ","End":"00:18.570","Text":"the quantity sold according to this formula and we\u0027re working in dollars."},{"Start":"00:18.570 ","End":"00:25.455","Text":"In part a, we have to find the total revenue function and its domain."},{"Start":"00:25.455 ","End":"00:31.020","Text":"Now, total revenue, the function name TR and sometimes we just use"},{"Start":"00:31.020 ","End":"00:39.765","Text":"R. What it\u0027s equal to is the number of units times the price per unit."},{"Start":"00:39.765 ","End":"00:43.790","Text":"So the number of units is x,"},{"Start":"00:43.790 ","End":"00:48.215","Text":"that\u0027s the quantity sometimes called q for quantity,"},{"Start":"00:48.215 ","End":"00:52.730","Text":"p of x is the price per unit multiply them that\u0027ll give us"},{"Start":"00:52.730 ","End":"00:58.325","Text":"our revenue or total revenue and so if I expand,"},{"Start":"00:58.325 ","End":"01:01.490","Text":"p of x is copying it from here,"},{"Start":"01:01.490 ","End":"01:11.320","Text":"minus nought 0.6 x plus 120 and then I can open the brackets and say minus nought 0.6"},{"Start":"01:11.320 ","End":"01:16.280","Text":"x squared plus 120 x. I\u0027ll just"},{"Start":"01:16.280 ","End":"01:21.890","Text":"indicate that r is a function of x. I could have put this brackets all along,"},{"Start":"01:21.890 ","End":"01:23.240","Text":"it really doesn\u0027t matter,"},{"Start":"01:23.240 ","End":"01:27.155","Text":"but R revenue function of x is this."},{"Start":"01:27.155 ","End":"01:30.770","Text":"All I have to do now is give you its domain."},{"Start":"01:30.770 ","End":"01:33.950","Text":"The domain doesn\u0027t come from here,"},{"Start":"01:33.950 ","End":"01:42.110","Text":"it just comes from real life where the domain x or quantity can\u0027t be negative."},{"Start":"01:42.110 ","End":"01:45.890","Text":"You can\u0027t produce a negative number of units so the domain is"},{"Start":"01:45.890 ","End":"01:50.180","Text":"just x bigger or equal to 0, could be 0."},{"Start":"01:50.180 ","End":"01:55.280","Text":"It might not manufacture anything and that\u0027s part A."},{"Start":"01:55.280 ","End":"01:58.309","Text":"Now in part B,"},{"Start":"01:58.309 ","End":"02:01.100","Text":"we\u0027re given that x is 20,"},{"Start":"02:01.100 ","End":"02:04.070","Text":"meaning there were 20 units produced and we want to"},{"Start":"02:04.070 ","End":"02:07.670","Text":"find what the price is and what the revenue is."},{"Start":"02:07.670 ","End":"02:08.990","Text":"We have both functions."},{"Start":"02:08.990 ","End":"02:11.735","Text":"The price function, which is the demand function,"},{"Start":"02:11.735 ","End":"02:16.310","Text":"is here so what I have to say is that x equals 20,"},{"Start":"02:16.310 ","End":"02:23.210","Text":"I need to find what is p of 20 and I need to know"},{"Start":"02:23.210 ","End":"02:30.780","Text":"then what is the revenue or total revenue for 20 equals,"},{"Start":"02:30.780 ","End":"02:32.460","Text":"that\u0027s like 2 questions in 1."},{"Start":"02:32.460 ","End":"02:34.710","Text":"As for p of 20,"},{"Start":"02:34.710 ","End":"02:39.465","Text":"let me substitute x equals 20 in the demand function."},{"Start":"02:39.465 ","End":"02:47.745","Text":"So we get p of 20 equals minus 0.6 times,"},{"Start":"02:47.745 ","End":"02:49.580","Text":"a little times like this,"},{"Start":"02:49.580 ","End":"02:51.860","Text":"not to confuse with the decimal point,"},{"Start":"02:51.860 ","End":"02:58.190","Text":"times 20 plus 120 and that is equal"},{"Start":"02:58.190 ","End":"03:05.825","Text":"to 0.6 times 20 is minus 12 plus 120,"},{"Start":"03:05.825 ","End":"03:14.360","Text":"which is 120 minus 12 is 108 and remember we\u0027re working in dollars,"},{"Start":"03:14.360 ","End":"03:18.860","Text":"so the price of a unit is $108."},{"Start":"03:18.860 ","End":"03:22.650","Text":"The idea is to use the revenue function."},{"Start":"03:23.470 ","End":"03:34.790","Text":"I just have to substitute this time in this function and so r of 20"},{"Start":"03:34.790 ","End":"03:43.235","Text":"is equal to minus 0.6 times 20 squared"},{"Start":"03:43.235 ","End":"03:50.330","Text":"plus 120 times 20"},{"Start":"03:50.330 ","End":"03:54.335","Text":"and so r of 20 equals."},{"Start":"03:54.335 ","End":"03:56.210","Text":"I don\u0027t think we need to calculate,"},{"Start":"03:56.210 ","End":"03:58.010","Text":"20 squared is 400,"},{"Start":"03:58.010 ","End":"04:02.190","Text":"400 times 6 is 2,400, but it\u0027s 0.6,"},{"Start":"04:02.190 ","End":"04:09.240","Text":"so it\u0027s 240 so we have minus 240 plus 12 times 2 is 24,"},{"Start":"04:09.240 ","End":"04:14.075","Text":"so it\u0027s 2,400 and if we subtract,"},{"Start":"04:14.075 ","End":"04:15.320","Text":"that gives us what,"},{"Start":"04:15.320 ","End":"04:22.335","Text":"2,160, and that would be in dollars."},{"Start":"04:22.335 ","End":"04:25.739","Text":"So this is the price this is the total revenue."},{"Start":"04:25.739 ","End":"04:27.070","Text":"That\u0027s just part B."},{"Start":"04:27.070 ","End":"04:28.318","Text":"Just as a check,"},{"Start":"04:28.318 ","End":"04:32.630","Text":"you could take this $108 and multiply it by"},{"Start":"04:32.630 ","End":"04:39.890","Text":"20 units and 108 times 20 is 2,160 so that\u0027s an extra check."},{"Start":"04:39.890 ","End":"04:44.405","Text":"Now part C, this time we\u0027re not given the quantity,"},{"Start":"04:44.405 ","End":"04:46.085","Text":"we are given the price,"},{"Start":"04:46.085 ","End":"04:48.605","Text":"and we have to find the revenue."},{"Start":"04:48.605 ","End":"04:54.005","Text":"Now what we have to first do is get x, the quantity."},{"Start":"04:54.005 ","End":"04:58.100","Text":"We\u0027ll do that from the demand function."},{"Start":"04:58.100 ","End":"05:07.020","Text":"Part C, p of x equals minus 0.6 x plus 120."},{"Start":"05:07.020 ","End":"05:09.410","Text":"But in our case, we\u0027re given the price,"},{"Start":"05:09.410 ","End":"05:14.190","Text":"we\u0027re given that it\u0027s $12 so I\u0027m just going to erase"},{"Start":"05:14.190 ","End":"05:17.190","Text":"this and write 12 in"},{"Start":"05:17.190 ","End":"05:21.850","Text":"its place and now this is an equation in x and we want to find out what x is."},{"Start":"05:21.850 ","End":"05:23.660","Text":"That\u0027s just your basic algebra."},{"Start":"05:23.660 ","End":"05:25.490","Text":"We\u0027ll move this over to that side,"},{"Start":"05:25.490 ","End":"05:34.735","Text":"this over to that side so I\u0027ve got 0.6 x equals 120 minus 12."},{"Start":"05:34.735 ","End":"05:39.260","Text":"Well this we can see already is 108,"},{"Start":"05:39.260 ","End":"05:48.335","Text":"so x is going to equal if we divide both sides by 0.6, 108 over 0.6."},{"Start":"05:48.335 ","End":"05:50.960","Text":"I don\u0027t think we need a calculator."},{"Start":"05:50.960 ","End":"05:54.058","Text":"Well, you could. Anyway,"},{"Start":"05:54.058 ","End":"05:56.200","Text":"comes out to 180."},{"Start":"05:56.200 ","End":"05:58.130","Text":"So that\u0027s the x,"},{"Start":"05:58.130 ","End":"05:59.300","Text":"but that\u0027s not what we\u0027re asked for."},{"Start":"05:59.300 ","End":"06:05.390","Text":"We\u0027re asked for the revenue so what we do is we plug in this 180,"},{"Start":"06:05.390 ","End":"06:06.740","Text":"which is now the quantity,"},{"Start":"06:06.740 ","End":"06:11.720","Text":"the number of units into the revenue function here."},{"Start":"06:11.720 ","End":"06:16.500","Text":"Let\u0027s plug it right into here and then we get that R of x in"},{"Start":"06:16.500 ","End":"06:21.320","Text":"our case R of 180 will equal minus"},{"Start":"06:21.320 ","End":"06:32.260","Text":"0.6 times 180 squared plus 120 times 180."},{"Start":"06:32.300 ","End":"06:36.015","Text":"I\u0027ll spare you the details of the computation,"},{"Start":"06:36.015 ","End":"06:41.745","Text":"it comes out to be 2,160 and that\u0027s also in dollars"},{"Start":"06:41.745 ","End":"06:47.990","Text":"and it\u0027s just a coincidence that when we produce 20 units or 180 units,"},{"Start":"06:47.990 ","End":"06:49.870","Text":"it comes out the same."},{"Start":"06:49.870 ","End":"06:54.250","Text":"Just fluke. So we\u0027ve answered A,"},{"Start":"06:54.250 ","End":"06:59.650","Text":"we\u0027ve answered B both parts and we\u0027ve answered C and so we\u0027re done."}],"ID":8347},{"Watched":false,"Name":"Exercise 10","Duration":"11m 7s","ChapterTopicVideoID":8193,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8193.jpeg","UploadDate":"2017-01-01T13:58:06.7770000","DurationForVideoObject":"PT11M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.035","Text":"In this exercise, we\u0027re given the total revenue function of a certain product."},{"Start":"00:07.035 ","End":"00:14.700","Text":"We call it R, but sometimes it\u0027s called the TR for total revenue."},{"Start":"00:14.700 ","End":"00:19.395","Text":"Also, remember that x is really q the quantity."},{"Start":"00:19.395 ","End":"00:21.735","Text":"This is the formula."},{"Start":"00:21.735 ","End":"00:24.405","Text":"Notice that it\u0027s quadratic,"},{"Start":"00:24.405 ","End":"00:26.490","Text":"I\u0027ll say more in a moment."},{"Start":"00:26.490 ","End":"00:28.065","Text":"The first part is,"},{"Start":"00:28.065 ","End":"00:31.900","Text":"what is the domain of the revenue function?"},{"Start":"00:31.910 ","End":"00:34.800","Text":"Now, if you just look at it mathematically,"},{"Start":"00:34.800 ","End":"00:36.255","Text":"x could be anything."},{"Start":"00:36.255 ","End":"00:39.090","Text":"But since it\u0027s a real-life problem, x,"},{"Start":"00:39.090 ","End":"00:42.055","Text":"which is the quantity produced,"},{"Start":"00:42.055 ","End":"00:49.105","Text":"can\u0027t be negative and so the domain is x bigger or equal to 0."},{"Start":"00:49.105 ","End":"00:51.855","Text":"Now onto part B,"},{"Start":"00:51.855 ","End":"00:54.210","Text":"I mentioned the word quadratic."},{"Start":"00:54.210 ","End":"00:55.935","Text":"If we think of R,"},{"Start":"00:55.935 ","End":"00:57.105","Text":"the revenue was like,"},{"Start":"00:57.105 ","End":"00:58.680","Text":"y in mathematics,"},{"Start":"00:58.680 ","End":"01:00.060","Text":"if we think of x and y,"},{"Start":"01:00.060 ","End":"01:09.240","Text":"then y is of the form ax squared plus bx plus c. That\u0027s the quadratic."},{"Start":"01:09.240 ","End":"01:17.630","Text":"In fact, we can say that a is minus 0.08, b is 40."},{"Start":"01:17.630 ","End":"01:22.410","Text":"There is no constant term which means that it\u0027s there but it\u0027s 0."},{"Start":"01:22.480 ","End":"01:25.775","Text":"I\u0027ll give you a reminder of the properties."},{"Start":"01:25.775 ","End":"01:27.545","Text":"Some of them have the parabola."},{"Start":"01:27.545 ","End":"01:29.375","Text":"There\u0027s 2 kinds of parabola."},{"Start":"01:29.375 ","End":"01:34.765","Text":"There\u0027s one that face up or open up,"},{"Start":"01:34.765 ","End":"01:36.540","Text":"and there\u0027s one that face down."},{"Start":"01:36.540 ","End":"01:41.445","Text":"Some people say open down and it upward-facing"},{"Start":"01:41.445 ","End":"01:48.005","Text":"when a is bigger than 0 and downward-facing when a is less than 0."},{"Start":"01:48.005 ","End":"01:54.125","Text":"The other thing I wanted to say about parabolas is they have a vertex."},{"Start":"01:54.125 ","End":"01:56.575","Text":"Here or here."},{"Start":"01:56.575 ","End":"01:58.260","Text":"Actually, in our case,"},{"Start":"01:58.260 ","End":"02:01.830","Text":"we are going to be a is negative."},{"Start":"02:01.830 ","End":"02:04.435","Text":"We\u0027re actually going to be in this case."},{"Start":"02:04.435 ","End":"02:06.360","Text":"Maybe I\u0027ll highlight it."},{"Start":"02:06.360 ","End":"02:09.155","Text":"This is what we have in our case."},{"Start":"02:09.155 ","End":"02:12.230","Text":"I also want to label this vertex,"},{"Start":"02:12.230 ","End":"02:15.140","Text":"just give names to the x and the y coordinates."},{"Start":"02:15.140 ","End":"02:17.540","Text":"I\u0027ll call it x_v,"},{"Start":"02:17.540 ","End":"02:18.935","Text":"x of the vertex,"},{"Start":"02:18.935 ","End":"02:21.155","Text":"and y of the vertex."},{"Start":"02:21.155 ","End":"02:24.420","Text":"There\u0027s a formula for x of the vertex."},{"Start":"02:24.420 ","End":"02:26.550","Text":"I will give it to you in a moment."},{"Start":"02:26.550 ","End":"02:31.730","Text":"Back here, there\u0027s 3 things I usually want to know about a parabola before sketching it."},{"Start":"02:31.730 ","End":"02:35.240","Text":"The first is whether it\u0027s facing upwards or downwards,"},{"Start":"02:35.240 ","End":"02:40.880","Text":"and I can say that this parabola faces downwards."},{"Start":"02:40.880 ","End":"02:44.630","Text":"The other thing is the vertex."},{"Start":"02:44.630 ","End":"02:47.690","Text":"I want to know what the coordinates are."},{"Start":"02:47.690 ","End":"02:50.075","Text":"I use the formula."},{"Start":"02:50.075 ","End":"02:52.040","Text":"Maybe I\u0027ll write it over here."},{"Start":"02:52.040 ","End":"02:59.160","Text":"In general, the x of the vertex is equal to minus b over 2a,"},{"Start":"02:59.160 ","End":"03:05.139","Text":"where b and a we have and we get the y by substituting in the equation."},{"Start":"03:05.139 ","End":"03:10.684","Text":"So x, v from this formula,"},{"Start":"03:10.684 ","End":"03:16.490","Text":"minus b is minus 40 over 2a,"},{"Start":"03:16.490 ","End":"03:20.580","Text":"which is minus 0.08."},{"Start":"03:21.370 ","End":"03:23.930","Text":"I\u0027ll do the computation for you."},{"Start":"03:23.930 ","End":"03:26.390","Text":"It comes out to 250,"},{"Start":"03:26.390 ","End":"03:29.450","Text":"positive because there\u0027s a minus over a minus."},{"Start":"03:29.450 ","End":"03:32.705","Text":"Then y of the vertex,"},{"Start":"03:32.705 ","End":"03:38.445","Text":"we get by substituting 250 in our formula."},{"Start":"03:38.445 ","End":"03:44.085","Text":"This will be minus 0.08 times"},{"Start":"03:44.085 ","End":"03:51.865","Text":"250 squared plus 40 times 250."},{"Start":"03:51.865 ","End":"03:54.770","Text":"Again, I\u0027ll do the computation for you."},{"Start":"03:54.770 ","End":"03:58.880","Text":"It comes out to be 5,000."},{"Start":"03:58.880 ","End":"04:04.250","Text":"Now, we know that the vertex is at"},{"Start":"04:04.250 ","End":"04:11.110","Text":"the point 250, 5000."},{"Start":"04:11.110 ","End":"04:13.850","Text":"I want to know the intersection with the axis."},{"Start":"04:13.850 ","End":"04:21.245","Text":"Let\u0027s see, the intersection with the x-axis is what we get when we say y equals 0,"},{"Start":"04:21.245 ","End":"04:29.045","Text":"we get that minus 0.08x squared plus 40x,"},{"Start":"04:29.045 ","End":"04:30.965","Text":"just copied it from here,"},{"Start":"04:30.965 ","End":"04:33.545","Text":"is equal to 0."},{"Start":"04:33.545 ","End":"04:36.980","Text":"We can factor this because x is in both of them."},{"Start":"04:36.980 ","End":"04:38.270","Text":"There is no constant term."},{"Start":"04:38.270 ","End":"04:46.910","Text":"It\u0027s x times minus 0.08x plus 40 is equal to 0."},{"Start":"04:46.910 ","End":"04:49.310","Text":"Just took x outside the brackets."},{"Start":"04:49.310 ","End":"04:52.055","Text":"Then we\u0027ll get 2 solutions."},{"Start":"04:52.055 ","End":"04:59.000","Text":"One of them will be when x equals 0 and the other will"},{"Start":"04:59.000 ","End":"05:07.250","Text":"be when minus 0.08x plus 40 equals 0."},{"Start":"05:07.250 ","End":"05:08.960","Text":"Because of course, when a product is 0,"},{"Start":"05:08.960 ","End":"05:11.075","Text":"one over the other must be 0."},{"Start":"05:11.075 ","End":"05:14.840","Text":"When x is 0 is already a value of x."},{"Start":"05:14.840 ","End":"05:18.305","Text":"But here we have to solve a simple linear equation."},{"Start":"05:18.305 ","End":"05:20.285","Text":"What x is equal to,"},{"Start":"05:20.285 ","End":"05:23.780","Text":"I\u0027ll take the 40 to the other side."},{"Start":"05:23.780 ","End":"05:31.340","Text":"That will be minus 40 and divide by the coefficient of x minus 0.08,"},{"Start":"05:31.340 ","End":"05:33.320","Text":"the minuses cancel,"},{"Start":"05:33.320 ","End":"05:39.265","Text":"and this just comes out to be 500."},{"Start":"05:39.265 ","End":"05:44.975","Text":"Notice that there\u0027s no need to find the y-intercept,"},{"Start":"05:44.975 ","End":"05:49.520","Text":"meaning we\u0027ve already found it because x equals 0, then y is 0."},{"Start":"05:49.520 ","End":"05:52.895","Text":"The 2 intercepts, 0,"},{"Start":"05:52.895 ","End":"05:58.880","Text":"0 and 500, 0."},{"Start":"05:58.880 ","End":"06:02.465","Text":"Now we\u0027ve got all the information we need for a sketch."},{"Start":"06:02.465 ","End":"06:09.740","Text":"The parabola faces down and its vertex is at this point."},{"Start":"06:09.740 ","End":"06:14.825","Text":"These are the intersections with the axis."},{"Start":"06:14.825 ","End":"06:19.265","Text":"Now, we\u0027ll do the sketch."},{"Start":"06:19.265 ","End":"06:22.040","Text":"The horizontal will be x,"},{"Start":"06:22.040 ","End":"06:23.480","Text":"which is the quantity,"},{"Start":"06:23.480 ","End":"06:28.385","Text":"the vertical, I can call it y or I could call it revenue."},{"Start":"06:28.385 ","End":"06:33.470","Text":"The intersection with the axis is 0,0,"},{"Start":"06:33.470 ","End":"06:36.140","Text":"and then I need 500 zeros."},{"Start":"06:36.140 ","End":"06:38.870","Text":"So let\u0027s just say that this is here,"},{"Start":"06:38.870 ","End":"06:42.470","Text":"500 and this is 0."},{"Start":"06:42.470 ","End":"06:46.565","Text":"The vertex is 250,"},{"Start":"06:46.565 ","End":"06:51.170","Text":"exactly halfway between the 0 and the 500,"},{"Start":"06:51.170 ","End":"06:54.845","Text":"let\u0027s say this is 5,000 for y."},{"Start":"06:54.845 ","End":"06:59.270","Text":"We want this point somewhere here."},{"Start":"06:59.270 ","End":"07:02.975","Text":"You get the general idea of the shape. It\u0027s not perfect."},{"Start":"07:02.975 ","End":"07:05.210","Text":"That part B,"},{"Start":"07:05.210 ","End":"07:07.310","Text":"that\u0027ll just clear some space."},{"Start":"07:07.310 ","End":"07:11.810","Text":"I\u0027ll continue over here with part C. We want"},{"Start":"07:11.810 ","End":"07:17.540","Text":"the demand function and we can get it from the revenue function."},{"Start":"07:17.540 ","End":"07:22.250","Text":"Because the formula is that the revenue function,"},{"Start":"07:22.250 ","End":"07:25.355","Text":"let\u0027s say r is in terms of x,"},{"Start":"07:25.355 ","End":"07:31.250","Text":"is equal to x times p of x."},{"Start":"07:31.250 ","End":"07:33.910","Text":"Now, we\u0027re given r of x,"},{"Start":"07:33.910 ","End":"07:37.080","Text":"which is it\u0027s still here."},{"Start":"07:37.080 ","End":"07:39.285","Text":"I can copy it from here."},{"Start":"07:39.285 ","End":"07:44.955","Text":"I have the minus 0.08x squared"},{"Start":"07:44.955 ","End":"07:51.650","Text":"plus 40x is x times the demand function,"},{"Start":"07:51.650 ","End":"07:53.330","Text":"which is p of x."},{"Start":"07:53.330 ","End":"07:55.850","Text":"It gives the price in terms of the quantity."},{"Start":"07:55.850 ","End":"08:02.000","Text":"All I have to do is divide both sides by x. X comes out of the brackets."},{"Start":"08:02.000 ","End":"08:12.140","Text":"We get 0.08x plus"},{"Start":"08:12.140 ","End":"08:19.320","Text":"40 is equal to p of x."},{"Start":"08:19.320 ","End":"08:23.490","Text":"Should have written that backwards."},{"Start":"08:23.490 ","End":"08:30.030","Text":"This is the answer for the first part of C. That\u0027s the demand function."},{"Start":"08:30.290 ","End":"08:37.250","Text":"Might as well say that its domain also is x bigger or equal to 0,"},{"Start":"08:37.250 ","End":"08:39.700","Text":"x is still the quantity."},{"Start":"08:39.700 ","End":"08:43.050","Text":"For the sketch, we see it\u0027s linear."},{"Start":"08:43.050 ","End":"08:45.000","Text":"We really only need 2 points."},{"Start":"08:45.000 ","End":"08:49.640","Text":"Let me just get some axes in borrowed the earliest sketches,"},{"Start":"08:49.640 ","End":"08:54.560","Text":"just fix it up a bit and we can make a little table."},{"Start":"08:54.560 ","End":"08:56.420","Text":"We only need 2 values."},{"Start":"08:56.420 ","End":"08:58.070","Text":"Sometimes I do a third,"},{"Start":"08:58.070 ","End":"08:59.495","Text":"if they\u0027re too close."},{"Start":"08:59.495 ","End":"09:04.530","Text":"We want x, this is now p. We need x"},{"Start":"09:04.530 ","End":"09:09.560","Text":"and p. The easiest point to compute always is when x is not."},{"Start":"09:09.560 ","End":"09:12.905","Text":"You don\u0027t produce anything."},{"Start":"09:12.905 ","End":"09:14.990","Text":"Put x equals not here,"},{"Start":"09:14.990 ","End":"09:18.280","Text":"this bit is not and we just get 40."},{"Start":"09:18.280 ","End":"09:22.560","Text":"X is not, p is 40."},{"Start":"09:22.560 ","End":"09:28.040","Text":"You know what? Why don\u0027t I go for the 500?"},{"Start":"09:28.040 ","End":"09:31.670","Text":"Because I noticed earlier that the revenue was 0."},{"Start":"09:31.670 ","End":"09:36.110","Text":"Then let\u0027s try x is 500,"},{"Start":"09:36.110 ","End":"09:38.540","Text":"and then I plug it in."},{"Start":"09:38.540 ","End":"09:48.980","Text":"Minus 0.08 times 500 just happens to be minus 40 plus 40, it\u0027s 0."},{"Start":"09:48.980 ","End":"09:51.620","Text":"Why is this not a surprise to me?"},{"Start":"09:51.620 ","End":"09:58.220","Text":"Because earlier we had the parabola and the revenue was 0 here."},{"Start":"09:58.220 ","End":"10:01.700","Text":"The quantity was 500,"},{"Start":"10:01.700 ","End":"10:03.230","Text":"but the revenue was 0,"},{"Start":"10:03.230 ","End":"10:06.940","Text":"which means that the price was 0."},{"Start":"10:06.940 ","End":"10:08.984","Text":"That makes sense."},{"Start":"10:08.984 ","End":"10:10.635","Text":"0, 40,"},{"Start":"10:10.635 ","End":"10:12.890","Text":"40 is the largest value."},{"Start":"10:12.890 ","End":"10:16.070","Text":"Let\u0027s say here we have 40."},{"Start":"10:16.070 ","End":"10:17.420","Text":"The point 0,"},{"Start":"10:17.420 ","End":"10:20.420","Text":"40 is then here."},{"Start":"10:20.420 ","End":"10:25.440","Text":"The point 500, 0 is here."},{"Start":"10:25.440 ","End":"10:28.025","Text":"You know what? Let\u0027s just throw in another one."},{"Start":"10:28.025 ","End":"10:31.250","Text":"Let\u0027s say we took a quantity of 200."},{"Start":"10:31.250 ","End":"10:38.840","Text":"200 times 0.08 would be what 16, I think."},{"Start":"10:38.840 ","End":"10:40.220","Text":"But with a minus,"},{"Start":"10:40.220 ","End":"10:50.535","Text":"minus 16 plus 40 would be 24. We have 200."},{"Start":"10:50.535 ","End":"10:56.835","Text":"Roughly 24 means more than halfway is here."},{"Start":"10:56.835 ","End":"11:00.270","Text":"It\u0027s somewhere around here."},{"Start":"11:00.270 ","End":"11:04.150","Text":"Hopefully, these are a straight line."},{"Start":"11:05.030 ","End":"11:08.560","Text":"Done with this exercise."}],"ID":8348},{"Watched":false,"Name":"Exercise 11","Duration":"6m 46s","ChapterTopicVideoID":8194,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8194.jpeg","UploadDate":"2017-01-01T13:59:55.5370000","DurationForVideoObject":"PT6M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.995","Text":"This exercise, we\u0027re given the demand function of a certain product."},{"Start":"00:04.995 ","End":"00:08.955","Text":"P stands for price and x is the quantity."},{"Start":"00:08.955 ","End":"00:13.515","Text":"For each quantity, we have the price as this formula."},{"Start":"00:13.515 ","End":"00:16.725","Text":"Let\u0027s take each of these 4 parts separately."},{"Start":"00:16.725 ","End":"00:18.675","Text":"Part a."},{"Start":"00:18.675 ","End":"00:22.320","Text":"We just have to find the domain of the function."},{"Start":"00:22.320 ","End":"00:26.520","Text":"Mathematically, there\u0027s no reason you can\u0027t substitute any value of x,"},{"Start":"00:26.520 ","End":"00:31.140","Text":"but it\u0027s a real-life question and x being the quantity can\u0027t be"},{"Start":"00:31.140 ","End":"00:38.355","Text":"negative so we just write x is not negative means x bigger or equal to 0."},{"Start":"00:38.355 ","End":"00:41.085","Text":"Now part b,"},{"Start":"00:41.085 ","End":"00:44.790","Text":"the total revenue function and"},{"Start":"00:44.790 ","End":"00:49.265","Text":"the average revenue function is first of all do the total revenue function."},{"Start":"00:49.265 ","End":"00:54.110","Text":"I\u0027ll call it TR(x) sometimes abbreviated to just R(x)."},{"Start":"00:54.110 ","End":"01:05.450","Text":"What it is, is the number of units sold times the price of each unit."},{"Start":"01:05.450 ","End":"01:07.865","Text":"But we have p(x)."},{"Start":"01:07.865 ","End":"01:11.900","Text":"p(x) you just have to copy it from there it\u0027s -0.4x+100 and"},{"Start":"01:11.900 ","End":"01:18.520","Text":"so if we multiply out, we get -0.4x^2+100x."},{"Start":"01:22.930 ","End":"01:25.745","Text":"That\u0027s the total revenue."},{"Start":"01:25.745 ","End":"01:31.740","Text":"Now, the average revenue"},{"Start":"01:34.570 ","End":"01:42.560","Text":"of x is equal to the total revenue,"},{"Start":"01:42.560 ","End":"01:48.120","Text":"but the average means per unit so we divide by the number of units;"},{"Start":"01:53.200 ","End":"02:04.860","Text":"TR(x), total revenue minus 0.4x^2+100x and then divide that by x."},{"Start":"02:04.860 ","End":"02:10.880","Text":"What we would get, since each of these divides by x,"},{"Start":"02:10.880 ","End":"02:21.155","Text":"we\u0027ll divide everything by x and we get minus 0.4, just x+100."},{"Start":"02:21.155 ","End":"02:26.360","Text":"Now, of course all this works when x is bigger than 0,"},{"Start":"02:26.360 ","End":"02:28.220","Text":"the whole average concept."},{"Start":"02:28.220 ","End":"02:30.935","Text":"You can\u0027t take the average of 0 items."},{"Start":"02:30.935 ","End":"02:34.025","Text":"It\u0027s not enough for x to be bigger or equal to,"},{"Start":"02:34.025 ","End":"02:36.455","Text":"has to be strictly positive."},{"Start":"02:36.455 ","End":"02:39.530","Text":"Part c, that\u0027s something else."},{"Start":"02:39.530 ","End":"02:43.055","Text":"That\u0027s the marginal revenue function."},{"Start":"02:43.055 ","End":"02:45.055","Text":"When anything is marginal,"},{"Start":"02:45.055 ","End":"02:46.970","Text":"we take its derivative."},{"Start":"02:46.970 ","End":"02:48.875","Text":"Take the derivative of the revenue."},{"Start":"02:48.875 ","End":"02:50.780","Text":"Well, here I called it TR."},{"Start":"02:50.780 ","End":"02:56.950","Text":"It\u0027s just the derivative of TR(x)."},{"Start":"02:56.950 ","End":"03:04.060","Text":"In other words, I have to take the derivative of where is it? Here it is."},{"Start":"03:04.060 ","End":"03:12.230","Text":"From here, I just take the derivative 2 times 0.4 is minus"},{"Start":"03:12.230 ","End":"03:20.570","Text":"0.8 and we lower the power by 100x, becomes just 100."},{"Start":"03:20.570 ","End":"03:23.860","Text":"This is the marginal."},{"Start":"03:23.860 ","End":"03:26.575","Text":"Now in part d,"},{"Start":"03:26.575 ","End":"03:30.200","Text":"we have to find the maximal total revenue."},{"Start":"03:30.200 ","End":"03:38.180","Text":"We need to ask which x maximizes."},{"Start":"03:38.180 ","End":"03:42.230","Text":"I\u0027m just going to copy the function here."},{"Start":"03:42.230 ","End":"03:50.960","Text":"Maximizes -0.4x^2+100x."},{"Start":"03:50.960 ","End":"03:52.990","Text":"That\u0027s another way of putting it."},{"Start":"03:52.990 ","End":"03:57.050","Text":"Now we\u0027re going to use our knowledge of parabolas and quadratics."},{"Start":"03:57.050 ","End":"04:03.240","Text":"After all, this is of the general form, ax^2+bx+c."},{"Start":"04:05.720 ","End":"04:08.220","Text":"Our marginal revenue,"},{"Start":"04:08.220 ","End":"04:10.129","Text":"we\u0027ll just call that y,"},{"Start":"04:10.129 ","End":"04:12.530","Text":"just simplicity in math,"},{"Start":"04:12.530 ","End":"04:14.615","Text":"we add x and y."},{"Start":"04:14.615 ","End":"04:20.960","Text":"This is a quadratic and we\u0027re going to use the property of quadratic curves,"},{"Start":"04:20.960 ","End":"04:24.560","Text":"which are parabolas instead of using calculus to"},{"Start":"04:24.560 ","End":"04:29.180","Text":"differentiate and set the derivative to 0 and so on."},{"Start":"04:29.180 ","End":"04:32.555","Text":"If a is negative,"},{"Start":"04:32.555 ","End":"04:38.360","Text":"then it faces down, which is what we have."},{"Start":"04:38.360 ","End":"04:40.520","Text":"Then the other way, if a was positive,"},{"Start":"04:40.520 ","End":"04:42.545","Text":"it would face up,"},{"Start":"04:42.545 ","End":"04:45.605","Text":"opens upwards, opens downwards."},{"Start":"04:45.605 ","End":"04:51.410","Text":"This is a, this is b, and c is 0."},{"Start":"04:51.410 ","End":"04:55.885","Text":"The maximum would occur at the vertex."},{"Start":"04:55.885 ","End":"05:01.175","Text":"If it\u0027s the other way round then this is the vertex and there\u0027s a formula."},{"Start":"05:01.175 ","End":"05:03.800","Text":"The x of the vertex,"},{"Start":"05:03.800 ","End":"05:10.335","Text":"or call it x(v) is given by -b/2a in general."},{"Start":"05:10.335 ","End":"05:12.930","Text":"In our case, this will be,"},{"Start":"05:12.930 ","End":"05:22.965","Text":"-b will be -100/2, a is -0.4."},{"Start":"05:22.965 ","End":"05:30.490","Text":"This is going to equal -100/-0.8."},{"Start":"05:30.490 ","End":"05:39.115","Text":"Of course the minuses cancel and this gives us 125."},{"Start":"05:39.115 ","End":"05:41.450","Text":"This will be the x that will give"},{"Start":"05:41.450 ","End":"05:46.400","Text":"the maximal total revenue because we have an upside down parabola,"},{"Start":"05:46.400 ","End":"05:49.100","Text":"there is a maximum, otherwise there\u0027d be a minimum."},{"Start":"05:49.100 ","End":"05:52.460","Text":"The last thing we have to find out is,"},{"Start":"05:52.460 ","End":"05:55.370","Text":"what is the maximal total revenue?"},{"Start":"05:55.370 ","End":"06:03.035","Text":"What we have to do is substitute this x in the revenue function, this one here."},{"Start":"06:03.035 ","End":"06:11.665","Text":"What we need is TR(125) and see what that equals."},{"Start":"06:11.665 ","End":"06:13.980","Text":"I\u0027ll do it over here."},{"Start":"06:13.980 ","End":"06:15.930","Text":"Just a computation,"},{"Start":"06:15.930 ","End":"06:19.240","Text":"-0.4 times"},{"Start":"06:22.370 ","End":"06:27.330","Text":"125^2+100 times 125."},{"Start":"06:27.330 ","End":"06:29.875","Text":"This is equal to,"},{"Start":"06:29.875 ","End":"06:35.315","Text":"I make it 625 and that would be dollars."},{"Start":"06:35.315 ","End":"06:41.550","Text":"Whoops, I missed a 0, it\u0027s actually $6,250."},{"Start":"06:42.650 ","End":"06:47.410","Text":"That\u0027s the answer to d and we\u0027re done."}],"ID":8349},{"Watched":false,"Name":"Exercise 12","Duration":"6m 31s","ChapterTopicVideoID":8195,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8195.jpeg","UploadDate":"2017-01-01T14:01:36.7230000","DurationForVideoObject":"PT6M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"There\u0027s a manufacturer who makes a product and it has"},{"Start":"00:03.870 ","End":"00:10.770","Text":"a demand function P of x. P is price and x is quantity."},{"Start":"00:10.770 ","End":"00:16.620","Text":"Sometimes we use the letter Q of 100 minus"},{"Start":"00:16.620 ","End":"00:24.345","Text":"0.06x and it\u0027s in dollars per unit and he also has a total cost function 200 plus 4x."},{"Start":"00:24.345 ","End":"00:28.005","Text":"Let\u0027s do a first and then we\u0027ll do b."},{"Start":"00:28.005 ","End":"00:32.730","Text":"In a, we\u0027re asked what the amount x is that he needs to"},{"Start":"00:32.730 ","End":"00:38.075","Text":"produce how many units in order to get a maximum profit?"},{"Start":"00:38.075 ","End":"00:44.960","Text":"Now in general, profit is equal to the revenue minus the cost,"},{"Start":"00:44.960 ","End":"00:47.315","Text":"how much you take in, and minus how much you spend."},{"Start":"00:47.315 ","End":"00:49.370","Text":"I\u0027ll just write it in words,"},{"Start":"00:49.370 ","End":"00:53.195","Text":"there\u0027s revenue or total revenue I should say,"},{"Start":"00:53.195 ","End":"00:56.315","Text":"minus cost or total cost."},{"Start":"00:56.315 ","End":"00:59.120","Text":"In terms of letters,"},{"Start":"00:59.120 ","End":"01:00.820","Text":"we\u0027ve used letter P for price,"},{"Start":"01:00.820 ","End":"01:04.400","Text":"so we use the letter Pi for profit."},{"Start":"01:04.400 ","End":"01:08.150","Text":"The profit of the function of x is the revenue,"},{"Start":"01:08.150 ","End":"01:10.205","Text":"which I\u0027ll call TR,"},{"Start":"01:10.205 ","End":"01:13.490","Text":"total revenue, sometimes just R,"},{"Start":"01:13.490 ","End":"01:17.660","Text":"minus the cost total of x."},{"Start":"01:17.660 ","End":"01:22.095","Text":"Now TC, we are given,"},{"Start":"01:22.095 ","End":"01:25.445","Text":"so we have to figure out what TR is."},{"Start":"01:25.445 ","End":"01:29.690","Text":"Now, TR as a function of x,"},{"Start":"01:29.690 ","End":"01:36.485","Text":"the revenue, is just how many items he produced times how many dollars he got per item."},{"Start":"01:36.485 ","End":"01:42.405","Text":"It\u0027s x items and the price of each is P of x."},{"Start":"01:42.405 ","End":"01:46.110","Text":"But we have what P of x is, it\u0027s over here."},{"Start":"01:46.110 ","End":"01:50.760","Text":"I went on, I just highlighted what P of x is."},{"Start":"01:50.760 ","End":"01:53.165","Text":"Here, I have x times,"},{"Start":"01:53.165 ","End":"01:58.880","Text":"I\u0027m just copying this now, 100 minus 0.06x."},{"Start":"01:58.880 ","End":"02:01.580","Text":"Then expand the brackets."},{"Start":"02:01.580 ","End":"02:08.085","Text":"A 100x minus 0.06x^2."},{"Start":"02:08.085 ","End":"02:09.700","Text":"Now, this TR,"},{"Start":"02:09.700 ","End":"02:12.260","Text":"which is what I need here is given by this,"},{"Start":"02:12.260 ","End":"02:13.820","Text":"I\u0027m going to highlight it."},{"Start":"02:13.820 ","End":"02:20.695","Text":"We\u0027re going to have to subtract the total cost is given here."},{"Start":"02:20.695 ","End":"02:25.715","Text":"Perhaps I\u0027ll highlight, this color is this color."},{"Start":"02:25.715 ","End":"02:30.845","Text":"This is what\u0027s over here. What do we get?"},{"Start":"02:30.845 ","End":"02:38.720","Text":"Now I\u0027m copying from here 100x minus 0.06x^2,"},{"Start":"02:38.720 ","End":"02:45.210","Text":"minus TC of x. I\u0027ll just put them both with a minus,"},{"Start":"02:45.210 ","End":"02:49.390","Text":"minus 200 minus 4x."},{"Start":"02:49.550 ","End":"02:52.500","Text":"If I collect like terms,"},{"Start":"02:52.500 ","End":"02:57.345","Text":"we get minus 0.06x^2."},{"Start":"02:57.345 ","End":"02:59.270","Text":"Then, xs, I have 2 of them."},{"Start":"02:59.270 ","End":"03:01.220","Text":"I have 100 minus 4,"},{"Start":"03:01.220 ","End":"03:05.258","Text":"so that\u0027s plus 96x,"},{"Start":"03:05.258 ","End":"03:09.150","Text":"and constants, minus 200."},{"Start":"03:09.150 ","End":"03:12.590","Text":"I\u0027ll highlight the profit function in a different color."},{"Start":"03:12.590 ","End":"03:15.650","Text":"Now, this is the 1 we want to maximize."},{"Start":"03:15.650 ","End":"03:20.495","Text":"At least, we want to find out what x should be in order for this to be maximum."},{"Start":"03:20.495 ","End":"03:23.720","Text":"Notice that this is a quadratic expression,"},{"Start":"03:23.720 ","End":"03:28.985","Text":"polynomial in x. I want to remind you about quadratics and parabolas."},{"Start":"03:28.985 ","End":"03:32.555","Text":"In algebra, you probably studied parabolas,"},{"Start":"03:32.555 ","End":"03:38.085","Text":"y equals ax^2 plus bx plus c in general,"},{"Start":"03:38.085 ","End":"03:39.800","Text":"and in our case,"},{"Start":"03:39.800 ","End":"03:43.700","Text":"this would be a which is minus 0.06,"},{"Start":"03:43.700 ","End":"03:46.085","Text":"this is b, which is 96,"},{"Start":"03:46.085 ","End":"03:48.200","Text":"and c is minus 200."},{"Start":"03:48.200 ","End":"03:51.785","Text":"If a is positive,"},{"Start":"03:51.785 ","End":"03:57.915","Text":"then the parabola faces up, opens upwards,"},{"Start":"03:57.915 ","End":"04:00.665","Text":"and if a is negative,"},{"Start":"04:00.665 ","End":"04:04.490","Text":"it faces downwards like this."},{"Start":"04:04.490 ","End":"04:08.164","Text":"Here, we have a situation with a minimum,"},{"Start":"04:08.164 ","End":"04:11.015","Text":"and here we have a situation with a maximum."},{"Start":"04:11.015 ","End":"04:12.470","Text":"Now, in our case,"},{"Start":"04:12.470 ","End":"04:15.320","Text":"a is less than 0,"},{"Start":"04:15.320 ","End":"04:17.780","Text":"so we\u0027re okay for maximum."},{"Start":"04:17.780 ","End":"04:21.470","Text":"The other thing that we know about these parabolas,"},{"Start":"04:21.470 ","End":"04:26.293","Text":"they have a vertex where the minimum is and the maximum is where the maximum is,"},{"Start":"04:26.293 ","End":"04:28.010","Text":"it\u0027s the tip,"},{"Start":"04:28.010 ","End":"04:31.490","Text":"and the x of the vertex,"},{"Start":"04:31.490 ","End":"04:33.170","Text":"let\u0027s just label it v,"},{"Start":"04:33.170 ","End":"04:34.654","Text":"this is the vertex."},{"Start":"04:34.654 ","End":"04:39.755","Text":"The x of the vertex is minus b over 2a."},{"Start":"04:39.755 ","End":"04:46.060","Text":"Back here, the x of the vertex is going to equal minus b,"},{"Start":"04:46.060 ","End":"04:55.440","Text":"which is minus 96 over 2a, is minus 0.12."},{"Start":"04:55.440 ","End":"05:00.035","Text":"The calculator is 800."},{"Start":"05:00.035 ","End":"05:03.335","Text":"That\u0027s the answer to part a."},{"Start":"05:03.335 ","End":"05:12.160","Text":"It has to produce 800 units per day to maximize profits."},{"Start":"05:12.160 ","End":"05:14.490","Text":"Now let\u0027s look at part b."},{"Start":"05:14.490 ","End":"05:17.190","Text":"Part b is the natural question that arises,"},{"Start":"05:17.190 ","End":"05:19.235","Text":"so this is how many he has to produce,"},{"Start":"05:19.235 ","End":"05:21.910","Text":"but what is the maximum profit?"},{"Start":"05:21.910 ","End":"05:26.885","Text":"Let\u0027s call the maximum profit Pi max,"},{"Start":"05:26.885 ","End":"05:28.625","Text":"which is Pi,"},{"Start":"05:28.625 ","End":"05:31.895","Text":"when you substitute 800."},{"Start":"05:31.895 ","End":"05:36.290","Text":"Now, all we have to do then is look at this profit function"},{"Start":"05:36.290 ","End":"05:41.070","Text":"which is here and just substitute."},{"Start":"05:41.070 ","End":"05:49.485","Text":"I\u0027ve got minus 0.06 800 squared,"},{"Start":"05:49.485 ","End":"05:58.870","Text":"plus 96 times 800 minus 200."},{"Start":"05:59.380 ","End":"06:01.790","Text":"I won\u0027t make you do the work."},{"Start":"06:01.790 ","End":"06:04.204","Text":"I\u0027ll just give you the answer."},{"Start":"06:04.204 ","End":"06:10.685","Text":"I make it 38,200"},{"Start":"06:10.685 ","End":"06:17.730","Text":"and we\u0027re working in dollars so this is the answer to part b."},{"Start":"06:17.730 ","End":"06:25.550","Text":"It\u0027ll produce 800 units to get the maximum profit of $38,200."},{"Start":"06:25.550 ","End":"06:31.990","Text":"I just forgot to write part b here. Now we\u0027re done."}],"ID":8350},{"Watched":false,"Name":"Exercise 13","Duration":"7m 3s","ChapterTopicVideoID":8196,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8196.jpeg","UploadDate":"2017-01-01T14:03:26.2900000","DurationForVideoObject":"PT7M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.000","Text":"In this exercise, it concerns a manufacturer who has a product,"},{"Start":"00:06.000 ","End":"00:08.820","Text":"and it has a constant demand function."},{"Start":"00:08.820 ","End":"00:11.955","Text":"The price is always 20,"},{"Start":"00:11.955 ","End":"00:14.775","Text":"say we\u0027re working in dollars,"},{"Start":"00:14.775 ","End":"00:17.880","Text":"no matter how much the quantity is,"},{"Start":"00:17.880 ","End":"00:19.080","Text":"how much the demand is,"},{"Start":"00:19.080 ","End":"00:21.680","Text":"x is sometimes called q."},{"Start":"00:21.680 ","End":"00:28.110","Text":"The total cost function TC is given by the following formula."},{"Start":"00:28.110 ","End":"00:33.908","Text":"In a, we need to find which quantity he needs to produce,"},{"Start":"00:33.908 ","End":"00:37.950","Text":"how many units, in order to maximize his profit."},{"Start":"00:37.950 ","End":"00:40.540","Text":"Then after we find out the x,"},{"Start":"00:40.540 ","End":"00:45.090","Text":"we want to know what actually is the maximum profit."},{"Start":"00:45.370 ","End":"00:51.630","Text":"Let\u0027s start with a word formula for profit."},{"Start":"00:52.930 ","End":"00:56.375","Text":"It\u0027s the total revenue."},{"Start":"00:56.375 ","End":"01:00.590","Text":"How much you receive from selling the product,"},{"Start":"01:00.590 ","End":"01:03.380","Text":"minus what your expenses are."},{"Start":"01:03.380 ","End":"01:06.935","Text":"The total cost."},{"Start":"01:06.935 ","End":"01:10.310","Text":"If we write that in symbols,"},{"Start":"01:10.310 ","End":"01:16.735","Text":"profit, in this course we use the letter Pi because p is reserved for price,"},{"Start":"01:16.735 ","End":"01:20.050","Text":"so Pi of quantity x,"},{"Start":"01:20.050 ","End":"01:22.615","Text":"is the total revenue,"},{"Start":"01:22.615 ","End":"01:29.595","Text":"TR of x minus the total cost for x units."},{"Start":"01:29.595 ","End":"01:31.350","Text":"Now as it stands,"},{"Start":"01:31.350 ","End":"01:35.070","Text":"we have TC exactly over here."},{"Start":"01:35.070 ","End":"01:38.710","Text":"I think I\u0027ll highlight it."},{"Start":"01:39.770 ","End":"01:44.010","Text":"TC, yeah,"},{"Start":"01:44.010 ","End":"01:46.180","Text":"I\u0027ve highlighted here also."},{"Start":"01:46.180 ","End":"01:51.635","Text":"Total revenue is not given but we can compute it."},{"Start":"01:51.635 ","End":"01:57.920","Text":"Now, total revenue is the number of units, x,"},{"Start":"01:57.920 ","End":"02:01.850","Text":"and they\u0027re sold at p of x, so p of x is 20,"},{"Start":"02:01.850 ","End":"02:04.745","Text":"I don\u0027t even have to put times p of x,"},{"Start":"02:04.745 ","End":"02:06.050","Text":"it\u0027s x times 20."},{"Start":"02:06.050 ","End":"02:09.835","Text":"In other words, 20 bucks a piece, x pieces,"},{"Start":"02:09.835 ","End":"02:16.965","Text":"20x dollars, that\u0027s total revenue. I should have said of x."},{"Start":"02:16.965 ","End":"02:21.730","Text":"That\u0027s better. I\u0027ll do that in this color,"},{"Start":"02:21.730 ","End":"02:24.870","Text":"and there\u0027s our total revenue."},{"Start":"02:25.300 ","End":"02:31.250","Text":"Profit Pi is going to be equal total revenue from here,"},{"Start":"02:31.250 ","End":"02:34.355","Text":"that\u0027s 20x dollars,"},{"Start":"02:34.355 ","End":"02:36.410","Text":"minus the cost,"},{"Start":"02:36.410 ","End":"02:38.030","Text":"so take that from here,"},{"Start":"02:38.030 ","End":"02:45.325","Text":"that\u0027s 300 plus 2x^2,"},{"Start":"02:45.325 ","End":"02:48.845","Text":"and just a little bit of simplification,"},{"Start":"02:48.845 ","End":"02:50.900","Text":"this will put them in order."},{"Start":"02:50.900 ","End":"02:54.470","Text":"I\u0027ll put the x^2 first,"},{"Start":"02:54.470 ","End":"02:56.860","Text":"and then I\u0027ll put the xs,"},{"Start":"02:56.860 ","End":"02:58.965","Text":"and then the numbers,"},{"Start":"02:58.965 ","End":"03:04.000","Text":"then I\u0027ll highlight the profit in a different color."},{"Start":"03:04.040 ","End":"03:07.185","Text":"Now, to find minimum or maximum,"},{"Start":"03:07.185 ","End":"03:08.595","Text":"in this case maximum,"},{"Start":"03:08.595 ","End":"03:12.335","Text":"we normally would use calculus,"},{"Start":"03:12.335 ","End":"03:15.310","Text":"but some of you may not have learned calculus."},{"Start":"03:15.310 ","End":"03:19.562","Text":"If you haven\u0027t, I hope you\u0027ve learned about the quadratic function and parabolas"},{"Start":"03:19.562 ","End":"03:23.795","Text":"because this is like a quadratic function,"},{"Start":"03:23.795 ","End":"03:34.090","Text":"where y equals ax^2 plus bx plus c. I\u0027ll change the color."},{"Start":"03:34.090 ","End":"03:39.665","Text":"This is of this form where this would be our a negative 2,"},{"Start":"03:39.665 ","End":"03:41.720","Text":"b would be 20,"},{"Start":"03:41.720 ","End":"03:45.110","Text":"and c would be minus 300."},{"Start":"03:45.110 ","End":"03:55.730","Text":"What we know about such curves is that if a is bigger than 0,"},{"Start":"03:55.730 ","End":"04:02.470","Text":"then the parabola faces upwards,"},{"Start":"04:03.320 ","End":"04:09.090","Text":"opens upwards, and if a is negative,"},{"Start":"04:09.090 ","End":"04:15.075","Text":"then it opens or faces downwards."},{"Start":"04:15.075 ","End":"04:19.420","Text":"In our case, our a is negative,"},{"Start":"04:20.410 ","End":"04:22.820","Text":"so we\u0027re in this case,"},{"Start":"04:22.820 ","End":"04:25.565","Text":"which is good because in this case,"},{"Start":"04:25.565 ","End":"04:27.760","Text":"we have a maximum."},{"Start":"04:27.760 ","End":"04:32.090","Text":"It occurs that here also there\u0027s a vertex,"},{"Start":"04:32.090 ","End":"04:33.500","Text":"but it\u0027s a minimum."},{"Start":"04:33.500 ","End":"04:37.170","Text":"Anyway, we\u0027re only concerned now with this 1,"},{"Start":"04:37.170 ","End":"04:40.140","Text":"the vertex I\u0027ll call it v for this point."},{"Start":"04:40.140 ","End":"04:45.600","Text":"There is a formula that the x value of"},{"Start":"04:45.600 ","End":"04:52.200","Text":"the vertex is given by minus b over 2a,"},{"Start":"04:52.200 ","End":"04:55.560","Text":"and that will do for the maximum,"},{"Start":"04:55.560 ","End":"04:57.060","Text":"at least we\u0027ll have the x of it."},{"Start":"04:57.060 ","End":"05:00.120","Text":"Here, we will do that,"},{"Start":"05:00.120 ","End":"05:05.290","Text":"and we\u0027ll get x_v"},{"Start":"05:05.290 ","End":"05:12.375","Text":"which is minus 20 over 2a is twice minus 2,"},{"Start":"05:12.375 ","End":"05:14.580","Text":"and we can do that in our heads."},{"Start":"05:14.580 ","End":"05:19.560","Text":"Minus 20 over minus 4 is 5."},{"Start":"05:19.560 ","End":"05:25.859","Text":"What that means is that the answer to part a"},{"Start":"05:25.859 ","End":"05:33.675","Text":"is the 5 units needed for the minimum."},{"Start":"05:33.675 ","End":"05:36.135","Text":"I have another color left,"},{"Start":"05:36.135 ","End":"05:40.955","Text":"so this color is for the answer for part a."},{"Start":"05:40.955 ","End":"05:47.060","Text":"Now, part b is easy because all we have to do is plug"},{"Start":"05:47.060 ","End":"05:54.105","Text":"in this value of x5 into the profit function, which is this."},{"Start":"05:54.105 ","End":"05:58.750","Text":"In other words, I want Pi of 5,"},{"Start":"05:58.750 ","End":"06:01.370","Text":"and see what this is equal to."},{"Start":"06:01.370 ","End":"06:08.465","Text":"This will be my, let\u0027s call it Pi max, Pi profit,"},{"Start":"06:08.465 ","End":"06:12.635","Text":"max for maximum is 5,"},{"Start":"06:12.635 ","End":"06:15.500","Text":"and so it\u0027s equal to,"},{"Start":"06:15.500 ","End":"06:16.655","Text":"I\u0027m plugging it here,"},{"Start":"06:16.655 ","End":"06:22.596","Text":"minus 2 times 5^2 plus 20"},{"Start":"06:22.596 ","End":"06:29.160","Text":"times 5 minus 300."},{"Start":"06:29.160 ","End":"06:31.440","Text":"Let\u0027s see. 5^2 is 25,"},{"Start":"06:31.440 ","End":"06:33.615","Text":"so that\u0027s minus 50."},{"Start":"06:33.615 ","End":"06:38.700","Text":"Another negative, that\u0027s all together minus 350 plus 100,"},{"Start":"06:38.700 ","End":"06:47.955","Text":"so what we end up with is minus 250 dollars."},{"Start":"06:47.955 ","End":"06:50.460","Text":"We put the dollar to the left to the minus,"},{"Start":"06:50.460 ","End":"06:52.260","Text":"so to the right of the minus,"},{"Start":"06:52.260 ","End":"06:54.810","Text":"and that you ask is the maximum."},{"Start":"06:54.810 ","End":"06:57.145","Text":"It\u0027s a loss, not a profit."},{"Start":"06:57.145 ","End":"06:58.745","Text":"But it\u0027s the best you could do."},{"Start":"06:58.745 ","End":"07:00.500","Text":"Anyway, I\u0027ll just highlight it,"},{"Start":"07:00.500 ","End":"07:03.690","Text":"and declare that we are done."}],"ID":8351},{"Watched":false,"Name":"Exercise 14","Duration":"7m 56s","ChapterTopicVideoID":8197,"CourseChapterTopicPlaylistID":56142,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8197.jpeg","UploadDate":"2017-01-01T14:05:29.8170000","DurationForVideoObject":"PT7M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:10.245","Text":"A manufacturer has a product with demand function as follows; x=5000-50p/3."},{"Start":"00:10.245 ","End":"00:16.005","Text":"The total cost function TC(x)=200 + 4x."},{"Start":"00:16.005 ","End":"00:18.525","Text":"Everything\u0027s in dollars here."},{"Start":"00:18.525 ","End":"00:23.480","Text":"But notice that the total cost is given in terms of x,"},{"Start":"00:23.480 ","End":"00:26.810","Text":"which is the quantity,"},{"Start":"00:26.810 ","End":"00:31.895","Text":"number of items or units produced."},{"Start":"00:31.895 ","End":"00:35.735","Text":"But this 1 is the other way round which we sometimes get"},{"Start":"00:35.735 ","End":"00:40.880","Text":"its quantity in terms of price and not price in terms of quantity."},{"Start":"00:40.880 ","End":"00:44.450","Text":"Notice that there\u0027s something reversed here."},{"Start":"00:44.450 ","End":"00:46.410","Text":"Now let\u0027s get onto part a."},{"Start":"00:46.410 ","End":"00:47.970","Text":"We\u0027ll do the 1 at the time."},{"Start":"00:47.970 ","End":"00:51.950","Text":"In a we want to find the quantity"},{"Start":"00:51.950 ","End":"00:56.819","Text":"x that the manufacturer needs to produce to maximize his profit,"},{"Start":"00:56.819 ","End":"01:03.600","Text":"and then later in b we\u0027ll actually find out what is that maximum profit."},{"Start":"01:04.130 ","End":"01:08.345","Text":"We have to remember how to find profit."},{"Start":"01:08.345 ","End":"01:12.005","Text":"Is that profit is the difference of 2 things."},{"Start":"01:12.005 ","End":"01:19.985","Text":"It\u0027s the total income or revenue minus what he spent,"},{"Start":"01:19.985 ","End":"01:24.540","Text":"which is the cost or total cost."},{"Start":"01:24.540 ","End":"01:27.020","Text":"If we put this in symbols, profit,"},{"Start":"01:27.020 ","End":"01:31.700","Text":"we use the Greek letter Pi because P stands for price."},{"Start":"01:31.700 ","End":"01:36.575","Text":"Pi(x) ="},{"Start":"01:36.575 ","End":"01:46.040","Text":"TR(x)-TC(x)."},{"Start":"01:46.040 ","End":"01:47.810","Text":"Now we have total cost here,"},{"Start":"01:47.810 ","End":"01:50.480","Text":"but we don\u0027t have total revenue."},{"Start":"01:50.480 ","End":"01:54.050","Text":"However, total revenue, there is a formula for it."},{"Start":"01:54.050 ","End":"02:01.280","Text":"The total revenue, how much he takes in is the number of items or"},{"Start":"02:01.280 ","End":"02:10.340","Text":"products times the price per each item, the unit price."},{"Start":"02:10.340 ","End":"02:13.280","Text":"But we don\u0027t have P in terms of x,"},{"Start":"02:13.280 ","End":"02:16.490","Text":"we have x in terms of p. What we want to do"},{"Start":"02:16.490 ","End":"02:20.810","Text":"now is a bit of algebra to reverse this equation."},{"Start":"02:20.810 ","End":"02:24.440","Text":"This 1 I wanted to write it the other way round."},{"Start":"02:24.440 ","End":"02:29.285","Text":"Let\u0027s do the algebra at the side and just move that formula here."},{"Start":"02:29.285 ","End":"02:30.560","Text":"I\u0027ll have room here."},{"Start":"02:30.560 ","End":"02:37.010","Text":"I\u0027m going to copy x=5000-50p/3."},{"Start":"02:37.010 ","End":"02:38.525","Text":"I want to extract p,"},{"Start":"02:38.525 ","End":"02:42.210","Text":"multiply both sides by 3."},{"Start":"02:42.210 ","End":"02:48.345","Text":"So I\u0027ve got 3x=5000-50p."},{"Start":"02:48.345 ","End":"02:51.930","Text":"Let me put the ps is on the left and the xs is on the right."},{"Start":"02:51.930 ","End":"02:58.740","Text":"I have 50p=5000-3x,"},{"Start":"02:58.740 ","End":"03:02.495","Text":"then all I have left to do is divide by 50."},{"Start":"03:02.495 ","End":"03:09.390","Text":"If I divide 5000 divided by 50 is 100."},{"Start":"03:10.310 ","End":"03:17.810","Text":"3/50, I\u0027ll just write it as a decimal is 0.06x."},{"Start":"03:17.810 ","End":"03:19.985","Text":"Now that I have this,"},{"Start":"03:19.985 ","End":"03:28.010","Text":"and I can write this as p in terms of x. I could just substitute that in"},{"Start":"03:28.010 ","End":"03:33.965","Text":"here and continue and get the total revenue is"},{"Start":"03:33.965 ","End":"03:41.610","Text":"x times 100 minus 0.06x."},{"Start":"03:42.280 ","End":"03:45.095","Text":"Now I can continue with this formula,"},{"Start":"03:45.095 ","End":"03:47.210","Text":"let me just highlight to make things clearer,"},{"Start":"03:47.210 ","End":"03:53.675","Text":"the total revenue I\u0027m going to get from here,"},{"Start":"03:53.675 ","End":"04:00.750","Text":"the total cost I can get from here."},{"Start":"04:00.850 ","End":"04:03.500","Text":"If I substitute these,"},{"Start":"04:03.500 ","End":"04:06.680","Text":"I can now figure out what is the profit."},{"Start":"04:06.680 ","End":"04:15.920","Text":"I\u0027ve got Pi profit of x is equal to this green color minus the magenta color."},{"Start":"04:15.920 ","End":"04:18.230","Text":"From here, I multiply out,"},{"Start":"04:18.230 ","End":"04:26.840","Text":"I\u0027ve got 100x-0.06x^2."},{"Start":"04:26.840 ","End":"04:35.420","Text":"Then I have to subtract this so I\u0027ll just put them both with a minus, -200-4x."},{"Start":"04:35.420 ","End":"04:39.680","Text":"I just collect like terms and order them,"},{"Start":"04:39.680 ","End":"04:46.660","Text":"so I\u0027ve got -0.06 x^2."},{"Start":"04:46.660 ","End":"04:55.165","Text":"Let\u0027s see plus 100x-4x, that\u0027s plus 96x-200."},{"Start":"04:55.165 ","End":"04:59.374","Text":"I think it\u0027ll highlight this to this is the profit function."},{"Start":"04:59.374 ","End":"05:03.710","Text":"Now we have the function that we need to maximize."},{"Start":"05:03.710 ","End":"05:10.220","Text":"We want to find the x that gives the largest value of the profit on what the profit is."},{"Start":"05:10.220 ","End":"05:13.880","Text":"We could do this with calculus and differentiation."},{"Start":"05:13.880 ","End":"05:16.790","Text":"We don\u0027t need to, because this is a quadratic function."},{"Start":"05:16.790 ","End":"05:24.815","Text":"Want to remind you when we have y=ax^2+bx+c,"},{"Start":"05:24.815 ","End":"05:28.115","Text":"then depending on the sign of a,"},{"Start":"05:28.115 ","End":"05:30.560","Text":"if a is negative,"},{"Start":"05:30.560 ","End":"05:32.825","Text":"we have a parabola like this,"},{"Start":"05:32.825 ","End":"05:35.885","Text":"facing down or opening downwards."},{"Start":"05:35.885 ","End":"05:37.520","Text":"If a is positive,"},{"Start":"05:37.520 ","End":"05:40.010","Text":"which it isn\u0027t, then when it would be this way round."},{"Start":"05:40.010 ","End":"05:45.980","Text":"In each case there\u0027s a vertex which is either the minimum or the maximum."},{"Start":"05:45.980 ","End":"05:50.765","Text":"In our case, we have that this is our a which is negative."},{"Start":"05:50.765 ","End":"05:55.225","Text":"We have a maximum here."},{"Start":"05:55.225 ","End":"05:57.740","Text":"If we label the vertex v,"},{"Start":"05:57.740 ","End":"06:04.280","Text":"there\u0027s a formula that the x_v is minus b/2a."},{"Start":"06:04.280 ","End":"06:06.110","Text":"It\u0027s the same formula here,"},{"Start":"06:06.110 ","End":"06:08.520","Text":"but we don\u0027t need this 1."},{"Start":"06:11.360 ","End":"06:15.480","Text":"Well, let\u0027s continue labeling b is 96."},{"Start":"06:15.480 ","End":"06:16.560","Text":"We don\u0027t even need c,"},{"Start":"06:16.560 ","End":"06:19.120","Text":"but it\u0027s minus 200."},{"Start":"06:19.670 ","End":"06:25.160","Text":"We get the x_v is minus b/2a is"},{"Start":"06:25.160 ","End":"06:34.020","Text":"minus 96 over 2(-0.06)."},{"Start":"06:34.930 ","End":"06:38.450","Text":"This is minus will cancel with the minus,"},{"Start":"06:38.450 ","End":"06:43.375","Text":"and we\u0027ll get 96/0.12."},{"Start":"06:43.375 ","End":"06:47.550","Text":"This comes out to be 800."},{"Start":"06:47.550 ","End":"06:51.595","Text":"So 800 is the answer to part a."},{"Start":"06:51.595 ","End":"06:58.700","Text":"That is the quantity or the number of units that have to be sold to maximum profit."},{"Start":"06:58.700 ","End":"07:04.845","Text":"All we have to do now in part b is find out what"},{"Start":"07:04.845 ","End":"07:11.665","Text":"the profit is for 800 units."},{"Start":"07:11.665 ","End":"07:14.360","Text":"Still have the formula written up here."},{"Start":"07:14.360 ","End":"07:18.965","Text":"What I want is Pi maximum,"},{"Start":"07:18.965 ","End":"07:21.080","Text":"which is the PI,"},{"Start":"07:21.080 ","End":"07:25.214","Text":"the profit for 800 units."},{"Start":"07:25.214 ","End":"07:29.555","Text":"That is equal to from here,"},{"Start":"07:29.555 ","End":"07:38.690","Text":"-0.06 times 800^2+96 times"},{"Start":"07:38.690 ","End":"07:47.420","Text":"800-200 comes out to be 38,200."},{"Start":"07:47.420 ","End":"07:51.320","Text":"I did the calculation for you and it\u0027s in dollars."},{"Start":"07:51.320 ","End":"07:54.095","Text":"Let concludes part b."},{"Start":"07:54.095 ","End":"07:57.360","Text":"We\u0027ve done with this exercise."}],"ID":8352}],"Thumbnail":null,"ID":56142},{"Name":"Common Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Linear Function","Duration":"3m 51s","ChapterTopicVideoID":5135,"CourseChapterTopicPlaylistID":56143,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5135.jpeg","UploadDate":"2020-09-30T13:15:56.6730000","DurationForVideoObject":"PT3M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this clip, we introduce the concept of a linear function."},{"Start":"00:04.260 ","End":"00:14.265","Text":"Linear function is a function of the form y or f of x is equal to mx plus n,"},{"Start":"00:14.265 ","End":"00:17.970","Text":"where m and n are just some numbers, some constants."},{"Start":"00:17.970 ","End":"00:21.615","Text":"Often in the literature 1c is written as,"},{"Start":"00:21.615 ","End":"00:26.700","Text":"say, y equals ax plus b very common letters."},{"Start":"00:26.700 ","End":"00:29.550","Text":"I\u0027ve seen all sorts of hybrid combinations."},{"Start":"00:29.550 ","End":"00:34.695","Text":"I\u0027ve seen y equals mx plus b. I\u0027ll give you an example."},{"Start":"00:34.695 ","End":"00:41.065","Text":"A linear function could be y equals 4x plus 10."},{"Start":"00:41.065 ","End":"00:42.440","Text":"That\u0027s a specific example."},{"Start":"00:42.440 ","End":"00:45.650","Text":"It doesn\u0027t really matter what letters we use in the general."},{"Start":"00:45.650 ","End":"00:47.705","Text":"Of course, in this case, if we referring to,"},{"Start":"00:47.705 ","End":"00:51.005","Text":"we\u0027ll use this as the model just to keep it standard."},{"Start":"00:51.005 ","End":"00:57.655","Text":"In this case, we would have that m is equal to 4 and n is equal to 10."},{"Start":"00:57.655 ","End":"00:59.540","Text":"Any pair of numbers we give for m and n,"},{"Start":"00:59.540 ","End":"01:01.145","Text":"we get a linear function."},{"Start":"01:01.145 ","End":"01:05.400","Text":"Here\u0027s another example, y equals minus 10x."},{"Start":"01:06.280 ","End":"01:08.730","Text":"What do we get? Here,"},{"Start":"01:08.730 ","End":"01:10.395","Text":"what is m and what is n?"},{"Start":"01:10.395 ","End":"01:13.150","Text":"Here m is equal to minus 10x."},{"Start":"01:13.760 ","End":"01:16.890","Text":"What about the n? There is no n,"},{"Start":"01:16.890 ","End":"01:20.580","Text":"so in other words, n is equal to 0."},{"Start":"01:20.580 ","End":"01:25.820","Text":"Yet another example, y is equal to 8."},{"Start":"01:25.820 ","End":"01:27.995","Text":"Here, we don\u0027t have anything with x,"},{"Start":"01:27.995 ","End":"01:30.835","Text":"so that means that m is 0."},{"Start":"01:30.835 ","End":"01:33.795","Text":"In this case, n is 8."},{"Start":"01:33.795 ","End":"01:37.875","Text":"You see if I write minus 10x plus 0 as here,"},{"Start":"01:37.875 ","End":"01:39.450","Text":"and that\u0027s minus 10x."},{"Start":"01:39.450 ","End":"01:41.250","Text":"If I write 0x plus 8,"},{"Start":"01:41.250 ","End":"01:42.570","Text":"there\u0027s no need for the 0x,"},{"Start":"01:42.570 ","End":"01:44.205","Text":"so it\u0027s just 8."},{"Start":"01:44.205 ","End":"01:47.270","Text":"They could be m or n could be 0,"},{"Start":"01:47.270 ","End":"01:49.970","Text":"and so it looks like a term is missing."},{"Start":"01:49.970 ","End":"01:56.270","Text":"In fact, you can get more extreme situations with almost nothing left in the equation."},{"Start":"01:56.270 ","End":"02:01.760","Text":"For example, y equals 0 is also a linear equation."},{"Start":"02:01.760 ","End":"02:05.749","Text":"It\u0027s just that m and n are both 0."},{"Start":"02:05.749 ","End":"02:08.240","Text":"Because if I put m and n, both 0 here,"},{"Start":"02:08.240 ","End":"02:11.540","Text":"I just get y equals 0 and this too is a linear equation."},{"Start":"02:11.540 ","End":"02:16.220","Text":"Now sometimes something that doesn\u0027t appear to be a linear equation is,"},{"Start":"02:16.220 ","End":"02:18.860","Text":"in fact, or can be brought into a linear form."},{"Start":"02:18.860 ","End":"02:25.930","Text":"For example, I could take the equation 4y plus 2x equals 10."},{"Start":"02:25.930 ","End":"02:28.790","Text":"It\u0027s an equation but it\u0027s not a function."},{"Start":"02:28.790 ","End":"02:32.870","Text":"But it turns out that you can actually make y as a linear function of"},{"Start":"02:32.870 ","End":"02:37.485","Text":"x just by doing a bit of algebra. Let\u0027s see."},{"Start":"02:37.485 ","End":"02:40.880","Text":"If we bring the 2x to the other side,"},{"Start":"02:40.880 ","End":"02:43.920","Text":"we will get that"},{"Start":"02:44.660 ","End":"02:51.395","Text":"4y is equal to minus 2x plus 10."},{"Start":"02:51.395 ","End":"02:53.285","Text":"Dividing by 4,"},{"Start":"02:53.285 ","End":"02:57.580","Text":"we get that y is equal to minus 2 over 4."},{"Start":"02:57.580 ","End":"02:59.196","Text":"Let\u0027s cancel the fraction."},{"Start":"02:59.196 ","End":"03:04.645","Text":"That will be minus 1.5 of x plus 10/4."},{"Start":"03:04.645 ","End":"03:07.890","Text":"10/4 is 5 over 2 or 2.5,"},{"Start":"03:07.890 ","End":"03:10.420","Text":"we\u0027ll write it as 5/2."},{"Start":"03:10.420 ","End":"03:17.000","Text":"This equation actually represented y as a linear function of x."},{"Start":"03:17.000 ","End":"03:21.555","Text":"We can say that and point out that m is equal to"},{"Start":"03:21.555 ","End":"03:28.785","Text":"minus 1/2 and that n is equal to 5/2."},{"Start":"03:28.785 ","End":"03:30.815","Text":"We haven\u0027t done very much so far."},{"Start":"03:30.815 ","End":"03:35.540","Text":"We\u0027ve just shown essentially what a linear function is and this is"},{"Start":"03:35.540 ","End":"03:42.470","Text":"the form mx plus n is the general linear function in x with,"},{"Start":"03:42.470 ","End":"03:44.964","Text":"as I say, some other textbooks,"},{"Start":"03:44.964 ","End":"03:46.370","Text":"so I\u0027m using different letters."},{"Start":"03:46.370 ","End":"03:50.190","Text":"But we don\u0027t let that worry us. That\u0027s it for now."}],"ID":5123},{"Watched":false,"Name":"The Quadratic Function","Duration":"3m 35s","ChapterTopicVideoID":5136,"CourseChapterTopicPlaylistID":56143,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5136.jpeg","UploadDate":"2020-09-30T13:16:32.9400000","DurationForVideoObject":"PT3M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"This clip is a brief introduction to the quadratic function."},{"Start":"00:05.085 ","End":"00:11.783","Text":"A quadratic function is a function of the form f of x is"},{"Start":"00:11.783 ","End":"00:18.660","Text":"equal to ax^2 plus bx plus c,"},{"Start":"00:18.660 ","End":"00:21.810","Text":"where a, b, and c are some constants."},{"Start":"00:21.810 ","End":"00:26.910","Text":"Now it\u0027s very important that a not to be 0,"},{"Start":"00:26.910 ","End":"00:29.700","Text":"because if a is 0,"},{"Start":"00:29.700 ","End":"00:31.230","Text":"it\u0027s not a quadratic function."},{"Start":"00:31.230 ","End":"00:32.700","Text":"There is no x^2 term,"},{"Start":"00:32.700 ","End":"00:34.455","Text":"it\u0027s a linear function."},{"Start":"00:34.455 ","End":"00:37.005","Text":"So a shouldn\u0027t be 0."},{"Start":"00:37.005 ","End":"00:40.425","Text":"I\u0027ll give you an example to start off with."},{"Start":"00:40.425 ","End":"00:43.885","Text":"How about, f of x"},{"Start":"00:43.885 ","End":"00:52.055","Text":"is equal to 4x^2 minus 10x plus 1."},{"Start":"00:52.055 ","End":"00:59.675","Text":"This is a quadratic function with the constants a being equal to 4,"},{"Start":"00:59.675 ","End":"01:04.655","Text":"b being minus 10,"},{"Start":"01:04.655 ","End":"01:07.280","Text":"and c being 1."},{"Start":"01:07.280 ","End":"01:12.185","Text":"Another example, and here I\u0027ll use the y notation,"},{"Start":"01:12.185 ","End":"01:21.560","Text":"y is equal to minus x^2 plus 20."},{"Start":"01:21.560 ","End":"01:25.395","Text":"Here, a is equal to minus 1,"},{"Start":"01:25.395 ","End":"01:27.900","Text":"that\u0027s the coefficient of x^2,"},{"Start":"01:27.900 ","End":"01:30.815","Text":"b is absent, it\u0027s missing."},{"Start":"01:30.815 ","End":"01:32.960","Text":"When it\u0027s missing, it\u0027s 0."},{"Start":"01:32.960 ","End":"01:36.920","Text":"Because you could say plus 0x here."},{"Start":"01:36.920 ","End":"01:39.514","Text":"As for c, it\u0027s just 20."},{"Start":"01:39.514 ","End":"01:42.140","Text":"Another example."},{"Start":"01:42.140 ","End":"01:52.275","Text":"Let\u0027s take a g of x for a change, different letter."},{"Start":"01:52.275 ","End":"01:57.040","Text":"g of x is 50x^2 minus 20x."},{"Start":"02:01.480 ","End":"02:04.520","Text":"Now here in the previous example,"},{"Start":"02:04.520 ","End":"02:05.870","Text":"we had b equals 0."},{"Start":"02:05.870 ","End":"02:13.015","Text":"Here c is missing the free co-efficient so we get a is equal to 50,"},{"Start":"02:13.015 ","End":"02:18.965","Text":"b would be minus 20 in this case."},{"Start":"02:18.965 ","End":"02:20.735","Text":"Since there is no other term,"},{"Start":"02:20.735 ","End":"02:27.070","Text":"it\u0027s like saying that c equals 0, plus 0 here."},{"Start":"02:27.950 ","End":"02:30.890","Text":"Let\u0027s take 1 more example."},{"Start":"02:30.890 ","End":"02:33.925","Text":"We\u0027ll take it in terms of y again."},{"Start":"02:33.925 ","End":"02:39.684","Text":"y is equal to minus 4x^2."},{"Start":"02:39.684 ","End":"02:41.085","Text":"Just like here,"},{"Start":"02:41.085 ","End":"02:44.085","Text":"b was 0 and here c was 0."},{"Start":"02:44.085 ","End":"02:46.500","Text":"Here both b and c are 0."},{"Start":"02:46.500 ","End":"02:48.030","Text":"We\u0027re only left with a."},{"Start":"02:48.030 ","End":"02:50.085","Text":"A is minus 4,"},{"Start":"02:50.085 ","End":"02:52.515","Text":"b is missing,"},{"Start":"02:52.515 ","End":"02:55.095","Text":"b term is missing,"},{"Start":"02:55.095 ","End":"02:59.870","Text":"and the c term is missing which means that c is 0."},{"Start":"02:59.870 ","End":"03:03.260","Text":"That\u0027s as far as the examples go."},{"Start":"03:03.260 ","End":"03:08.360","Text":"In the rest of the tutorial on the quadratic function,"},{"Start":"03:08.360 ","End":"03:11.240","Text":"we\u0027ll learn things like what is the meaning of a,"},{"Start":"03:11.240 ","End":"03:14.645","Text":"b, and c, the graph of the quadratic function"},{"Start":"03:14.645 ","End":"03:18.650","Text":"which happens to be in the shape of a parabola."},{"Start":"03:18.650 ","End":"03:19.700","Text":"I\u0027ll just write the word,"},{"Start":"03:19.700 ","End":"03:21.695","Text":"but we\u0027ll encounter it later."},{"Start":"03:21.695 ","End":"03:25.100","Text":"We\u0027ll learn about the vertex of the parabola,"},{"Start":"03:25.100 ","End":"03:30.199","Text":"we\u0027ll learn about the intersection of the quadratic function with the axes,"},{"Start":"03:30.199 ","End":"03:33.980","Text":"the x and y, and all sorts of other fun stuff like that."},{"Start":"03:33.980 ","End":"03:36.900","Text":"But we\u0027re done for the intro."}],"ID":5124},{"Watched":false,"Name":"The Exponential Function","Duration":"13m 9s","ChapterTopicVideoID":5137,"CourseChapterTopicPlaylistID":56143,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5137.jpeg","UploadDate":"2020-09-30T12:53:28.2700000","DurationForVideoObject":"PT13M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this section we\u0027ll talk about the exponential function."},{"Start":"00:04.050 ","End":"00:07.690","Text":"The exponential function looks like,"},{"Start":"00:08.060 ","End":"00:12.885","Text":"f(x) is equal to a^x,"},{"Start":"00:12.885 ","End":"00:15.675","Text":"where a is a constant, or parameter."},{"Start":"00:15.675 ","End":"00:17.949","Text":"But, there is a restriction,"},{"Start":"00:17.949 ","End":"00:22.875","Text":"and the restriction is that a has to be bigger than 0,"},{"Start":"00:22.875 ","End":"00:26.830","Text":"and it also should not equal 1."},{"Start":"00:27.470 ","End":"00:31.685","Text":"In other words, it\u0027s a positive number not equal to 1,"},{"Start":"00:31.685 ","End":"00:33.350","Text":"and this is the function."},{"Start":"00:33.350 ","End":"00:39.640","Text":"The reason its called exponential is that the variable x is the exponent."},{"Start":"00:39.640 ","End":"00:42.045","Text":"A also has a name, it\u0027s the base."},{"Start":"00:42.045 ","End":"00:44.830","Text":"We have base, and exponent."},{"Start":"00:45.920 ","End":"00:48.300","Text":"Just as an aside,"},{"Start":"00:48.300 ","End":"00:50.914","Text":"in a lot of books, and a lot of colleges,"},{"Start":"00:50.914 ","End":"00:59.780","Text":"they use b^x rather than a^x because it has a nice mnemonic because"},{"Start":"00:59.780 ","End":"01:10.029","Text":"then x is like exponent and b is base."},{"Start":"01:10.140 ","End":"01:13.580","Text":"But here, we\u0027ll use the letter a."},{"Start":"01:13.680 ","End":"01:17.170","Text":"Some examples might be,"},{"Start":"01:17.170 ","End":"01:22.980","Text":"we could take f(x) is equal to 4^x,"},{"Start":"01:22.980 ","End":"01:24.635","Text":"or we could take,"},{"Start":"01:24.635 ","End":"01:30.670","Text":"let\u0027s say we use the y notation, y=1/2^x,"},{"Start":"01:30.670 ","End":"01:33.880","Text":"or y equals, I don\u0027t know,"},{"Start":"01:34.100 ","End":"01:37.535","Text":"4.7^x, something to the power of x."},{"Start":"01:37.535 ","End":"01:40.930","Text":"In case you\u0027re wondering about these restrictions about"},{"Start":"01:40.930 ","End":"01:44.320","Text":"why a has to be positive and not equal to 1,"},{"Start":"01:44.320 ","End":"01:46.930","Text":"I\u0027ll just mention this,"},{"Start":"01:46.930 ","End":"01:50.870","Text":"if a=1, then we just get the constant function."},{"Start":"01:50.870 ","End":"01:53.330","Text":"Think about it, if a is 1,"},{"Start":"01:53.330 ","End":"01:56.540","Text":"1^x is always 1, so if a is 1,"},{"Start":"01:56.540 ","End":"01:59.250","Text":"we get a constant function."},{"Start":"02:01.090 ","End":"02:04.820","Text":"What happens if a is negative?"},{"Start":"02:04.820 ","End":"02:06.470","Text":"If a is negative,"},{"Start":"02:06.470 ","End":"02:09.540","Text":"we have problems with the domain."},{"Start":"02:15.410 ","End":"02:20.870","Text":"For instance, if we take a is equal to minus 4,"},{"Start":"02:20.870 ","End":"02:23.825","Text":"then we have a problem because it isn\u0027t defined for all x."},{"Start":"02:23.825 ","End":"02:25.400","Text":"You put x equals 1/2,"},{"Start":"02:25.400 ","End":"02:27.625","Text":"minus 4 to the power of 1/2,"},{"Start":"02:27.625 ","End":"02:29.505","Text":"square root of minus 4,"},{"Start":"02:29.505 ","End":"02:30.855","Text":"there are various problems."},{"Start":"02:30.855 ","End":"02:34.130","Text":"In short, a^x is the exponential function,"},{"Start":"02:34.130 ","End":"02:37.384","Text":"but there are some restrictions that you have to observe."},{"Start":"02:37.384 ","End":"02:40.640","Text":"Fortunately with the exponential function,"},{"Start":"02:40.640 ","End":"02:46.100","Text":"there\u0027s only 1 parameter and that\u0027s a in contrast to say the quadratic function,"},{"Start":"02:46.100 ","End":"02:50.975","Text":"which has an a, a b, and a c. We only have to worry about a."},{"Start":"02:50.975 ","End":"02:55.045","Text":"Let\u0027s talk about it. The parameter a."},{"Start":"02:55.045 ","End":"02:58.150","Text":"Now there\u0027s really 2 main possibilities for a,"},{"Start":"02:58.150 ","End":"02:59.770","Text":"because if I look at this restriction,"},{"Start":"02:59.770 ","End":"03:10.550","Text":"I\u0027ll see that I have either a bigger than 1 or a can be between 0 and 1."},{"Start":"03:10.550 ","End":"03:19.470","Text":"The main property that distinguishes these 2 cases is that if a is bigger than 1,"},{"Start":"03:19.470 ","End":"03:23.380","Text":"then the function is increasing."},{"Start":"03:23.960 ","End":"03:27.300","Text":"If a is between 0 and 1,"},{"Start":"03:27.300 ","End":"03:30.520","Text":"then f is decreasing."},{"Start":"03:30.600 ","End":"03:35.500","Text":"I\u0027d like to illustrate this to you graphically,"},{"Start":"03:35.500 ","End":"03:38.515","Text":"so let\u0027s make some room here."},{"Start":"03:38.515 ","End":"03:43.265","Text":"Let me draw the axis first."},{"Start":"03:43.265 ","End":"03:45.643","Text":"Here are the axis,"},{"Start":"03:45.643 ","End":"03:49.565","Text":"and I\u0027ll draw both of them on the same graph."},{"Start":"03:49.565 ","End":"03:51.860","Text":"I\u0027ll take an example of a bigger than 1."},{"Start":"03:51.860 ","End":"03:54.440","Text":"Let\u0027s say a=2."},{"Start":"03:54.440 ","End":"03:58.470","Text":"We\u0027re going to draw here y=2^x."},{"Start":"04:00.730 ","End":"04:05.025","Text":"If we just make a quick table in our heads,"},{"Start":"04:05.025 ","End":"04:07.890","Text":"we can see if x is 0,"},{"Start":"04:07.890 ","End":"04:09.675","Text":"2^0 is 1,"},{"Start":"04:09.675 ","End":"04:13.750","Text":"this might be 0, 1."},{"Start":"04:15.020 ","End":"04:17.745","Text":"Say 1 is here,"},{"Start":"04:17.745 ","End":"04:21.690","Text":"then if x is 1,"},{"Start":"04:21.690 ","End":"04:29.250","Text":"y is 2, so 1, 2."},{"Start":"04:29.250 ","End":"04:32.175","Text":"If x is 2,"},{"Start":"04:32.175 ","End":"04:39.545","Text":"then y equals 2^2 which is 4 and increases steeply."},{"Start":"04:39.545 ","End":"04:46.260","Text":"We can go on the other side and say x is equal to minus 1,"},{"Start":"04:46.260 ","End":"04:52.180","Text":"2 to the minus 1 would be 1/2."},{"Start":"04:52.370 ","End":"04:54.900","Text":"If I take x is minus 2,"},{"Start":"04:54.900 ","End":"04:56.955","Text":"2 to the minus 2."},{"Start":"04:56.955 ","End":"04:59.745","Text":"Say minus 2 is here,"},{"Start":"04:59.745 ","End":"05:08.325","Text":"minus, this is 1/2 and we get even less,"},{"Start":"05:08.325 ","End":"05:12.730","Text":"smaller, we get 1/4."},{"Start":"05:14.410 ","End":"05:17.570","Text":"Computer has a mind of its own."},{"Start":"05:17.570 ","End":"05:22.205","Text":"We get 1/4. Let\u0027s write it on this side,"},{"Start":"05:22.205 ","End":"05:25.880","Text":"1/4, which would be this point here."},{"Start":"05:25.880 ","End":"05:29.225","Text":"If I join a line through them."},{"Start":"05:29.225 ","End":"05:32.495","Text":"There we are. Excuse me, it\u0027s a little crooked."},{"Start":"05:32.495 ","End":"05:34.865","Text":"That\u0027s the general idea of y=2^x,"},{"Start":"05:34.865 ","End":"05:37.700","Text":"and we notice that it\u0027s increasing."},{"Start":"05:37.700 ","End":"05:43.040","Text":"Its, try again,"},{"Start":"05:43.040 ","End":"05:52.740","Text":"increasing, increasing, increasing,"},{"Start":"05:52.740 ","End":"05:55.690","Text":"more and more steeply in fact."},{"Start":"05:56.270 ","End":"06:02.395","Text":"That\u0027s typical for a bigger than 1."},{"Start":"06:02.395 ","End":"06:05.690","Text":"Let\u0027s now do an example of a less than 1,"},{"Start":"06:05.690 ","End":"06:09.275","Text":"say a=1/2, and I\u0027ll do that in a different color."},{"Start":"06:09.275 ","End":"06:17.055","Text":"Now I\u0027m going to do y equals,"},{"Start":"06:17.055 ","End":"06:20.330","Text":"let\u0027s say a 1/2 to the power of x,"},{"Start":"06:20.330 ","End":"06:22.370","Text":"where a is 1/2."},{"Start":"06:22.370 ","End":"06:26.059","Text":"Again, we can sketch a few points."},{"Start":"06:26.059 ","End":"06:30.855","Text":"When x is 0, y is again equal to 1."},{"Start":"06:30.855 ","End":"06:37.640","Text":"In fact, that\u0027s true for all of the exponential functions that f(0) is equal to 1,"},{"Start":"06:37.640 ","End":"06:39.650","Text":"they all pass through this point."},{"Start":"06:39.650 ","End":"06:42.910","Text":"This time when x is 1,"},{"Start":"06:42.910 ","End":"06:45.720","Text":"y is equal to 1/2."},{"Start":"06:45.720 ","End":"06:49.989","Text":"We get this point here,"},{"Start":"06:51.470 ","End":"06:57.065","Text":"sorry about that, which is actually the mirror image of this point."},{"Start":"06:57.065 ","End":"07:00.890","Text":"If I let a equal 2,"},{"Start":"07:00.890 ","End":"07:03.140","Text":"then y equals 1/2^2,"},{"Start":"07:03.140 ","End":"07:04.295","Text":"which is a 1/4,"},{"Start":"07:04.295 ","End":"07:09.935","Text":"which is exactly the mirror image of that one there."},{"Start":"07:09.935 ","End":"07:12.125","Text":"If x is minus 1,"},{"Start":"07:12.125 ","End":"07:16.175","Text":"1/2 to the power of minus 1 is 1 over 1/2,"},{"Start":"07:16.175 ","End":"07:19.220","Text":"which is 2, which gives us the mirror image,"},{"Start":"07:19.220 ","End":"07:22.380","Text":"in fact, of this point."},{"Start":"07:22.720 ","End":"07:27.942","Text":"Likewise, if x equals minus 2,"},{"Start":"07:27.942 ","End":"07:34.095","Text":"1/2 to the power of minus 2 is 4."},{"Start":"07:34.095 ","End":"07:39.495","Text":"That\u0027s somewhere opposite this point here."},{"Start":"07:39.495 ","End":"07:43.390","Text":"If I try to sketch this,"},{"Start":"07:43.390 ","End":"07:46.565","Text":"this is the best I can do."},{"Start":"07:46.565 ","End":"07:49.550","Text":"We see it\u0027s decreasing."},{"Start":"07:49.550 ","End":"07:52.145","Text":"I can draw some arrows here,"},{"Start":"07:52.145 ","End":"07:54.869","Text":"so it gets smaller and smaller."},{"Start":"08:00.010 ","End":"08:04.400","Text":"I can say a bit more about the parameter a other than the fact that it"},{"Start":"08:04.400 ","End":"08:08.880","Text":"determines whether we\u0027re increasing or decreasing."},{"Start":"08:09.530 ","End":"08:13.860","Text":"If a gets larger,"},{"Start":"08:13.860 ","End":"08:16.340","Text":"or in other words, as a increases,"},{"Start":"08:16.340 ","End":"08:18.410","Text":"the graph just becomes steeper."},{"Start":"08:18.410 ","End":"08:23.035","Text":"For example, if I take a equals"},{"Start":"08:23.035 ","End":"08:31.545","Text":"4 and I look at y equals 4^x and I use a different color,"},{"Start":"08:31.545 ","End":"08:34.050","Text":"then when x is 0,"},{"Start":"08:34.050 ","End":"08:35.535","Text":"y is still 1."},{"Start":"08:35.535 ","End":"08:36.945","Text":"But when x is 1,"},{"Start":"08:36.945 ","End":"08:41.460","Text":"y is already equal to 4"},{"Start":"08:41.460 ","End":"08:47.325","Text":"and if x is minus 1, y is 1/4."},{"Start":"08:47.325 ","End":"08:51.230","Text":"What we get, this won\u0027t be very accurate,"},{"Start":"08:51.230 ","End":"08:53.500","Text":"but we will get"},{"Start":"08:53.500 ","End":"09:02.379","Text":"a much steeper graph and it goes down to 0 quicker over here."},{"Start":"09:02.379 ","End":"09:06.895","Text":"Similarly on the other side,"},{"Start":"09:06.895 ","End":"09:12.275","Text":"if I let y"},{"Start":"09:12.275 ","End":"09:19.040","Text":"equals 1/4^x,"},{"Start":"09:19.040 ","End":"09:21.170","Text":"as a gets smaller,"},{"Start":"09:21.170 ","End":"09:26.380","Text":"it gets steeper in the other way."},{"Start":"09:26.380 ","End":"09:28.680","Text":"When x is minus 1,"},{"Start":"09:28.680 ","End":"09:31.125","Text":"then y is 4,"},{"Start":"09:31.125 ","End":"09:36.255","Text":"it always goes through here and then it at 1 it becomes a 1/4,"},{"Start":"09:36.255 ","End":"09:41.800","Text":"so we get something like this."},{"Start":"09:41.800 ","End":"09:46.240","Text":"In other words, a determines not only"},{"Start":"09:46.240 ","End":"09:51.020","Text":"which way whether it\u0027s increasing or decreasing, but how steeply."},{"Start":"09:51.570 ","End":"09:55.750","Text":"Now I\u0027d like to make some comments, observations,"},{"Start":"09:55.750 ","End":"10:02.395","Text":"let\u0027s call them observations about the exponential function."},{"Start":"10:02.395 ","End":"10:06.150","Text":"In general, observations."},{"Start":"10:06.830 ","End":"10:09.275","Text":"One observation is,"},{"Start":"10:09.275 ","End":"10:11.795","Text":"if you look at this, at the graphs,"},{"Start":"10:11.795 ","End":"10:15.110","Text":"they\u0027re all positive, they\u0027re always above the x-axis."},{"Start":"10:15.110 ","End":"10:21.340","Text":"I didn\u0027t even need this bit of the drawing, always positive."},{"Start":"10:22.010 ","End":"10:27.020","Text":"I\u0027m talking about the exponential function, it\u0027s always positive."},{"Start":"10:27.950 ","End":"10:30.810","Text":"Because it\u0027s always positive,"},{"Start":"10:30.810 ","End":"10:33.765","Text":"if it\u0027s going to be 0,"},{"Start":"10:33.765 ","End":"10:36.900","Text":"it\u0027s never going to intersect the x-axis,"},{"Start":"10:36.900 ","End":"10:41.475","Text":"there is no x-intercept."},{"Start":"10:41.475 ","End":"10:46.875","Text":"In another words, does not intersect the x-axis."},{"Start":"10:46.875 ","End":"10:49.740","Text":"As for the y-intercept,"},{"Start":"10:49.740 ","End":"10:53.370","Text":"the y-intercept is always the same,"},{"Start":"10:53.370 ","End":"10:54.840","Text":"it\u0027s always equal to 1."},{"Start":"10:54.840 ","End":"10:58.601","Text":"When x is 0, a^0 is 1."},{"Start":"10:58.601 ","End":"11:07.255","Text":"Intercept is always 1 or we can just say,"},{"Start":"11:07.255 ","End":"11:09.500","Text":"sometimes we give the coordinates of the point,"},{"Start":"11:09.500 ","End":"11:10.910","Text":"it\u0027s always 0,"},{"Start":"11:10.910 ","End":"11:15.510","Text":"1, which is this point here."},{"Start":"11:15.510 ","End":"11:20.050","Text":"Wherever a is, the y-intercept is 0, 1."},{"Start":"11:21.190 ","End":"11:25.520","Text":"There\u0027s one last thing I\u0027d like to mention before we go on"},{"Start":"11:25.520 ","End":"11:29.570","Text":"to the next section on logarithmic functions."},{"Start":"11:29.570 ","End":"11:34.550","Text":"Is that there is a very special exponential function."},{"Start":"11:34.550 ","End":"11:39.545","Text":"Just remark here."},{"Start":"11:39.545 ","End":"11:47.835","Text":"There is a special exponential function,"},{"Start":"11:47.835 ","End":"11:49.740","Text":"there\u0027s one special one,"},{"Start":"11:49.740 ","End":"11:56.785","Text":"and it is the function y equals e^x."},{"Start":"11:56.785 ","End":"12:01.265","Text":"Some of you may have heard of the number e and some of you may not."},{"Start":"12:01.265 ","End":"12:03.350","Text":"But e, although it\u0027s a letter,"},{"Start":"12:03.350 ","End":"12:05.575","Text":"is a constant number,"},{"Start":"12:05.575 ","End":"12:10.540","Text":"pretty much like Pi is a number."},{"Start":"12:12.290 ","End":"12:15.625","Text":"In contrast, by analogy,"},{"Start":"12:15.625 ","End":"12:24.655","Text":"Pi is a special number which is approximately equal to 3.1416, let\u0027s say."},{"Start":"12:24.655 ","End":"12:27.958","Text":"E is another special number in mathematics,"},{"Start":"12:27.958 ","End":"12:36.615","Text":"and e is approximately 2.718, and it goes on."},{"Start":"12:36.615 ","End":"12:40.355","Text":"You don\u0027t have to know exactly what it is or where it comes from,"},{"Start":"12:40.355 ","End":"12:42.590","Text":"the calculator knows about it."},{"Start":"12:42.590 ","End":"12:44.720","Text":"You just have to know it\u0027s a special number."},{"Start":"12:44.720 ","End":"12:50.430","Text":"Why we use this number e is that when you get down to the calculus,"},{"Start":"12:50.430 ","End":"12:54.920","Text":"many formulae for integration,"},{"Start":"12:54.920 ","End":"12:56.690","Text":"and differentiation, and that sort of thing,"},{"Start":"12:56.690 ","End":"12:58.880","Text":"everything comes up much simpler."},{"Start":"12:58.880 ","End":"13:01.775","Text":"Later, perhaps after the logarithms,"},{"Start":"13:01.775 ","End":"13:05.390","Text":"I\u0027ll explain what this number e is and where it comes from."},{"Start":"13:05.390 ","End":"13:07.865","Text":"That\u0027s all I\u0027ll say on not matter for now."},{"Start":"13:07.865 ","End":"13:10.860","Text":"Next up, logarithmic functions."}],"ID":5125},{"Watched":false,"Name":"The Logarithmic Function","Duration":"10m 50s","ChapterTopicVideoID":5138,"CourseChapterTopicPlaylistID":56143,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5138.jpeg","UploadDate":"2020-09-30T13:08:37.0630000","DurationForVideoObject":"PT10M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.989","Text":"In this clip, we\u0027ll talk a bit about the logarithmic function."},{"Start":"00:03.989 ","End":"00:14.190","Text":"This is a function of the form f of x equals log to the base of some number,"},{"Start":"00:14.190 ","End":"00:20.760","Text":"a, some parameter constant, logarithm of x."},{"Start":"00:20.760 ","End":"00:25.005","Text":"That\u0027s all. But there are some restrictions on the a,"},{"Start":"00:25.005 ","End":"00:28.800","Text":"just like they were restrictions on a in the exponential function."},{"Start":"00:28.800 ","End":"00:30.810","Text":"Here, as there,"},{"Start":"00:30.810 ","End":"00:39.575","Text":"we have to have a positive and also we disqualify the value 1."},{"Start":"00:39.575 ","End":"00:41.735","Text":"That\u0027s the restriction,"},{"Start":"00:41.735 ","End":"00:47.400","Text":"and there is another restriction on the domain of definition."},{"Start":"00:47.400 ","End":"00:52.385","Text":"We require that x be positive."},{"Start":"00:52.385 ","End":"00:55.610","Text":"You can only take the logarithm of a positive number."},{"Start":"00:55.610 ","End":"01:00.079","Text":"Some examples would be using the y notation."},{"Start":"01:00.079 ","End":"01:06.005","Text":"We could take y equals log to the base 2 of x."},{"Start":"01:06.005 ","End":"01:16.240","Text":"You could have y equals log to the base 1/3 of x."},{"Start":"01:16.240 ","End":"01:21.095","Text":"You could have, going back to functional notation,"},{"Start":"01:21.095 ","End":"01:30.320","Text":"f of x equals log base 4.7 of x and so on."},{"Start":"01:30.320 ","End":"01:33.170","Text":"But, you must remember not to use the forbidden values,"},{"Start":"01:33.170 ","End":"01:37.595","Text":"not to put anything negative and not to use the value 1."},{"Start":"01:37.595 ","End":"01:40.625","Text":"Also, notice that there\u0027s only 1 parameter."},{"Start":"01:40.625 ","End":"01:42.280","Text":"We have just a."},{"Start":"01:42.280 ","End":"01:46.040","Text":"1 parameter a determines which logarithmic function we have,"},{"Start":"01:46.040 ","End":"01:48.170","Text":"and this is in contrast to, say,"},{"Start":"01:48.170 ","End":"01:51.350","Text":"the quadratic function where you have 3 parameters, a, b,"},{"Start":"01:51.350 ","End":"01:56.580","Text":"and c. We\u0027d like to see what the significance of a is."},{"Start":"01:56.580 ","End":"02:00.950","Text":"The main significance is in the shape of the graph."},{"Start":"02:00.950 ","End":"02:06.360","Text":"If a is bigger than 1,"},{"Start":"02:07.490 ","End":"02:11.429","Text":"then f is increasing,"},{"Start":"02:11.429 ","End":"02:14.920","Text":"F meaning this function is increasing."},{"Start":"02:18.040 ","End":"02:22.640","Text":"If a is between 0 and 1 and"},{"Start":"02:22.640 ","End":"02:29.510","Text":"the only possibilities here are between either greater than 1 or between 0 and 1,"},{"Start":"02:29.510 ","End":"02:34.860","Text":"then if f is the opposite, it\u0027s decreasing."},{"Start":"02:36.580 ","End":"02:40.550","Text":"Let\u0027s draw a sketch of the graph."},{"Start":"02:40.550 ","End":"02:42.035","Text":"We can make a table."},{"Start":"02:42.035 ","End":"02:44.630","Text":"Let\u0027s take an example."},{"Start":"02:44.630 ","End":"02:49.730","Text":"For a greater than 1 will take a equals 2."},{"Start":"02:49.730 ","End":"02:55.960","Text":"If I take a equals 2,"},{"Start":"02:55.960 ","End":"02:58.860","Text":"then I need some axes here."},{"Start":"03:01.510 ","End":"03:06.440","Text":"Let\u0027s make a table for the case where a is equal to 2."},{"Start":"03:06.440 ","End":"03:15.780","Text":"In other words, the function is f of x equals log to the base 2 of x."},{"Start":"03:17.380 ","End":"03:22.080","Text":"Let\u0027s see. I got 2 columns,"},{"Start":"03:22.080 ","End":"03:25.360","Text":"1 for x and 1 for y."},{"Start":"03:25.360 ","End":"03:27.740","Text":"We\u0027re only going to take x as positive,"},{"Start":"03:27.740 ","End":"03:31.955","Text":"that\u0027s also why I drew this thing so short here,"},{"Start":"03:31.955 ","End":"03:33.965","Text":"because of the domain."},{"Start":"03:33.965 ","End":"03:36.755","Text":"The domain, remember, was x bigger than 0,"},{"Start":"03:36.755 ","End":"03:38.810","Text":"so there\u0027s not going to be anything here,"},{"Start":"03:38.810 ","End":"03:43.715","Text":"it\u0027s going to be just empty for whatever value a we take."},{"Start":"03:43.715 ","End":"03:47.360","Text":"If we take a equals 2,"},{"Start":"03:47.360 ","End":"03:51.530","Text":"so when x equals, let\u0027s say, 1,"},{"Start":"03:51.530 ","End":"03:55.680","Text":"the log of 1 is 0,"},{"Start":"03:55.680 ","End":"03:57.365","Text":"and that\u0027s true for any a."},{"Start":"03:57.365 ","End":"04:03.680","Text":"In fact, all the graphs of log of x wherever a is going to pass through here;"},{"Start":"04:03.680 ","End":"04:05.255","Text":"this is going to be a special point,"},{"Start":"04:05.255 ","End":"04:07.445","Text":"meeting point of all the graphs."},{"Start":"04:07.445 ","End":"04:10.385","Text":"If x is 2,"},{"Start":"04:10.385 ","End":"04:14.180","Text":"log to the base 2 of 2 is 1."},{"Start":"04:14.180 ","End":"04:15.740","Text":"If it\u0027s 4,"},{"Start":"04:15.740 ","End":"04:20.145","Text":"it\u0027s going to be log to the base 2 of 4 is 2,"},{"Start":"04:20.145 ","End":"04:24.615","Text":"log to the base 2 of 8 is 3."},{"Start":"04:24.615 ","End":"04:26.510","Text":"We can go a bit in the other direction of"},{"Start":"04:26.510 ","End":"04:32.470","Text":"the log of 1/2 because it\u0027s 2 to the power of minus 1,"},{"Start":"04:32.470 ","End":"04:35.010","Text":"the log would be minus 1."},{"Start":"04:35.010 ","End":"04:37.440","Text":"If we took x as 1/4,"},{"Start":"04:37.440 ","End":"04:44.365","Text":"log to the base 2 of 1/4 is minus 2 because 2 to the minus 2 is 1/4."},{"Start":"04:44.365 ","End":"04:46.880","Text":"If we plot these like this,"},{"Start":"04:46.880 ","End":"04:49.510","Text":"let me do a freehand sketch,"},{"Start":"04:49.510 ","End":"04:52.225","Text":"so through here, here,"},{"Start":"04:52.225 ","End":"04:56.965","Text":"here, here, and so on,"},{"Start":"04:56.965 ","End":"05:00.920","Text":"up to here, a very crude rough sketch."},{"Start":"05:00.920 ","End":"05:02.030","Text":"You get the idea though,"},{"Start":"05:02.030 ","End":"05:03.785","Text":"it\u0027s an increasing function."},{"Start":"05:03.785 ","End":"05:07.475","Text":"We go up and up,"},{"Start":"05:07.475 ","End":"05:11.020","Text":"and up, and up, and so on."},{"Start":"05:11.020 ","End":"05:15.520","Text":"Now, let\u0027s take an example of the other case where a is between 0 and 1."},{"Start":"05:15.520 ","End":"05:20.335","Text":"If I take a equals 1/2 and make a table,"},{"Start":"05:20.335 ","End":"05:25.390","Text":"we have 1/2 is 1."},{"Start":"05:25.390 ","End":"05:30.550","Text":"In fact, it\u0027s going to be the mirror image of the previous 1,"},{"Start":"05:30.550 ","End":"05:32.500","Text":"the mirror image about the x-axis."},{"Start":"05:32.500 ","End":"05:35.710","Text":"Try a freehand sketch."},{"Start":"05:35.710 ","End":"05:40.150","Text":"Something that goes from high here to"},{"Start":"05:40.150 ","End":"05:45.730","Text":"this central point where they all go through and continuing,"},{"Start":"05:45.730 ","End":"05:48.887","Text":"excuse me for the bad drawing,"},{"Start":"05:48.887 ","End":"05:52.430","Text":"but the idea is what\u0027s important here."},{"Start":"05:52.430 ","End":"05:56.450","Text":"A basically determines which way the graph goes,"},{"Start":"05:56.450 ","End":"05:59.390","Text":"increasing or decreasing, bigger than 1, less than 1."},{"Start":"05:59.390 ","End":"06:05.030","Text":"I can say a little bit more that if a gets even bigger,"},{"Start":"06:05.030 ","End":"06:07.265","Text":"say I took a equals 3,"},{"Start":"06:07.265 ","End":"06:09.710","Text":"then it would be shallower."},{"Start":"06:09.710 ","End":"06:11.810","Text":"It would pass through here,"},{"Start":"06:11.810 ","End":"06:14.910","Text":"but it would be shallower."},{"Start":"06:14.930 ","End":"06:19.485","Text":"For example, the 0.31 would be on it."},{"Start":"06:19.485 ","End":"06:25.505","Text":"If we took as the last example that a equals 1/3,"},{"Start":"06:25.505 ","End":"06:29.030","Text":"we\u0027d actually get the mirror image"},{"Start":"06:29.030 ","End":"06:36.380","Text":"of this 1 and it would start somewhere here,"},{"Start":"06:36.380 ","End":"06:40.130","Text":"it would still go through this central point"},{"Start":"06:40.130 ","End":"06:44.945","Text":"and just be a bit less steep or something like this."},{"Start":"06:44.945 ","End":"06:49.230","Text":"That would be for a equals 1/3."},{"Start":"06:51.380 ","End":"06:55.280","Text":"A not only determines increasing, decreasing,"},{"Start":"06:55.280 ","End":"06:59.090","Text":"but how steeply, and you can see by the numbers which direction it goes in."},{"Start":"06:59.090 ","End":"07:02.930","Text":"If I put some arrows here at this and then in the case"},{"Start":"07:02.930 ","End":"07:08.180","Text":"of between 0 and 1, then it\u0027s decreasing."},{"Start":"07:08.180 ","End":"07:10.160","Text":"Similarly the purple one\u0027s decreasing,"},{"Start":"07:10.160 ","End":"07:13.265","Text":"the orange one\u0027s increasing and so on."},{"Start":"07:13.265 ","End":"07:16.970","Text":"What I would like to add is that mostly we"},{"Start":"07:16.970 ","End":"07:20.720","Text":"won\u0027t be concerning ourselves with all these different values of a."},{"Start":"07:20.720 ","End":"07:29.585","Text":"There\u0027s 1 special value of a that is used most commonly in mathematics for calculus."},{"Start":"07:29.585 ","End":"07:32.640","Text":"That number, see if you can guess,"},{"Start":"07:32.640 ","End":"07:36.980","Text":"we hinted at it in the exponential function section."},{"Start":"07:36.980 ","End":"07:39.950","Text":"Well, I\u0027ll tell you, it\u0027s when a equals e,"},{"Start":"07:39.950 ","End":"07:44.990","Text":"by which I mean that the function would be f of x"},{"Start":"07:44.990 ","End":"07:51.410","Text":"equals log to the base e of x."},{"Start":"07:51.410 ","End":"07:54.920","Text":"We already mentioned e that it\u0027s just a constant,"},{"Start":"07:54.920 ","End":"07:59.120","Text":"a number that happens to equal 2.718,"},{"Start":"07:59.120 ","End":"08:04.730","Text":"approximately, because it falls into the category of a bigger than 1,"},{"Start":"08:04.730 ","End":"08:07.940","Text":"so it\u0027s increasing so it\u0027s 1 of these."},{"Start":"08:07.940 ","End":"08:11.045","Text":"In fact, because e is between 2 and 3,"},{"Start":"08:11.045 ","End":"08:13.700","Text":"the e^x function would be,"},{"Start":"08:13.700 ","End":"08:15.815","Text":"it\u0027s getting a bit crowded here,"},{"Start":"08:15.815 ","End":"08:22.325","Text":"but something like this."},{"Start":"08:22.325 ","End":"08:27.570","Text":"That would be where a equals e,"},{"Start":"08:27.570 ","End":"08:32.145","Text":"which means that this is the function y"},{"Start":"08:32.145 ","End":"08:38.390","Text":"equals log to the base e of x, the red 1."},{"Start":"08:38.390 ","End":"08:42.530","Text":"Because it\u0027s so useful and so special,"},{"Start":"08:42.530 ","End":"08:44.585","Text":"it\u0027s given a special name."},{"Start":"08:44.585 ","End":"08:53.930","Text":"The logarithm to the base e is given the name natural log, which is Ln."},{"Start":"08:53.930 ","End":"08:56.810","Text":"We write natural log of x."},{"Start":"08:56.810 ","End":"08:59.960","Text":"When you see Ln function,"},{"Start":"08:59.960 ","End":"09:01.250","Text":"it just means the log,"},{"Start":"09:01.250 ","End":"09:04.080","Text":"but to the special base e."},{"Start":"09:04.250 ","End":"09:11.510","Text":"All the usual rules of logarithms apply to the natural log because e is just some number."},{"Start":"09:11.510 ","End":"09:14.210","Text":"I\u0027m talking about all these logarithm rules,"},{"Start":"09:14.210 ","End":"09:23.400","Text":"such as the natural log of 1 is equal to 0 such that"},{"Start":"09:23.400 ","End":"09:30.031","Text":"the natural log of"},{"Start":"09:30.031 ","End":"09:40.290","Text":"x_1"},{"Start":"09:40.290 ","End":"09:40.291","Text":"times"},{"Start":"09:40.291 ","End":"09:42.610","Text":"x_2."},{"Start":"09:42.910 ","End":"09:51.120","Text":"Yeah, that will do. That will be the natural log of x_1 plus natural log of x_2."},{"Start":"09:51.120 ","End":"09:54.070","Text":"All the usual laws of the logarithms,"},{"Start":"09:54.070 ","End":"10:04.020","Text":"natural log of x_1 over x_2 is natural log of x_1 minus natural log of x_2."},{"Start":"10:04.020 ","End":"10:07.269","Text":"Also continuing over here with specific numbers,"},{"Start":"10:07.269 ","End":"10:09.360","Text":"natural log of 1 is 0,"},{"Start":"10:09.360 ","End":"10:12.520","Text":"and the natural log, say,"},{"Start":"10:12.520 ","End":"10:19.535","Text":"of e would equal 1 because e^1 is e. In general,"},{"Start":"10:19.535 ","End":"10:25.730","Text":"natural log of e^n,"},{"Start":"10:25.730 ","End":"10:29.930","Text":"n is unusually a whole number but it also could be a fraction,"},{"Start":"10:29.930 ","End":"10:34.130","Text":"this is equal to n natural log of e which is 1,"},{"Start":"10:34.130 ","End":"10:35.975","Text":"so this is just equal to n,"},{"Start":"10:35.975 ","End":"10:42.034","Text":"and sure all the usual rules of logarithms apply."},{"Start":"10:42.034 ","End":"10:44.990","Text":"I don\u0027t think there\u0027s anything more I want to say about"},{"Start":"10:44.990 ","End":"10:50.280","Text":"the natural logarithm or about the logarithms. We\u0027re done."}],"ID":5126},{"Watched":false,"Name":"Basic Functions","Duration":"18m 35s","ChapterTopicVideoID":5139,"CourseChapterTopicPlaylistID":56143,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5139.jpeg","UploadDate":"2020-09-30T13:15:06.5170000","DurationForVideoObject":"PT18M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"In this clip, I\u0027d like to talk about some basic functions."},{"Start":"00:03.945 ","End":"00:09.390","Text":"The idea is to build up a stock of a few basic functions that"},{"Start":"00:09.390 ","End":"00:15.075","Text":"you should know how they look even without having to sketch from scratch to plot points."},{"Start":"00:15.075 ","End":"00:18.450","Text":"What we\u0027re going to start with in our collection will be Point Number 1,"},{"Start":"00:18.450 ","End":"00:26.790","Text":"will be functions like y=x^2,"},{"Start":"00:26.790 ","End":"00:29.550","Text":"y=x^4, then x^6 and so on."},{"Start":"00:29.550 ","End":"00:38.895","Text":"In general, we\u0027re going to talk about y=x^n where n is even,"},{"Start":"00:38.895 ","End":"00:41.360","Text":"an even whole number bigger,"},{"Start":"00:41.360 ","End":"00:42.515","Text":"or equal to 2,"},{"Start":"00:42.515 ","End":"00:45.020","Text":"from 2 onwards ^4,"},{"Start":"00:45.020 ","End":"00:48.270","Text":"x^6, x^8, x^10, and so on."},{"Start":"00:48.270 ","End":"00:50.415","Text":"The first one in the series,"},{"Start":"00:50.415 ","End":"00:55.775","Text":"x^2 is a basic parabola and you should be familiar with that."},{"Start":"00:55.775 ","End":"00:58.370","Text":"If I tell you y=x^2,"},{"Start":"00:58.370 ","End":"01:03.410","Text":"you could very quickly just make a set of axes and draw a U shape,"},{"Start":"01:03.410 ","End":"01:04.880","Text":"should be more symmetrical,"},{"Start":"01:04.880 ","End":"01:06.350","Text":"should have gone through 0,0."},{"Start":"01:06.350 ","End":"01:07.985","Text":"But, you know the general shape."},{"Start":"01:07.985 ","End":"01:10.220","Text":"What you don\u0027t know maybe is x^4,"},{"Start":"01:10.220 ","End":"01:11.870","Text":"x^6, and so on."},{"Start":"01:11.870 ","End":"01:17.210","Text":"Well, in general, they all look very similar with a little bit of difference."},{"Start":"01:17.210 ","End":"01:23.465","Text":"The main difference, if I take x^4 for example,"},{"Start":"01:23.465 ","End":"01:26.015","Text":"y=x^4 will look remarkably similar,"},{"Start":"01:26.015 ","End":"01:28.405","Text":"at least in the general U shape."},{"Start":"01:28.405 ","End":"01:29.990","Text":"It looks very much like this."},{"Start":"01:29.990 ","End":"01:33.650","Text":"The main difference being that it goes up steeper,"},{"Start":"01:33.650 ","End":"01:36.230","Text":"so it will be closer to the y axis,"},{"Start":"01:36.230 ","End":"01:38.915","Text":"and at the bottom it\u0027s a bit flatter and so on."},{"Start":"01:38.915 ","End":"01:40.220","Text":"If I take x^6,"},{"Start":"01:40.220 ","End":"01:41.630","Text":"or any of the higher ones,"},{"Start":"01:41.630 ","End":"01:44.360","Text":"they\u0027ll all have the same basic shape."},{"Start":"01:44.360 ","End":"01:47.230","Text":"They\u0027ll all go up steeply,"},{"Start":"01:47.230 ","End":"01:51.990","Text":"here a bit flatter but still going through 0,0 and up steeply again."},{"Start":"01:51.990 ","End":"01:54.345","Text":"This will be in general,"},{"Start":"01:54.345 ","End":"01:57.155","Text":"y=x^n where n is like here,"},{"Start":"01:57.155 ","End":"01:59.625","Text":"even and greater, or equal to 2,"},{"Start":"01:59.625 ","End":"02:03.150","Text":"should go through the 0,0."},{"Start":"02:03.150 ","End":"02:04.865","Text":"Now let\u0027s move on to the next step."},{"Start":"02:04.865 ","End":"02:08.809","Text":"We\u0027re going to talk about the odd ones instead of the even ones."},{"Start":"02:08.809 ","End":"02:13.575","Text":"This time, it\u0027s going to be y=x^3,"},{"Start":"02:13.575 ","End":"02:19.775","Text":"y=x^5, y=x^7 and so on."},{"Start":"02:19.775 ","End":"02:27.150","Text":"y=x^n where n is odd, 3,5,7,9, etc."},{"Start":"02:27.150 ","End":"02:29.910","Text":"and bigger or equal to 3."},{"Start":"02:29.910 ","End":"02:31.890","Text":"On the very first one,"},{"Start":"02:31.890 ","End":"02:35.330","Text":"y=x^3, you\u0027re probably familiar with."},{"Start":"02:35.330 ","End":"02:38.855","Text":"The x^3 goes like this,"},{"Start":"02:38.855 ","End":"02:46.135","Text":"flattens out here and then it rises again steeply, y=x^3."},{"Start":"02:46.135 ","End":"02:51.350","Text":"Note also that when x is 0, y is 0."},{"Start":"02:51.350 ","End":"02:54.035","Text":"You can tell from this equation,"},{"Start":"02:54.035 ","End":"02:59.000","Text":"and also that when x is 1, 1^n is 1,"},{"Start":"02:59.000 ","End":"03:02.430","Text":"and likewise -1^n is -1,"},{"Start":"03:02.430 ","End":"03:08.365","Text":"provided n is odd so these 3 points appear on all of these odd family."},{"Start":"03:08.365 ","End":"03:11.730","Text":"I\u0027m going to show you the x^5 next,"},{"Start":"03:11.730 ","End":"03:13.530","Text":"and in general, x^n."},{"Start":"03:13.530 ","End":"03:17.790","Text":"As you can see, they all look very similar."},{"Start":"03:17.790 ","End":"03:23.885","Text":"The x^5 also goes through these 3 points that I wrote here."},{"Start":"03:23.885 ","End":"03:27.735","Text":"It just goes up steeper because it\u0027s x^5."},{"Start":"03:27.735 ","End":"03:31.365","Text":"For example, if x is 2,"},{"Start":"03:31.365 ","End":"03:33.345","Text":"here it\u0027s 8,"},{"Start":"03:33.345 ","End":"03:35.685","Text":"2^3 but here it\u0027s 32."},{"Start":"03:35.685 ","End":"03:37.290","Text":"In general, x^n,"},{"Start":"03:37.290 ","End":"03:39.110","Text":"as n grows larger,"},{"Start":"03:39.110 ","End":"03:40.978","Text":"it will be higher up,"},{"Start":"03:40.978 ","End":"03:42.740","Text":"and therefore it will be narrower,"},{"Start":"03:42.740 ","End":"03:46.505","Text":"and closer to the axis but the general picture is exactly the same."},{"Start":"03:46.505 ","End":"03:48.860","Text":"You just draw a line,"},{"Start":"03:48.860 ","End":"03:52.690","Text":"curved line goes through the origin and upwards."},{"Start":"03:52.690 ","End":"03:57.170","Text":"If you can manage it, it\u0027s more correct to make it flat here,"},{"Start":"03:57.170 ","End":"03:59.735","Text":"it flattens out, and then gets steeper,"},{"Start":"03:59.735 ","End":"04:01.940","Text":"flattens out, gets steeper."},{"Start":"04:01.940 ","End":"04:04.460","Text":"Onto the next one."},{"Start":"04:04.460 ","End":"04:12.200","Text":"In Part 3, we\u0027ll continue with y=x^ negative 1,"},{"Start":"04:12.200 ","End":"04:16.540","Text":"y=x^ negative 3,"},{"Start":"04:16.540 ","End":"04:21.110","Text":"y=x^ negative 5 etc."},{"Start":"04:21.110 ","End":"04:22.685","Text":"In general,"},{"Start":"04:22.685 ","End":"04:32.280","Text":"y=x^n but"},{"Start":"04:32.280 ","End":"04:34.275","Text":"n gets negative values,"},{"Start":"04:34.275 ","End":"04:35.970","Text":"n is equal to -1,"},{"Start":"04:35.970 ","End":"04:38.954","Text":"-3, -5, etc."},{"Start":"04:38.954 ","End":"04:44.645","Text":"I could have just said n is a negative odd number."},{"Start":"04:44.645 ","End":"04:47.795","Text":"Let\u0027s take the first one now,"},{"Start":"04:47.795 ","End":"04:56.005","Text":"x^ -1 is just 1 over x so what we need is y=1/x."},{"Start":"04:56.005 ","End":"05:01.145","Text":"You may not be familiar with this so this time let\u0027s make a little table."},{"Start":"05:01.145 ","End":"05:05.010","Text":"We\u0027ll make a table and get,"},{"Start":"05:06.550 ","End":"05:11.330","Text":"x and y, and we\u0027ll take convenient values."},{"Start":"05:11.330 ","End":"05:16.050","Text":"For example, it\u0027s easy to take x=1,"},{"Start":"05:16.050 ","End":"05:19.850","Text":"and y=1 but I want something a bit less than 1 also so let\u0027s try a 1/2,"},{"Start":"05:19.850 ","End":"05:27.620","Text":"y is 1/ 1/2 is 2 and the reversal also if x is 2, y is 1/2."},{"Start":"05:27.620 ","End":"05:30.335","Text":"Also want to try some negatives."},{"Start":"05:30.335 ","End":"05:33.545","Text":"Let\u0027s just put a little dotted line here."},{"Start":"05:33.545 ","End":"05:39.450","Text":"Same thing should hold for negatives if we have -1/2 here,"},{"Start":"05:39.450 ","End":"05:41.940","Text":"whenever we -1/2 will be -2,"},{"Start":"05:41.940 ","End":"05:44.685","Text":"x is -1, y is -1."},{"Start":"05:44.685 ","End":"05:49.335","Text":"If x is -2, y is -1/2."},{"Start":"05:49.335 ","End":"05:51.660","Text":"If I plot all these,"},{"Start":"05:51.660 ","End":"05:59.940","Text":"then I\u0027ll get something like this where these are the points I computed 1,1,"},{"Start":"06:03.940 ","End":"06:09.810","Text":"2,1/2 and so on -1,-1."},{"Start":"06:09.810 ","End":"06:13.800","Text":"Actually, it turns out that these 2,"},{"Start":"06:13.800 ","End":"06:19.620","Text":"if I put in the 1,1 and this one here,"},{"Start":"06:19.620 ","End":"06:21.280","Text":"-1, -1,"},{"Start":"06:21.280 ","End":"06:24.500","Text":"these actually appear in the whole series."},{"Start":"06:24.500 ","End":"06:27.120","Text":"If I drew the next one,"},{"Start":"06:27.730 ","End":"06:36.980","Text":"y=x^-3, we get very similar except that it adheres more closely to the axis."},{"Start":"06:36.980 ","End":"06:41.330","Text":"It clings, it\u0027s closer to the axis over here."},{"Start":"06:41.330 ","End":"06:43.850","Text":"But, of course it still goes through these 2 points."},{"Start":"06:43.850 ","End":"06:46.220","Text":"All of them go through 1,1"},{"Start":"06:46.220 ","End":"06:52.070","Text":"and -1,-1 but then they just go closer to the axis more quickly."},{"Start":"06:52.070 ","End":"06:54.990","Text":"This could be 1/x^3,"},{"Start":"06:54.990 ","End":"07:01.070","Text":"or it could be in general 1/x^n for some n in series we talked about."},{"Start":"07:01.070 ","End":"07:05.855","Text":"There\u0027s something I almost forgot to mention about these family of functions."},{"Start":"07:05.855 ","End":"07:08.990","Text":"They don\u0027t intersect the axes."},{"Start":"07:08.990 ","End":"07:12.905","Text":"Let\u0027s use this one to represent the whole x^n,"},{"Start":"07:12.905 ","End":"07:15.470","Text":"not just 1/x, 1/x^3."},{"Start":"07:15.470 ","End":"07:21.380","Text":"Notice that the content dissect the y axis because for the y-axis,"},{"Start":"07:21.380 ","End":"07:25.115","Text":"if I\u0027m looking for the y-intercept for the y-axis,"},{"Start":"07:25.115 ","End":"07:28.940","Text":"then I have to substitute x=0."},{"Start":"07:28.940 ","End":"07:33.280","Text":"I get y=1/0^n,"},{"Start":"07:33.280 ","End":"07:34.770","Text":"and that\u0027s not possible."},{"Start":"07:34.770 ","End":"07:37.864","Text":"That\u0027s 1/0, so it doesn\u0027t intersect the y-axis."},{"Start":"07:37.864 ","End":"07:41.285","Text":"As for the x-axis or the x-intercept,"},{"Start":"07:41.285 ","End":"07:43.595","Text":"then I have to put y=0,"},{"Start":"07:43.595 ","End":"07:48.830","Text":"so I put y=0 then I get 0=1"},{"Start":"07:48.830 ","End":"07:56.760","Text":"over x^n and if I multiply both sides by x^n, I get 0."},{"Start":"07:56.760 ","End":"08:05.310","Text":"This gives me 0=1 which is certainly not possible so no x intercept,"},{"Start":"08:05.310 ","End":"08:08.210","Text":"and no y intercept for this family of functions."},{"Start":"08:08.210 ","End":"08:11.940","Text":"Now we can move onto the next set."},{"Start":"08:12.400 ","End":"08:17.120","Text":"Now we come to the next set of basic functions."},{"Start":"08:17.120 ","End":"08:20.120","Text":"These are the functions as written here,"},{"Start":"08:20.120 ","End":"08:23.735","Text":"x^-2, x^-4, x^-6."},{"Start":"08:23.735 ","End":"08:29.045","Text":"These are the x to the power of something that\u0027s negative and even,"},{"Start":"08:29.045 ","End":"08:32.030","Text":"even of course I imply a whole number."},{"Start":"08:32.030 ","End":"08:35.525","Text":"If it\u0027s convenient, we can use the other form,"},{"Start":"08:35.525 ","End":"08:39.035","Text":"like x^-2 is 1/x^2."},{"Start":"08:39.035 ","End":"08:44.990","Text":"Here we have 1/x^4,"},{"Start":"08:44.990 ","End":"08:49.145","Text":"1 over x^6, and so on."},{"Start":"08:49.145 ","End":"08:52.723","Text":"They all look very similar in their basic shape,"},{"Start":"08:52.723 ","End":"08:55.680","Text":"and I\u0027ll show you what they look like."},{"Start":"08:55.680 ","End":"09:01.790","Text":"Let\u0027s draw. Some axes."},{"Start":"09:02.730 ","End":"09:10.240","Text":"That\u0027s a bit better. Now, I\u0027ll try and freehand the 1/x^2."},{"Start":"09:15.050 ","End":"09:24.180","Text":"On the other side, exactly the mirror image. Is good enough."},{"Start":"09:24.180 ","End":"09:27.720","Text":"Notice this could be 1 over x^2,"},{"Start":"09:27.720 ","End":"09:30.910","Text":"it could be any one of them really."},{"Start":"09:31.430 ","End":"09:35.370","Text":"Notice it\u0027s always above the axis."},{"Start":"09:35.370 ","End":"09:41.310","Text":"That\u0027s because 1 over x^2 is 1 over a positive number."},{"Start":"09:41.310 ","End":"09:43.260","Text":"That\u0027s always positive."},{"Start":"09:43.260 ","End":"09:47.295","Text":"I think it\u0027s always floating above the x-axis."},{"Start":"09:47.295 ","End":"09:51.900","Text":"Notice the domain x cannot be 0."},{"Start":"09:51.900 ","End":"09:55.920","Text":"This is not included."},{"Start":"09:55.920 ","End":"09:59.400","Text":"I mean it doesn\u0027t touch the y-axis."},{"Start":"09:59.400 ","End":"10:02.860","Text":"There is no y-intercept."},{"Start":"10:03.440 ","End":"10:09.000","Text":"I can say there\u0027s no x-intercept either because this thing is always strictly positive."},{"Start":"10:09.000 ","End":"10:11.385","Text":"1 over something can\u0027t even be 0."},{"Start":"10:11.385 ","End":"10:14.790","Text":"I do want to say a word about how they do"},{"Start":"10:14.790 ","End":"10:19.080","Text":"differ even though this is supposed to represent all of them."},{"Start":"10:19.080 ","End":"10:23.205","Text":"Let\u0027s just say, although this does represent the family,"},{"Start":"10:23.205 ","End":"10:30.180","Text":"that this one is specifically y equals 1 over x^2."},{"Start":"10:30.180 ","End":"10:37.160","Text":"Let\u0027s superimpose y equals 1 over x^4,"},{"Start":"10:37.160 ","End":"10:40.000","Text":"but I\u0027ll do that one in orange."},{"Start":"10:40.970 ","End":"10:43.830","Text":"I\u0027ll even write it in orange,"},{"Start":"10:43.830 ","End":"10:47.115","Text":"y equals 1 over x^4."},{"Start":"10:47.115 ","End":"10:57.210","Text":"Well, the way it differs is that above this 1, it\u0027s higher."},{"Start":"10:57.210 ","End":"10:59.464","Text":"It still goes through here,"},{"Start":"10:59.464 ","End":"11:01.440","Text":"and then below that part,"},{"Start":"11:01.440 ","End":"11:03.960","Text":"it actually goes to 0 quicker,"},{"Start":"11:03.960 ","End":"11:05.910","Text":"meaning it\u0027s lower than this one."},{"Start":"11:05.910 ","End":"11:07.950","Text":"Well, the picture explains what I mean."},{"Start":"11:07.950 ","End":"11:16.020","Text":"Likewise here, this bit of 1 over x^4 is above 1 over x^2 in this part,"},{"Start":"11:16.020 ","End":"11:18.700","Text":"and here it\u0027s below."},{"Start":"11:19.760 ","End":"11:25.290","Text":"That\u0027s it for these negative even powers."},{"Start":"11:25.290 ","End":"11:29.700","Text":"Onto the next set, well,"},{"Start":"11:29.700 ","End":"11:34.935","Text":"we\u0027re continuing with the x^n functions."},{"Start":"11:34.935 ","End":"11:39.330","Text":"This time the single function where n equals 1/2,"},{"Start":"11:39.330 ","End":"11:42.615","Text":"which gives us y equals square root of x."},{"Start":"11:42.615 ","End":"11:45.180","Text":"You may, or may not be familiar with this,"},{"Start":"11:45.180 ","End":"11:47.748","Text":"why don\u0027t we even sketch it together,"},{"Start":"11:47.748 ","End":"11:49.980","Text":"and we\u0027ll do the table and everything."},{"Start":"11:49.980 ","End":"11:58.540","Text":"Let\u0027s make a little table here with some x and y."},{"Start":"11:59.210 ","End":"12:05.370","Text":"Now, the domain of this function is x bigger or equal to 0 because the square root,"},{"Start":"12:05.370 ","End":"12:07.110","Text":"we can\u0027t take of a negative number,"},{"Start":"12:07.110 ","End":"12:11.346","Text":"we can take 0 and we can take positive,"},{"Start":"12:11.346 ","End":"12:16.701","Text":"so we are only going to get stuff on the right side of the y-axis."},{"Start":"12:16.701 ","End":"12:20.880","Text":"Notice that if I put a number here, positive number,"},{"Start":"12:20.880 ","End":"12:23.964","Text":"the square root of it is also positive,"},{"Start":"12:23.964 ","End":"12:28.664","Text":"so we\u0027re really only going to see stuff in the first quadrant."},{"Start":"12:28.664 ","End":"12:33.615","Text":"0 is the most convenient to start off with,"},{"Start":"12:33.615 ","End":"12:37.259","Text":"that\u0027s actually very first value we\u0027re allowed to substitute."},{"Start":"12:37.259 ","End":"12:39.390","Text":"Y also is 0,"},{"Start":"12:39.390 ","End":"12:41.520","Text":"if it\u0027s the square root."},{"Start":"12:41.520 ","End":"12:43.905","Text":"Plot them in a moment."},{"Start":"12:43.905 ","End":"12:47.070","Text":"1 is also easy to compute."},{"Start":"12:47.070 ","End":"12:49.170","Text":"Square root of 1 is 1."},{"Start":"12:49.170 ","End":"12:52.380","Text":"We could put stuff like 2 and 3,"},{"Start":"12:52.380 ","End":"12:54.240","Text":"but a bit messy,"},{"Start":"12:54.240 ","End":"12:55.941","Text":"need to look it up on the calculator."},{"Start":"12:55.941 ","End":"13:00.300","Text":"I\u0027m going to go for numbers that will give us nice whole square roots."},{"Start":"13:00.300 ","End":"13:03.347","Text":"4 would be next and its square root would be 2."},{"Start":"13:03.347 ","End":"13:07.710","Text":"Maybe we can fit in up to 9, which would be 3."},{"Start":"13:07.710 ","End":"13:09.480","Text":"Let\u0027s see if we have 0,"},{"Start":"13:09.480 ","End":"13:12.220","Text":"we\u0027re going to need 1."},{"Start":"13:12.380 ","End":"13:17.560","Text":"Then we\u0027re not going to need 2, 3, 4."},{"Start":"13:17.750 ","End":"13:22.450","Text":"We\u0027ll have 5,"},{"Start":"13:22.450 ","End":"13:25.320","Text":"6, 7, 8, 9, 10."},{"Start":"13:25.320 ","End":"13:27.360","Text":"9, we\u0027re going to take."},{"Start":"13:27.360 ","End":"13:30.270","Text":"We\u0027ll need here 0, we have that,"},{"Start":"13:30.270 ","End":"13:34.880","Text":"and we\u0027ll need 1,"},{"Start":"13:34.880 ","End":"13:37.270","Text":"2, and 3."},{"Start":"13:38.920 ","End":"13:42.050","Text":"We\u0027ll get some, I\u0027ll make it in a color,"},{"Start":"13:42.050 ","End":"13:44.980","Text":"let\u0027s go for red."},{"Start":"13:44.980 ","End":"13:50.190","Text":"0, 0, 1, 1,"},{"Start":"13:50.190 ","End":"13:56.010","Text":"4 gives us 2 and 9 gives us 3,"},{"Start":"13:56.010 ","End":"13:57.885","Text":"which is somewhere here."},{"Start":"13:57.885 ","End":"14:01.230","Text":"That\u0027s what we get, then plot some points."},{"Start":"14:01.230 ","End":"14:07.320","Text":"It actually starts out vertically and then gradually gets flatter and flatter."},{"Start":"14:07.320 ","End":"14:10.140","Text":"Of course, it goes on to infinity."},{"Start":"14:10.140 ","End":"14:14.100","Text":"Sometimes I even write an arrow on the graph itself to say hey,"},{"Start":"14:14.100 ","End":"14:15.190","Text":"it doesn\u0027t just stop here,"},{"Start":"14:15.190 ","End":"14:17.085","Text":"it goes on forever."},{"Start":"14:17.085 ","End":"14:20.145","Text":"That\u0027s really all I have say."},{"Start":"14:20.145 ","End":"14:22.440","Text":"Number 6, again,"},{"Start":"14:22.440 ","End":"14:25.410","Text":"in the series of the x^n,"},{"Start":"14:25.410 ","End":"14:27.270","Text":"this time n equals 1/3,"},{"Start":"14:27.270 ","End":"14:29.595","Text":"which gives us the cube root of x."},{"Start":"14:29.595 ","End":"14:30.870","Text":"Little bit less famous,"},{"Start":"14:30.870 ","End":"14:34.410","Text":"but still useful to add to our stock of basic functions."},{"Start":"14:34.410 ","End":"14:37.335","Text":"It\u0027s a bit different than the square root of x."},{"Start":"14:37.335 ","End":"14:40.625","Text":"For one thing, its domain is all of x."},{"Start":"14:40.625 ","End":"14:43.910","Text":"Let\u0027s just make a sketch,"},{"Start":"14:43.910 ","End":"14:46.685","Text":"draw table, plot the points."},{"Start":"14:46.685 ","End":"14:53.250","Text":"Here we have, put x here and y here."},{"Start":"14:53.250 ","End":"14:58.140","Text":"When x is 0,"},{"Start":"14:58.140 ","End":"15:01.545","Text":"the cube root of 0 is 0."},{"Start":"15:01.545 ","End":"15:03.840","Text":"Let\u0027s keep going with the positive."},{"Start":"15:03.840 ","End":"15:05.910","Text":"Let\u0027s try taking x=1."},{"Start":"15:05.910 ","End":"15:08.730","Text":"Cube root of 1 is 1."},{"Start":"15:08.730 ","End":"15:11.520","Text":"2, 3, 4 and so on a messy,"},{"Start":"15:11.520 ","End":"15:14.220","Text":"comes out cube root needs the calculator."},{"Start":"15:14.220 ","End":"15:19.185","Text":"Let\u0027s go straight to 8 because cube root of 8 is a nice whole number,"},{"Start":"15:19.185 ","End":"15:22.170","Text":"it comes out 2 and so on."},{"Start":"15:22.170 ","End":"15:24.210","Text":"To get to 3, we\u0027d have to go up to 27,"},{"Start":"15:24.210 ","End":"15:25.710","Text":"I don\u0027t think I have room for that."},{"Start":"15:25.710 ","End":"15:28.440","Text":"But we go on to the negative numbers."},{"Start":"15:28.440 ","End":"15:32.100","Text":"Exactly the same, cube root of a negative is a negative."},{"Start":"15:32.100 ","End":"15:33.525","Text":"Take minus 1,"},{"Start":"15:33.525 ","End":"15:35.595","Text":"the cube root of that is minus 1."},{"Start":"15:35.595 ","End":"15:38.550","Text":"Remember, minus times minus times minus is also minus."},{"Start":"15:38.550 ","End":"15:40.455","Text":"There is no problem with negatives here."},{"Start":"15:40.455 ","End":"15:46.830","Text":"Likewise, we could take minus 8 here and get minus 2."},{"Start":"15:46.830 ","End":"15:48.765","Text":"Let\u0027s plot those."},{"Start":"15:48.765 ","End":"15:50.370","Text":"Just like we did with the square root,"},{"Start":"15:50.370 ","End":"15:53.298","Text":"I\u0027d like to do it in red."},{"Start":"15:53.298 ","End":"15:56.925","Text":"0,0 gives us this point here."},{"Start":"15:56.925 ","End":"16:00.530","Text":"Let\u0027s say that this is 1,"},{"Start":"16:00.530 ","End":"16:01.930","Text":"so we get here 1,"},{"Start":"16:01.930 ","End":"16:04.455","Text":"that will be 1 and that will be 1."},{"Start":"16:04.455 ","End":"16:07.545","Text":"To get to 2,"},{"Start":"16:07.545 ","End":"16:09.060","Text":"we need to go all the way up to 8."},{"Start":"16:09.060 ","End":"16:10.620","Text":"I don\u0027t know exactly where that is,"},{"Start":"16:10.620 ","End":"16:12.990","Text":"but let\u0027s say that 8 is somewhere here,"},{"Start":"16:12.990 ","End":"16:16.950","Text":"and then we get this point here, 8, 2."},{"Start":"16:16.950 ","End":"16:19.095","Text":"This is actually an odd function."},{"Start":"16:19.095 ","End":"16:22.005","Text":"We get exactly the same on the other side, rotated."},{"Start":"16:22.005 ","End":"16:23.685","Text":"For minus 1,"},{"Start":"16:23.685 ","End":"16:26.355","Text":"we get this point here say."},{"Start":"16:26.355 ","End":"16:29.190","Text":"To get to minus 2,"},{"Start":"16:29.190 ","End":"16:31.710","Text":"we have to go all the way up to minus 8."},{"Start":"16:31.710 ","End":"16:34.215","Text":"Let\u0027s say that\u0027s somewhere here."},{"Start":"16:34.215 ","End":"16:36.480","Text":"Now, we just have to connect them."},{"Start":"16:36.480 ","End":"16:38.340","Text":"Here it is,"},{"Start":"16:38.340 ","End":"16:39.750","Text":"the cube root of x,"},{"Start":"16:39.750 ","End":"16:42.990","Text":"goes along like this,"},{"Start":"16:42.990 ","End":"16:45.330","Text":"then it starts going upwards."},{"Start":"16:45.330 ","End":"16:48.450","Text":"Actually, tangent is vertical at this point and then"},{"Start":"16:48.450 ","End":"16:51.720","Text":"it gets flatter and flatter again and so on to infinity."},{"Start":"16:51.720 ","End":"16:53.130","Text":"That\u0027s the next to last one."},{"Start":"16:53.130 ","End":"16:58.455","Text":"The last one I\u0027d like to show you will be the absolute value of x coming up."},{"Start":"16:58.455 ","End":"17:01.470","Text":"Here we are with basic function number 7,"},{"Start":"17:01.470 ","End":"17:03.654","Text":"y equals absolute value of x,"},{"Start":"17:03.654 ","End":"17:07.094","Text":"very famous function,"},{"Start":"17:07.094 ","End":"17:09.600","Text":"the one you should certainly know about."},{"Start":"17:09.600 ","End":"17:13.200","Text":"Let\u0027s just do it with a table."},{"Start":"17:13.200 ","End":"17:14.580","Text":"Let\u0027s start right away."},{"Start":"17:14.580 ","End":"17:17.520","Text":"Now, notice that x can be anything,"},{"Start":"17:17.520 ","End":"17:19.440","Text":"there\u0027s no restriction on the domain."},{"Start":"17:19.440 ","End":"17:23.460","Text":"Domain is all x because whether x is positive or negative,"},{"Start":"17:23.460 ","End":"17:26.232","Text":"absolute value of x is defined,"},{"Start":"17:26.232 ","End":"17:27.840","Text":"but y will always be positive,"},{"Start":"17:27.840 ","End":"17:30.450","Text":"which is why I drew this piece a bit shorter because I know"},{"Start":"17:30.450 ","End":"17:34.230","Text":"everything\u0027s going to be positive or at least non-negative."},{"Start":"17:34.230 ","End":"17:38.220","Text":"When x is 0, the absolute value is defined as 0."},{"Start":"17:38.220 ","End":"17:41.445","Text":"Then if we take 1 or minus 1,"},{"Start":"17:41.445 ","End":"17:43.545","Text":"the absolute value is going to be 1."},{"Start":"17:43.545 ","End":"17:46.245","Text":"If we take 2 or minus 2,"},{"Start":"17:46.245 ","End":"17:48.870","Text":"the absolute value is going to be 2 and so on."},{"Start":"17:48.870 ","End":"17:50.385","Text":"I\u0027ll be able to do one more."},{"Start":"17:50.385 ","End":"17:53.880","Text":"If we have 3 or minus 3,"},{"Start":"17:53.880 ","End":"17:57.030","Text":"then the absolute value is going to be 3."},{"Start":"17:57.030 ","End":"17:58.650","Text":"If I plot all these points,"},{"Start":"17:58.650 ","End":"18:01.885","Text":"0,0 is here, 1,1 would be here,"},{"Start":"18:01.885 ","End":"18:03.645","Text":"2,2 is here,"},{"Start":"18:03.645 ","End":"18:05.545","Text":"3,3 is here,"},{"Start":"18:05.545 ","End":"18:07.545","Text":"likewise on the minus 1."},{"Start":"18:07.545 ","End":"18:10.110","Text":"This is also 1, 2, 3."},{"Start":"18:10.270 ","End":"18:14.120","Text":"Minus 1 is also 1."},{"Start":"18:14.120 ","End":"18:20.430","Text":"Minus 2,2, minus 3,3."},{"Start":"18:20.430 ","End":"18:23.190","Text":"It\u0027s supposed to come out a straight line."},{"Start":"18:23.190 ","End":"18:25.440","Text":"There\u0027s a rough sketch, this is what it will look like."},{"Start":"18:25.440 ","End":"18:28.995","Text":"Of course, it will go on indefinitely to infinity."},{"Start":"18:28.995 ","End":"18:31.550","Text":"Make introduction to absolute value of x,"},{"Start":"18:31.550 ","End":"18:36.300","Text":"and that\u0027s the last one in the set of basic functions we\u0027ll be learning about."}],"ID":5127}],"Thumbnail":null,"ID":56143},{"Name":"Translation and Reflection of Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vertical Translation of Basic Functions","Duration":"4m 21s","ChapterTopicVideoID":5150,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5150.jpeg","UploadDate":"2020-09-30T13:13:58.6730000","DurationForVideoObject":"PT4M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.280","Text":"In this clip and in the next 1 we\u0027ll be talking about translation of functions."},{"Start":"00:06.280 ","End":"00:11.410","Text":"In this section, it will be vertical and then the next we\u0027ll do the horizontal,"},{"Start":"00:11.410 ","End":"00:16.075","Text":"vertical meaning up and down and translation is another way of saying shift,"},{"Start":"00:16.075 ","End":"00:18.250","Text":"but perhaps it\u0027s best to start with an example."},{"Start":"00:18.250 ","End":"00:19.930","Text":"Let\u0027s take a function,"},{"Start":"00:19.930 ","End":"00:23.680","Text":"any function, let\u0027s say y equals x^2,"},{"Start":"00:23.680 ","End":"00:31.725","Text":"so we\u0027ll take y equals x^2."},{"Start":"00:31.725 ","End":"00:33.885","Text":"Here\u0027s the y equals x^2,"},{"Start":"00:33.885 ","End":"00:40.470","Text":"which I got by simply just plotting a few points,"},{"Start":"00:40.470 ","End":"00:42.240","Text":"0, 0, 1, 1, 2, 4,"},{"Start":"00:42.240 ","End":"00:44.625","Text":"etc, and joining them with a line."},{"Start":"00:44.625 ","End":"00:46.335","Text":"The point is not to be accurate,"},{"Start":"00:46.335 ","End":"00:49.568","Text":"it\u0027s to get the general idea."},{"Start":"00:49.568 ","End":"00:54.905","Text":"Now someone asks me to take this function and move it upwards 1 unit,"},{"Start":"00:54.905 ","End":"00:56.225","Text":"to shift it upwards,"},{"Start":"00:56.225 ","End":"00:58.700","Text":"what we call a vertical translation upwards."},{"Start":"00:58.700 ","End":"01:00.230","Text":"What would it look like?"},{"Start":"01:00.230 ","End":"01:04.080","Text":"Well, I could just take all these points and move them 1 up,"},{"Start":"01:04.100 ","End":"01:07.220","Text":"then I get these points and likewise,"},{"Start":"01:07.220 ","End":"01:11.350","Text":"I could connect them with a continuous line,"},{"Start":"01:11.350 ","End":"01:13.515","Text":"something like this,"},{"Start":"01:13.515 ","End":"01:16.775","Text":"so that if this is where y equals 0,"},{"Start":"01:16.775 ","End":"01:18.455","Text":"here y equals 1,"},{"Start":"01:18.455 ","End":"01:20.735","Text":"if here y equals 1, here will be 2,"},{"Start":"01:20.735 ","End":"01:22.760","Text":"each time we add 1,"},{"Start":"01:22.760 ","End":"01:25.345","Text":"where we move it upwards,"},{"Start":"01:25.345 ","End":"01:29.265","Text":"so that this distance here is 1,"},{"Start":"01:29.265 ","End":"01:33.850","Text":"this is moving upwards by 1 unit."},{"Start":"01:34.080 ","End":"01:38.440","Text":"The question is, what is the equation of this new function?"},{"Start":"01:38.440 ","End":"01:39.955","Text":"Well, quite clearly,"},{"Start":"01:39.955 ","End":"01:42.365","Text":"if I moved it up with 1,"},{"Start":"01:42.365 ","End":"01:46.140","Text":"the equation would be y equals x^2 plus 1,"},{"Start":"01:46.140 ","End":"01:49.490","Text":"whatever was here was x^2 and then we add 1."},{"Start":"01:49.490 ","End":"01:53.035","Text":"At every point, we go upwards by 1 unit,"},{"Start":"01:53.035 ","End":"01:55.360","Text":"upwards by 1 unit,"},{"Start":"01:55.360 ","End":"01:58.390","Text":"and so on, here,"},{"Start":"01:58.390 ","End":"02:01.895","Text":"here, everywhere we go up by 1 unit."},{"Start":"02:01.895 ","End":"02:05.340","Text":"That\u0027s upward translation or upward shifting,"},{"Start":"02:05.340 ","End":"02:07.240","Text":"now what about the reverse direction?"},{"Start":"02:07.240 ","End":"02:14.000","Text":"Now, I\u0027m asked to move the original function y equals x^2 down by 2 units, let\u0027s say."},{"Start":"02:14.000 ","End":"02:18.309","Text":"Again, first of all I\u0027d plot some points,"},{"Start":"02:18.309 ","End":"02:23.955","Text":"something like this and then I join them up with a line here."},{"Start":"02:23.955 ","End":"02:28.115","Text":"The way we get from the black to the blue is by going 2 units down."},{"Start":"02:28.115 ","End":"02:33.161","Text":"Here I go downwards 2 units, downwards 2 units,"},{"Start":"02:33.161 ","End":"02:34.490","Text":"and so on, here,"},{"Start":"02:34.490 ","End":"02:39.070","Text":"and here, and everywhere."},{"Start":"02:39.070 ","End":"02:44.285","Text":"The equation of this graph quite clearly would be"},{"Start":"02:44.285 ","End":"02:51.120","Text":"the original x^2 to get to the point on the black and then move it down by 2."},{"Start":"02:51.360 ","End":"02:54.385","Text":"Now I\u0027d like to generalize this concept,"},{"Start":"02:54.385 ","End":"02:56.575","Text":"what if it wasn\u0027t x^2?"},{"Start":"02:56.575 ","End":"02:59.200","Text":"Suppose it was just any old function."},{"Start":"02:59.200 ","End":"03:03.640","Text":"If it was just some f(x),"},{"Start":"03:03.640 ","End":"03:05.380","Text":"surely the same principle holds."},{"Start":"03:05.380 ","End":"03:06.770","Text":"There\u0027s nothing special about x^2,"},{"Start":"03:06.770 ","End":"03:09.040","Text":"so any function that I take,"},{"Start":"03:09.040 ","End":"03:11.050","Text":"I can raise and lower,"},{"Start":"03:11.050 ","End":"03:13.690","Text":"translate upwards or downwards, shift up,"},{"Start":"03:13.690 ","End":"03:15.820","Text":"shift down many names,"},{"Start":"03:15.820 ","End":"03:18.175","Text":"the same principle would hold."},{"Start":"03:18.175 ","End":"03:21.715","Text":"Now I\u0027d like to take the generalization still further."},{"Start":"03:21.715 ","End":"03:24.400","Text":"What if it wasn\u0027t 1 unit I raised it by,"},{"Start":"03:24.400 ","End":"03:26.710","Text":"and what if it wasn\u0027t 2 units I lowered it by?"},{"Start":"03:26.710 ","End":"03:32.520","Text":"Suppose it with some general k upwards or l downwards,"},{"Start":"03:32.520 ","End":"03:38.800","Text":"surely I could replace 1 by k and 2 by l. Here,"},{"Start":"03:38.800 ","End":"03:42.685","Text":"I\u0027ve replaced the numbers by k and l generically,"},{"Start":"03:42.685 ","End":"03:44.850","Text":"k upwards, l downwards."},{"Start":"03:44.850 ","End":"03:46.290","Text":"This brings us to the point k,"},{"Start":"03:46.290 ","End":"03:49.150","Text":"this is the point minus l because it\u0027s downwards."},{"Start":"03:49.150 ","End":"03:51.470","Text":"As a summary,"},{"Start":"03:51.470 ","End":"03:56.980","Text":"what I can tell you about the vertical translation and functions is that if you"},{"Start":"03:56.980 ","End":"04:02.770","Text":"take a function that you know and you want to shift it upwards by k units,"},{"Start":"04:02.770 ","End":"04:06.190","Text":"you just add plus k to the formula."},{"Start":"04:06.190 ","End":"04:11.010","Text":"If you want to shift some function f(x) downwards by l units,"},{"Start":"04:11.010 ","End":"04:13.630","Text":"you just write minus l after the function,"},{"Start":"04:13.630 ","End":"04:15.505","Text":"and that gives you the new function."},{"Start":"04:15.505 ","End":"04:17.605","Text":"We\u0027re done with the vertical,"},{"Start":"04:17.605 ","End":"04:21.240","Text":"now let\u0027s move on to the horizontal, left and right."}],"ID":5128},{"Watched":false,"Name":"Horizontal Translation of Basic Functions","Duration":"6m 11s","ChapterTopicVideoID":5151,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5151.jpeg","UploadDate":"2020-09-30T12:59:29.9970000","DurationForVideoObject":"PT6M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:05.085","Text":"Here we are once again with translation of functions."},{"Start":"00:05.085 ","End":"00:08.069","Text":"You remember that translation means shifting,"},{"Start":"00:08.069 ","End":"00:12.240","Text":"only this time it will be horizontal rather than vertical, by which I mean,"},{"Start":"00:12.240 ","End":"00:14.220","Text":"we\u0027ll be shifting left and right,"},{"Start":"00:14.220 ","End":"00:16.905","Text":"whereas previously we were shifting up and down."},{"Start":"00:16.905 ","End":"00:19.875","Text":"It\u0027s a bit different as you will soon see."},{"Start":"00:19.875 ","End":"00:26.895","Text":"Once again, let\u0027s take some function and it might as well be y=x^2,"},{"Start":"00:26.895 ","End":"00:29.055","Text":"which we use as a sample function,"},{"Start":"00:29.055 ","End":"00:31.515","Text":"though this will apply in general."},{"Start":"00:31.515 ","End":"00:35.730","Text":"Someone asks me, could I please move it"},{"Start":"00:35.730 ","End":"00:41.075","Text":"4 units to the right and tell me what the new function will be."},{"Start":"00:41.075 ","End":"00:43.190","Text":"As before, what I would do,"},{"Start":"00:43.190 ","End":"00:46.520","Text":"we take these points that I used in the sketching and move them all"},{"Start":"00:46.520 ","End":"00:50.940","Text":"4 units to the right, something like this."},{"Start":"00:50.940 ","End":"00:55.903","Text":"Here\u0027s what I get. I plotted the points and I joined them with a line,"},{"Start":"00:55.903 ","End":"01:00.705","Text":"and now I\u0027ve got this orange-colored function."},{"Start":"01:00.705 ","End":"01:03.140","Text":"I\u0027ll just tell you what it is."},{"Start":"01:03.140 ","End":"01:06.455","Text":"The rule is that when we move 4 units to the right,"},{"Start":"01:06.455 ","End":"01:09.095","Text":"we subtract 4 from the x."},{"Start":"01:09.095 ","End":"01:19.490","Text":"This function will be y=x-4^2."},{"Start":"01:19.490 ","End":"01:22.460","Text":"You\u0027ll soon see why minus as many people"},{"Start":"01:22.460 ","End":"01:24.995","Text":"want to put a plus here because it goes to the right."},{"Start":"01:24.995 ","End":"01:27.440","Text":"Let\u0027s just check it on a couple of values."},{"Start":"01:27.440 ","End":"01:30.395","Text":"Let\u0027s check it, for example, when x is 4,"},{"Start":"01:30.395 ","End":"01:32.485","Text":"if we put x=4 here,"},{"Start":"01:32.485 ","End":"01:34.220","Text":"x-4 is 0,"},{"Start":"01:34.220 ","End":"01:35.689","Text":"like this number was here,"},{"Start":"01:35.689 ","End":"01:37.115","Text":"and squared is 0."},{"Start":"01:37.115 ","End":"01:39.095","Text":"That makes sense."},{"Start":"01:39.095 ","End":"01:43.760","Text":"Coincidentally, I noticed that these actually do cross,"},{"Start":"01:43.760 ","End":"01:44.990","Text":"so in actual world,"},{"Start":"01:44.990 ","End":"01:51.650","Text":"might as well make use of this luck that we have the 0.24 common to both of them."},{"Start":"01:51.650 ","End":"01:59.830","Text":"I\u0027ll actually write it again in black because the same 0.24 is also on the black graph."},{"Start":"01:59.830 ","End":"02:01.940","Text":"If we put 2,4 here,"},{"Start":"02:01.940 ","End":"02:03.790","Text":"we see 2^2 is 4."},{"Start":"02:03.790 ","End":"02:05.835","Text":"But on the orange 1,"},{"Start":"02:05.835 ","End":"02:13.830","Text":"if we put in 2, it\u0027s 2-4."},{"Start":"02:13.830 ","End":"02:15.940","Text":"When x is 2,"},{"Start":"02:17.240 ","End":"02:20.110","Text":"suppose x is 2 here,"},{"Start":"02:20.110 ","End":"02:23.345","Text":"I\u0027m just writing this in small for myself."},{"Start":"02:23.345 ","End":"02:27.215","Text":"X is 2 and 2-4 is -2."},{"Start":"02:27.215 ","End":"02:30.755","Text":"-2^2 is also 4, so that fits also."},{"Start":"02:30.755 ","End":"02:38.510","Text":"But you could even try another 1."},{"Start":"02:38.510 ","End":"02:40.325","Text":"Let\u0027s say x=5,"},{"Start":"02:40.325 ","End":"02:45.084","Text":"5-4 is 1,"},{"Start":"02:45.084 ","End":"02:47.825","Text":"1^2 is 1, works."},{"Start":"02:47.825 ","End":"02:51.700","Text":"Because you see what\u0027s happening is the -4 is transporting it to here."},{"Start":"02:51.700 ","End":"02:53.830","Text":"If I want to know where 5 goes,"},{"Start":"02:53.830 ","End":"02:57.700","Text":"I go 4 to the left and see where it goes on the black parabola,"},{"Start":"02:57.700 ","End":"03:00.125","Text":"and I copy that back here."},{"Start":"03:00.125 ","End":"03:02.962","Text":"In any event, you don\u0027t have to understand the reason,"},{"Start":"03:02.962 ","End":"03:05.215","Text":"just remember to do it the right way."},{"Start":"03:05.215 ","End":"03:07.450","Text":"If you put it as a plus 4,"},{"Start":"03:07.450 ","End":"03:09.685","Text":"shift 4 to the right."},{"Start":"03:09.685 ","End":"03:13.510","Text":"You put, instead of x, x-4."},{"Start":"03:13.510 ","End":"03:16.180","Text":"Now, let\u0027s try going the other way."},{"Start":"03:16.180 ","End":"03:17.485","Text":"Obviously, we\u0027ve done right."},{"Start":"03:17.485 ","End":"03:19.675","Text":"Now, we want to see what happens when we shift it to the left."},{"Start":"03:19.675 ","End":"03:21.790","Text":"Now, the instruction is,"},{"Start":"03:21.790 ","End":"03:28.415","Text":"move this parabola 5 units to the left please."},{"Start":"03:28.415 ","End":"03:34.265","Text":"I would take these points and join them up."},{"Start":"03:34.265 ","End":"03:37.440","Text":"I will do it in blue this time."},{"Start":"03:37.840 ","End":"03:41.840","Text":"Here\u0027s what I get. I plotted the points and I"},{"Start":"03:41.840 ","End":"03:46.060","Text":"already joined them with a line, roughly a parabola."},{"Start":"03:46.060 ","End":"03:48.680","Text":"I\u0027ll tell you what the equation of this is."},{"Start":"03:48.680 ","End":"03:52.775","Text":"As you can guess, if we moved it to the right-hand and you got a -4,"},{"Start":"03:52.775 ","End":"03:55.295","Text":"you\u0027re probably guessing and you\u0027d be right."},{"Start":"03:55.295 ","End":"03:59.790","Text":"This is y=x +5^2."},{"Start":"03:59.870 ","End":"04:02.450","Text":"We can check, for example,"},{"Start":"04:02.450 ","End":"04:05.585","Text":"if you put in x=-5,"},{"Start":"04:05.585 ","End":"04:07.892","Text":"that 5 is 0,"},{"Start":"04:07.892 ","End":"04:10.465","Text":"0^2 is 0, which is just fine."},{"Start":"04:10.465 ","End":"04:15.870","Text":"Try, let\u0027s say, x=-4."},{"Start":"04:15.870 ","End":"04:19.145","Text":"If I add 5, it\u0027s plus 1, squared is 1."},{"Start":"04:19.145 ","End":"04:20.795","Text":"You\u0027ll check it all works out."},{"Start":"04:20.795 ","End":"04:25.160","Text":"The plus 5 is because each point here,"},{"Start":"04:25.160 ","End":"04:29.840","Text":"I take the x value and go and look at the black parabola,"},{"Start":"04:29.840 ","End":"04:32.900","Text":"but I have to go 5 to the right to get to the corresponding point,"},{"Start":"04:32.900 ","End":"04:35.045","Text":"and then I look up the y and put it back here."},{"Start":"04:35.045 ","End":"04:36.560","Text":"That\u0027s the general idea."},{"Start":"04:36.560 ","End":"04:37.910","Text":"But if you don\u0027t follow that,"},{"Start":"04:37.910 ","End":"04:40.260","Text":"just remember it as a rule."},{"Start":"04:40.610 ","End":"04:47.615","Text":"Let\u0027s generalize this like we did with the vertical translation."},{"Start":"04:47.615 ","End":"04:49.425","Text":"The horizontal translation."},{"Start":"04:49.425 ","End":"04:55.040","Text":"First of all, we generalize from x^2 to an arbitrary function f. Let\u0027s do that here also."},{"Start":"04:55.040 ","End":"04:57.410","Text":"Supposing we just had any old function"},{"Start":"04:57.410 ","End":"05:03.380","Text":"f. If we have a function f of x and we want to move it 4 units to the right,"},{"Start":"05:03.380 ","End":"05:06.605","Text":"the new function is f of x-4."},{"Start":"05:06.605 ","End":"05:09.875","Text":"If we moved it 5 units to the left,"},{"Start":"05:09.875 ","End":"05:13.250","Text":"the new function is f of x plus 5."},{"Start":"05:13.250 ","End":"05:15.500","Text":"If we have a specific formula for f and we can"},{"Start":"05:15.500 ","End":"05:17.870","Text":"actually compute what this function is or this,"},{"Start":"05:17.870 ","End":"05:20.045","Text":"but this is at the general level."},{"Start":"05:20.045 ","End":"05:22.520","Text":"Once again, we\u0027re also going to generalize because"},{"Start":"05:22.520 ","End":"05:24.680","Text":"someone might not have said 4 units to the right,"},{"Start":"05:24.680 ","End":"05:27.500","Text":"might have said k units to the right,"},{"Start":"05:27.500 ","End":"05:30.260","Text":"and to the left might\u0027ve said l units to the left."},{"Start":"05:30.260 ","End":"05:33.120","Text":"We want to generalize that too."},{"Start":"05:33.490 ","End":"05:35.929","Text":"Now, we can generalize."},{"Start":"05:35.929 ","End":"05:38.585","Text":"I\u0027ve replaced the numbers,"},{"Start":"05:38.585 ","End":"05:43.965","Text":"the 5 bits l and the k for 4."},{"Start":"05:43.965 ","End":"05:45.520","Text":"What we can say is this,"},{"Start":"05:45.520 ","End":"05:50.525","Text":"if you\u0027re given a function and you want to move it k units to the right,"},{"Start":"05:50.525 ","End":"05:53.405","Text":"you replace x by x-k."},{"Start":"05:53.405 ","End":"05:54.975","Text":"If you want to move it,"},{"Start":"05:54.975 ","End":"05:57.285","Text":"shift it, translate it,"},{"Start":"05:57.285 ","End":"05:59.945","Text":"l units to the left,"},{"Start":"05:59.945 ","End":"06:03.395","Text":"then you replace x by x+l."},{"Start":"06:03.395 ","End":"06:08.461","Text":"That\u0027s the rule. For the rest of the lesson,"},{"Start":"06:08.461 ","End":"06:11.100","Text":"we\u0027ll just do some examples."}],"ID":5129},{"Watched":false,"Name":"Examples","Duration":"7m 40s","ChapterTopicVideoID":5152,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5152.jpeg","UploadDate":"2020-09-30T13:08:46.5430000","DurationForVideoObject":"PT7M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.695","Text":"The first example we\u0027ll do is the following."},{"Start":"00:04.695 ","End":"00:15.540","Text":"Well, it says is sketch the graph of y equals the square root of x+2."},{"Start":"00:15.540 ","End":"00:17.430","Text":"How I go about this,"},{"Start":"00:17.430 ","End":"00:20.400","Text":"is I look at my stock of basic functions."},{"Start":"00:20.400 ","End":"00:23.685","Text":"I see square root of x is the closest,"},{"Start":"00:23.685 ","End":"00:28.785","Text":"and I have to somehow shift it up and down to get this 1."},{"Start":"00:28.785 ","End":"00:33.630","Text":"First of all, I sketch the basic y equals the square root of x,"},{"Start":"00:33.630 ","End":"00:36.000","Text":"and we\u0027ll get something like this."},{"Start":"00:36.000 ","End":"00:40.530","Text":"Then we notice that what we have is not this, but this +2."},{"Start":"00:40.530 ","End":"00:44.975","Text":"We remember that when we have a function of x+a constant,"},{"Start":"00:44.975 ","End":"00:47.690","Text":"we just raise it upwards by 2 units."},{"Start":"00:47.690 ","End":"00:50.100","Text":"We\u0027ll get something like this."},{"Start":"00:50.260 ","End":"00:55.655","Text":"We take this graph of y equals the square root of x,"},{"Start":"00:55.655 ","End":"00:57.560","Text":"and then we just, +2 means,"},{"Start":"00:57.560 ","End":"00:59.390","Text":"raise it 2 upwards,"},{"Start":"00:59.390 ","End":"01:00.890","Text":"take this whole thing,"},{"Start":"01:00.890 ","End":"01:02.465","Text":"raise it 2 upwards,"},{"Start":"01:02.465 ","End":"01:04.310","Text":"and we get this."},{"Start":"01:04.310 ","End":"01:06.425","Text":"That\u0027s all there is to it."},{"Start":"01:06.425 ","End":"01:12.535","Text":"Now on to the next exercise where it\u0027s going to be number 2."},{"Start":"01:12.535 ","End":"01:20.110","Text":"What we\u0027re going to ask for is instead of y equals the square root of x+2,"},{"Start":"01:20.110 ","End":"01:24.660","Text":"we want the square root of x+2, like this."},{"Start":"01:24.660 ","End":"01:26.630","Text":"Let me erase what\u0027s irrelevant."},{"Start":"01:26.630 ","End":"01:33.049","Text":"I remember that when we replace x by x+a number,"},{"Start":"01:33.049 ","End":"01:36.935","Text":"it has the effect of moving it to the left."},{"Start":"01:36.935 ","End":"01:41.720","Text":"The +2 means that I need to take the graph part,"},{"Start":"01:41.720 ","End":"01:44.960","Text":"this part and move it 2 units to the left."},{"Start":"01:44.960 ","End":"01:47.555","Text":"I end up getting something like this,"},{"Start":"01:47.555 ","End":"01:49.130","Text":"instead of at 0,"},{"Start":"01:49.130 ","End":"01:50.965","Text":"it starts at -2."},{"Start":"01:50.965 ","End":"01:53.255","Text":"Otherwise looks pretty much the same."},{"Start":"01:53.255 ","End":"01:54.859","Text":"Next example."},{"Start":"01:54.859 ","End":"01:57.950","Text":"Let\u0027s erase what we don\u0027t need."},{"Start":"01:57.950 ","End":"01:59.780","Text":"We don\u0027t need this."},{"Start":"01:59.780 ","End":"02:03.335","Text":"We\u0027re going to have next is number 3."},{"Start":"02:03.335 ","End":"02:06.275","Text":"What I want is to sketch"},{"Start":"02:06.275 ","End":"02:15.680","Text":"y=x-3^2+4."},{"Start":"02:15.680 ","End":"02:19.025","Text":"First, let me erase all this stuff that I don\u0027t need,"},{"Start":"02:19.025 ","End":"02:21.184","Text":"and now let us look at this function."},{"Start":"02:21.184 ","End":"02:25.010","Text":"Well, it looks like it\u0027s a variation on y=x^2,"},{"Start":"02:25.010 ","End":"02:27.050","Text":"but there\u0027s a lot of stuff going on here."},{"Start":"02:27.050 ","End":"02:28.910","Text":"There\u0027s a -3 and a+4,"},{"Start":"02:28.910 ","End":"02:31.070","Text":"so let\u0027s take it 1 step at a time."},{"Start":"02:31.070 ","End":"02:36.239","Text":"Let\u0027s first of all, start with y=x^2."},{"Start":"02:37.130 ","End":"02:43.465","Text":"Here we have y=x^2 sketched and we build this thing up in steps."},{"Start":"02:43.465 ","End":"02:50.715","Text":"Next step is to go for y=x-3^2 and forget about the 4 for a moment."},{"Start":"02:50.715 ","End":"02:55.735","Text":"x-3^2, we replace the x by x-3."},{"Start":"02:55.735 ","End":"03:00.115","Text":"When we replace x by x minus a number,"},{"Start":"03:00.115 ","End":"03:04.945","Text":"it means that we move the function to the right by that number."},{"Start":"03:04.945 ","End":"03:09.835","Text":"If we take this parabola and move it 3 units to the right,"},{"Start":"03:09.835 ","End":"03:11.260","Text":"we get something like this,"},{"Start":"03:11.260 ","End":"03:13.400","Text":"the blue graph,"},{"Start":"03:13.400 ","End":"03:17.430","Text":"and this is y=x-3^2."},{"Start":"03:17.430 ","End":"03:21.875","Text":"Finally, we\u0027ll take the +4 into consideration,"},{"Start":"03:21.875 ","End":"03:25.100","Text":"which means that we take this function and if we add 4,"},{"Start":"03:25.100 ","End":"03:27.095","Text":"it means that it\u0027s upwards."},{"Start":"03:27.095 ","End":"03:32.145","Text":"What we want now is to go upwards by 4 units."},{"Start":"03:32.145 ","End":"03:37.530","Text":"We end up like something in the red by just taking this blue 1 and raising it up,"},{"Start":"03:37.530 ","End":"03:41.690","Text":"so the red 1 is the sketch of what we want."},{"Start":"03:41.690 ","End":"03:49.485","Text":"Example 4 is to sketch y=x+2^3-1."},{"Start":"03:49.485 ","End":"03:52.130","Text":"If we look at our stock of basic functions,"},{"Start":"03:52.130 ","End":"03:57.710","Text":"we see that x^3 is the 1 that we need because this is basically something cubed,"},{"Start":"03:57.710 ","End":"04:02.555","Text":"but the x has got some constant added to it and we have a constant at the end."},{"Start":"04:02.555 ","End":"04:06.740","Text":"Let\u0027s first of all sketch y=x^3."},{"Start":"04:06.740 ","End":"04:10.010","Text":"This is roughly something like this."},{"Start":"04:10.010 ","End":"04:17.070","Text":"Next step is to notice that if we just take this part here,"},{"Start":"04:17.180 ","End":"04:20.540","Text":"that\u0027s very much like y=x^3,"},{"Start":"04:20.540 ","End":"04:23.635","Text":"except that x is replaced by x+2."},{"Start":"04:23.635 ","End":"04:31.280","Text":"What we have to do for this 1 is to shift in this direction by 2 units."},{"Start":"04:31.280 ","End":"04:37.115","Text":"We get something like this where you notice that the origin has shifted to the point -2,"},{"Start":"04:37.115 ","End":"04:39.800","Text":"0 after we move the whole thing 2 to the left."},{"Start":"04:39.800 ","End":"04:45.245","Text":"The next thing we have to do is to take into account this -1 here,"},{"Start":"04:45.245 ","End":"04:47.120","Text":"which means we need to take"},{"Start":"04:47.120 ","End":"04:51.320","Text":"this blue graph and shift it 1 down and we get something like that."},{"Start":"04:51.320 ","End":"04:53.900","Text":"To summarize, we went from here to here by"},{"Start":"04:53.900 ","End":"04:57.095","Text":"shifting the graph 2 to the left and we got the blue 1,"},{"Start":"04:57.095 ","End":"05:00.260","Text":"and then we went from here to here by shifting the blue 1,"},{"Start":"05:00.260 ","End":"05:02.725","Text":"1 unit down and got the red 1,"},{"Start":"05:02.725 ","End":"05:05.265","Text":"and there we are."},{"Start":"05:05.265 ","End":"05:07.890","Text":"In example number 5,"},{"Start":"05:07.890 ","End":"05:12.685","Text":"we need to sketch y=1 over x-4."},{"Start":"05:12.685 ","End":"05:15.020","Text":"If we look into our stock of basic functions,"},{"Start":"05:15.020 ","End":"05:17.915","Text":"we see that the closest thing is 1 over x,"},{"Start":"05:17.915 ","End":"05:20.670","Text":"so let\u0027s sketch that first."},{"Start":"05:20.800 ","End":"05:23.495","Text":"It looks something like this."},{"Start":"05:23.495 ","End":"05:25.280","Text":"But it wasn\u0027t 1 over x,"},{"Start":"05:25.280 ","End":"05:27.200","Text":"it was 1 over x-4."},{"Start":"05:27.200 ","End":"05:30.125","Text":"If we replace x by x-4,"},{"Start":"05:30.125 ","End":"05:34.684","Text":"it has the effect of moving the graph 4 units to the right,"},{"Start":"05:34.684 ","End":"05:37.130","Text":"not to the left as some people get confused."},{"Start":"05:37.130 ","End":"05:41.285","Text":"To help sketch the graph when it\u0027s moved 4 units to the right,"},{"Start":"05:41.285 ","End":"05:42.890","Text":"I make use of the axes."},{"Start":"05:42.890 ","End":"05:47.051","Text":"I noticed that the graph tends to pull towards,"},{"Start":"05:47.051 ","End":"05:48.920","Text":"sticks to the axis,"},{"Start":"05:48.920 ","End":"05:51.653","Text":"and especially the y-axis,"},{"Start":"05:51.653 ","End":"05:54.230","Text":"this part gets closer to the y-axis,"},{"Start":"05:54.230 ","End":"05:56.765","Text":"this part gets closer to the y-axis."},{"Start":"05:56.765 ","End":"06:00.200","Text":"If I take the y-axis and shift it also 4 units to"},{"Start":"06:00.200 ","End":"06:04.100","Text":"the right and get an imaginary line through x=4,"},{"Start":"06:04.100 ","End":"06:08.625","Text":"I can get a dotted imaginary axis,"},{"Start":"06:08.625 ","End":"06:11.095","Text":"and that helps me with the sketch."},{"Start":"06:11.095 ","End":"06:14.795","Text":"I\u0027ll get something like this and we\u0027re done."},{"Start":"06:14.795 ","End":"06:17.585","Text":"Number 6 is our last example."},{"Start":"06:17.585 ","End":"06:23.015","Text":"We have to sketch y=1 over x+4+2."},{"Start":"06:23.015 ","End":"06:27.440","Text":"This is so similar to our previous example that I\u0027m not starting from scratch,"},{"Start":"06:27.440 ","End":"06:31.014","Text":"I\u0027m starting right off with 1 over x."},{"Start":"06:31.014 ","End":"06:38.599","Text":"Whereas in the previous 1 we had x-4 and we shifted 4 units to the right."},{"Start":"06:38.599 ","End":"06:41.555","Text":"In this 1, because it\u0027s x+4,"},{"Start":"06:41.555 ","End":"06:47.675","Text":"the+means that we have to take this 1 over x and shift it 4 units to the left."},{"Start":"06:47.675 ","End":"06:49.300","Text":"We\u0027re just ignoring the 2 for the moment,"},{"Start":"06:49.300 ","End":"06:53.480","Text":"so we\u0027re just going to look at y=1 over x+4."},{"Start":"06:53.480 ","End":"06:55.310","Text":"We\u0027ll get something like this."},{"Start":"06:55.310 ","End":"06:59.495","Text":"Now we have to take into account that we have a+2 here."},{"Start":"06:59.495 ","End":"07:02.735","Text":"We\u0027ll use the same method as we did before,"},{"Start":"07:02.735 ","End":"07:09.650","Text":"when we used this imaginary vertical guideline by moving the y-axis 4 units to the left."},{"Start":"07:09.650 ","End":"07:17.985","Text":"This time we\u0027ll take the x-axis and move it 2 units up, like this."},{"Start":"07:17.985 ","End":"07:25.175","Text":"Now we can use the blue and the red as guidelines to draw in the new function."},{"Start":"07:25.175 ","End":"07:27.510","Text":"We get something like this."},{"Start":"07:27.560 ","End":"07:31.545","Text":"Now this red 1 will be what we wanted,"},{"Start":"07:31.545 ","End":"07:41.230","Text":"y=1 over x+4+2, and we\u0027re done."}],"ID":5130},{"Watched":false,"Name":"Reflection about the X-axis","Duration":"5m 56s","ChapterTopicVideoID":5153,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5153.jpeg","UploadDate":"2020-09-30T13:11:30.0870000","DurationForVideoObject":"PT5M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.045","Text":"In this clip, I\u0027m going to talk about how to reflect a function in the x-axis."},{"Start":"00:06.045 ","End":"00:12.480","Text":"I\u0027m going to start with my usual example of a function y equals x^2,"},{"Start":"00:12.480 ","End":"00:14.085","Text":"and I\u0027ve already sketched it."},{"Start":"00:14.085 ","End":"00:18.090","Text":"What do I mean by reflecting it in the x-axis?"},{"Start":"00:18.090 ","End":"00:21.300","Text":"What I mean is imagining that the x-axis is a mirror,"},{"Start":"00:21.300 ","End":"00:23.865","Text":"and I want the mirror image of this function"},{"Start":"00:23.865 ","End":"00:27.700","Text":"that will be somewhere down here in the other direction."},{"Start":"00:27.980 ","End":"00:31.335","Text":"I could just give you a very quick rough sketch,"},{"Start":"00:31.335 ","End":"00:33.650","Text":"but I\u0027d like to be a little bit more precise."},{"Start":"00:33.650 ","End":"00:37.620","Text":"I happen to draw this by taking a few sample points."},{"Start":"00:37.620 ","End":"00:40.570","Text":"This is 1,1,"},{"Start":"00:40.570 ","End":"00:43.530","Text":"this is 2,4, 0,0 and so on."},{"Start":"00:43.530 ","End":"00:48.515","Text":"What I would begin by doing would be to draw the mirror image of these points."},{"Start":"00:48.515 ","End":"00:50.090","Text":"That would be here."},{"Start":"00:50.090 ","End":"00:51.620","Text":"This would be a mirror image through here."},{"Start":"00:51.620 ","End":"00:53.480","Text":"This point if I reflected it,"},{"Start":"00:53.480 ","End":"00:56.380","Text":"would come out to be here."},{"Start":"00:56.380 ","End":"01:00.950","Text":"This is -1,1 and this would come out here,"},{"Start":"01:00.950 ","End":"01:02.390","Text":"-1, -1,"},{"Start":"01:02.390 ","End":"01:05.150","Text":"and -2, -4."},{"Start":"01:05.150 ","End":"01:08.770","Text":"Then I draw a line through this."},{"Start":"01:08.770 ","End":"01:14.250","Text":"I\u0027d get something like this green graph here."},{"Start":"01:14.500 ","End":"01:18.340","Text":"Now, what\u0027s the formula of this green graph?"},{"Start":"01:18.340 ","End":"01:19.760","Text":"Well if you notice,"},{"Start":"01:19.760 ","End":"01:21.500","Text":"anytime I take some value,"},{"Start":"01:21.500 ","End":"01:26.680","Text":"let\u0027s take this 1 for example where x was 2 and y was 4."},{"Start":"01:26.680 ","End":"01:30.980","Text":"When I reflect the 4 in the x-axis it becomes -4."},{"Start":"01:30.980 ","End":"01:32.270","Text":"In fact, any number,"},{"Start":"01:32.270 ","End":"01:33.860","Text":"any value of y I take,"},{"Start":"01:33.860 ","End":"01:35.315","Text":"if I negate it,"},{"Start":"01:35.315 ","End":"01:38.360","Text":"takes it minus or if it\u0027s minus then take the plus,"},{"Start":"01:38.360 ","End":"01:41.750","Text":"and I get the mirror image in the x-axis."},{"Start":"01:41.750 ","End":"01:45.290","Text":"In other words, I have to replace the y by -y."},{"Start":"01:45.290 ","End":"01:48.755","Text":"Instead of x^2 I get -x^2."},{"Start":"01:48.755 ","End":"01:52.820","Text":"This is y equals -x^2."},{"Start":"01:52.820 ","End":"01:56.180","Text":"Again, for example, if I took x equals 2,"},{"Start":"01:56.180 ","End":"01:59.510","Text":"x^2 is 4 which gives me the point 2,4,"},{"Start":"01:59.510 ","End":"02:04.150","Text":"-x^2 is -4 which gives me the point 2, -4."},{"Start":"02:04.150 ","End":"02:08.420","Text":"That\u0027s all there is to reflection except that it works for any function,"},{"Start":"02:08.420 ","End":"02:10.295","Text":"not just for x^2."},{"Start":"02:10.295 ","End":"02:15.570","Text":"If I replace x^2 by a general f(x), like this,"},{"Start":"02:15.570 ","End":"02:16.780","Text":"y equals f(x),"},{"Start":"02:16.780 ","End":"02:20.915","Text":"then the mirror image is simply y=-f(x)."},{"Start":"02:20.915 ","End":"02:23.660","Text":"I will just do some examples now."},{"Start":"02:23.660 ","End":"02:26.360","Text":"Here\u0027s our first example."},{"Start":"02:26.360 ","End":"02:32.450","Text":"We have to sketch the graph of y equals minus the square root of x."},{"Start":"02:32.450 ","End":"02:34.280","Text":"Now, according to what we learned,"},{"Start":"02:34.280 ","End":"02:36.605","Text":"if we take as a basic function,"},{"Start":"02:36.605 ","End":"02:39.050","Text":"the square root of x,"},{"Start":"02:39.050 ","End":"02:42.005","Text":"we see we just have a minus stuck in front of it."},{"Start":"02:42.005 ","End":"02:44.420","Text":"When we have a minus stuck in front of something,"},{"Start":"02:44.420 ","End":"02:47.225","Text":"it means that we reflect it in the x-axis."},{"Start":"02:47.225 ","End":"02:49.445","Text":"First let\u0027s draw the square root of x,"},{"Start":"02:49.445 ","End":"02:52.685","Text":"and this looks something as follows."},{"Start":"02:52.685 ","End":"02:59.930","Text":"I helped myself by using some points that I plotted like 0,0, 1,1, 4,2."},{"Start":"02:59.930 ","End":"03:03.125","Text":"This 1 is reflected onto itself."},{"Start":"03:03.125 ","End":"03:05.060","Text":"This 1 goes over here,"},{"Start":"03:05.060 ","End":"03:07.370","Text":"instead of being plus 1 it goes to -1."},{"Start":"03:07.370 ","End":"03:11.195","Text":"This 1 at 4, instead of going to plus 2 goes to -2."},{"Start":"03:11.195 ","End":"03:15.170","Text":"Then I draw a line and I end up with this green curve,"},{"Start":"03:15.170 ","End":"03:19.820","Text":"and this will be y equals minus the square root"},{"Start":"03:19.820 ","End":"03:25.150","Text":"of x. I\u0027ve taken care of this minus now by reflecting it in, that\u0027s all."},{"Start":"03:25.150 ","End":"03:29.405","Text":"Number 2 is a slightly more complicated example."},{"Start":"03:29.405 ","End":"03:33.710","Text":"In it we have both horizontal shift,"},{"Start":"03:33.710 ","End":"03:38.195","Text":"the vertical shift, and the reflection in the x-axis all thrown in together."},{"Start":"03:38.195 ","End":"03:39.440","Text":"Let\u0027s look at it first."},{"Start":"03:39.440 ","End":"03:44.240","Text":"y equals minus, x-4^2 plus 5."},{"Start":"03:44.240 ","End":"03:47.720","Text":"We examine our stock of basic functions and easy to see"},{"Start":"03:47.720 ","End":"03:50.940","Text":"that this is based on the x^2 function."},{"Start":"03:50.940 ","End":"03:55.165","Text":"Let\u0027s start with a sketch of y=x^2."},{"Start":"03:55.165 ","End":"04:00.005","Text":"I\u0027d like to plot a few grid points first and then draw the curve through it."},{"Start":"04:00.005 ","End":"04:06.710","Text":"Anyway, this is a favorite function, y=x^2."},{"Start":"04:06.710 ","End":"04:14.405","Text":"What we\u0027ll do now is we\u0027ll just set up x squared go 1 step further and take (x-4)^2."},{"Start":"04:14.405 ","End":"04:23.565","Text":"We immediately recognize that the -4 causes it to shift 4 units to the right."},{"Start":"04:23.565 ","End":"04:27.650","Text":"Everyone gets confused with a minus thinking it\u0027s the left and there I did it too."},{"Start":"04:27.650 ","End":"04:32.720","Text":"Now, this is replacing x by x-4 causes a shift of 4 to the right."},{"Start":"04:32.720 ","End":"04:38.240","Text":"Yes, 4 in this direction."},{"Start":"04:38.240 ","End":"04:41.630","Text":"What I\u0027m going to do is take these help grid points"},{"Start":"04:41.630 ","End":"04:43.970","Text":"and draw them and draw a curve through it,"},{"Start":"04:43.970 ","End":"04:47.490","Text":"and I should end up with something like this."},{"Start":"04:47.620 ","End":"04:53.645","Text":"Now we come to the relevant part for this section,"},{"Start":"04:53.645 ","End":"04:56.240","Text":"that\u0027s this minus here."},{"Start":"04:56.240 ","End":"05:00.500","Text":"Now the minus is going to take this function that we just"},{"Start":"05:00.500 ","End":"05:04.550","Text":"sketched and reflect it about the x-axis."},{"Start":"05:04.550 ","End":"05:06.860","Text":"We\u0027ll end up getting something like this."},{"Start":"05:06.860 ","End":"05:09.680","Text":"This is what we get when we take"},{"Start":"05:09.680 ","End":"05:14.355","Text":"the blue and reflect it with the minus and we get the orange 1."},{"Start":"05:14.355 ","End":"05:18.280","Text":"Now there\u0027s still 1 more thing that we have to do,"},{"Start":"05:18.280 ","End":"05:23.330","Text":"and that thing is to add the plus 5 that will complete the picture."},{"Start":"05:23.330 ","End":"05:27.590","Text":"As you know, when we take a function of x and we add 5,"},{"Start":"05:27.590 ","End":"05:31.040","Text":"has the effect of raising the whole thing 5 upwards."},{"Start":"05:31.040 ","End":"05:34.520","Text":"I\u0027m indicating the up 5 with a red and I\u0027ll draw it in red."},{"Start":"05:34.520 ","End":"05:37.910","Text":"I didn\u0027t know how to indicate the reflection about the x-axis."},{"Start":"05:37.910 ","End":"05:40.175","Text":"I thought we thought maybe a little arrow like that."},{"Start":"05:40.175 ","End":"05:41.630","Text":"Just to summarize so far,"},{"Start":"05:41.630 ","End":"05:43.790","Text":"we went 4 to the right,"},{"Start":"05:43.790 ","End":"05:47.015","Text":"then we reflected, and now we\u0027re going 5 upwards."},{"Start":"05:47.015 ","End":"05:49.295","Text":"We\u0027ll get something like this."},{"Start":"05:49.295 ","End":"05:52.070","Text":"This is our final answer."},{"Start":"05:52.070 ","End":"05:56.850","Text":"The red 1 is the 1 that we wanted onto the next."}],"ID":5131},{"Watched":false,"Name":"Reflection about the Y-axis","Duration":"3m 50s","ChapterTopicVideoID":5154,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5154.jpeg","UploadDate":"2020-09-30T13:12:38.1530000","DurationForVideoObject":"PT3M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.670","Text":"In this section, we\u0027re going to study about reflection of a function in the y-axis."},{"Start":"00:05.670 ","End":"00:09.240","Text":"This comes in contrast to the previous section where we"},{"Start":"00:09.240 ","End":"00:13.770","Text":"learnt about reflection of a function in the x-axis."},{"Start":"00:13.770 ","End":"00:20.470","Text":"I\u0027d like to remind you of what we had when we reflected the function in the x-axis."},{"Start":"00:20.750 ","End":"00:24.690","Text":"I kept a sketch in 1 of the exercises."},{"Start":"00:24.690 ","End":"00:28.470","Text":"Here\u0027s an illustration of the x-axis just as a reminder."},{"Start":"00:28.470 ","End":"00:33.659","Text":"We took each point on the graph and found its mirror image through the x-axis,"},{"Start":"00:33.659 ","End":"00:35.940","Text":"which means that it\u0027s the same distance away,"},{"Start":"00:35.940 ","End":"00:37.530","Text":"but on the other side, for example,"},{"Start":"00:37.530 ","End":"00:38.880","Text":"if this was the 0.4,"},{"Start":"00:38.880 ","End":"00:42.040","Text":"2 this was the 0.4, minus 2."},{"Start":"00:42.040 ","End":"00:46.010","Text":"Very similar thing happens in the case of the y-axis."},{"Start":"00:46.010 ","End":"00:47.330","Text":"That\u0027s return there."},{"Start":"00:47.330 ","End":"00:51.230","Text":"In this case, if we reflect in the y-axis in the similar way,"},{"Start":"00:51.230 ","End":"00:53.855","Text":"we\u0027ll get the following picture."},{"Start":"00:53.855 ","End":"00:58.415","Text":"Here\u0027s what it looks like as you see the mirror image."},{"Start":"00:58.415 ","End":"01:02.330","Text":"I\u0027ll give you its formula and then we\u0027ll explain."},{"Start":"01:02.330 ","End":"01:08.584","Text":"The formula is Y equals the square root of minus X."},{"Start":"01:08.584 ","End":"01:11.465","Text":"Let\u0027s see if it works in this case."},{"Start":"01:11.465 ","End":"01:14.455","Text":"For example, if X is minus 4,"},{"Start":"01:14.455 ","End":"01:17.595","Text":"then minus X is plus 4."},{"Start":"01:17.595 ","End":"01:19.575","Text":"The square root of 4 is 2,"},{"Start":"01:19.575 ","End":"01:22.515","Text":"and this is the 0.42 that\u0027s fine."},{"Start":"01:22.515 ","End":"01:24.900","Text":"If we take X to be minus 1,"},{"Start":"01:24.900 ","End":"01:26.835","Text":"minus X is plus 1,"},{"Start":"01:26.835 ","End":"01:28.080","Text":"the square root is 1."},{"Start":"01:28.080 ","End":"01:29.935","Text":"We see it works."},{"Start":"01:29.935 ","End":"01:33.175","Text":"Now how do we generalize this?"},{"Start":"01:33.175 ","End":"01:37.760","Text":"Let\u0027s compare this also to the reflection in the x-axis."},{"Start":"01:37.760 ","End":"01:40.010","Text":"In the case of reflection in the x-axis,"},{"Start":"01:40.010 ","End":"01:43.925","Text":"we just put minus in front of the whole function."},{"Start":"01:43.925 ","End":"01:47.765","Text":"Here we put minus only in front of the X."},{"Start":"01:47.765 ","End":"01:51.245","Text":"In general, if the function wasn\u0027t the square root of X,"},{"Start":"01:51.245 ","End":"01:53.210","Text":"but any f(x),"},{"Start":"01:53.210 ","End":"01:56.105","Text":"what we would get would be the following."},{"Start":"01:56.105 ","End":"01:59.600","Text":"Instead of f(x), we have f(-x),"},{"Start":"01:59.600 ","End":"02:05.105","Text":"just replace x by minus x and that\u0027s all there is to it."},{"Start":"02:05.105 ","End":"02:08.509","Text":"Now we come to the example section."},{"Start":"02:08.509 ","End":"02:13.190","Text":"The first example is to sketch the function Y"},{"Start":"02:13.190 ","End":"02:17.735","Text":"equals minus the square root of minus x plus 3."},{"Start":"02:17.735 ","End":"02:21.650","Text":"Before I get started, I\u0027d just like to make a general remark that often,"},{"Start":"02:21.650 ","End":"02:24.470","Text":"you don\u0027t actually need to do a reflection in the y-axis."},{"Start":"02:24.470 ","End":"02:28.009","Text":"If you\u0027re lucky, the function might be symmetrical about the y-axis."},{"Start":"02:28.009 ","End":"02:30.500","Text":"For example, when Y equals x^2,"},{"Start":"02:30.500 ","End":"02:31.550","Text":"if you reflect it,"},{"Start":"02:31.550 ","End":"02:33.080","Text":"you just get itself."},{"Start":"02:33.080 ","End":"02:36.125","Text":"That\u0027s true in general for even functions."},{"Start":"02:36.125 ","End":"02:42.590","Text":"In this example, we have to identify a basic function that we\u0027re going to start with."},{"Start":"02:42.590 ","End":"02:46.455","Text":"That\u0027s clearly square root of X is what we\u0027re going to choose."},{"Start":"02:46.455 ","End":"02:49.020","Text":"Only here we have the square root of minus X."},{"Start":"02:49.020 ","End":"02:51.290","Text":"Now, remember that in the theory section"},{"Start":"02:51.290 ","End":"02:53.600","Text":"just a little bit before this, we already did that."},{"Start":"02:53.600 ","End":"02:56.075","Text":"Why don\u0027t I just copy it from there?"},{"Start":"02:56.075 ","End":"02:59.450","Text":"Next thing to do is to take care of this minus which"},{"Start":"02:59.450 ","End":"03:03.120","Text":"comes before and we ignore the rest meanwhile."},{"Start":"03:03.120 ","End":"03:05.450","Text":"We\u0027re going to build it up bit by bit."},{"Start":"03:05.450 ","End":"03:08.720","Text":"This minus, if we remember the section on"},{"Start":"03:08.720 ","End":"03:13.610","Text":"the reflection in the x-axis means that we reflect"},{"Start":"03:13.610 ","End":"03:22.410","Text":"this square root of minus x in the x-axis and this is what we get the graphing orange."},{"Start":"03:22.410 ","End":"03:27.410","Text":"Finally, we have to take care of the plus 3 here"},{"Start":"03:27.410 ","End":"03:32.900","Text":"and the plus 3 has the effect of raising it 3 units up."},{"Start":"03:32.900 ","End":"03:36.425","Text":"We need to take the orange bit,"},{"Start":"03:36.425 ","End":"03:38.945","Text":"and make it go up 3."},{"Start":"03:38.945 ","End":"03:42.200","Text":"We\u0027ll end up with something like this."},{"Start":"03:42.200 ","End":"03:45.560","Text":"This function in red is what we\u0027re looking for,"},{"Start":"03:45.560 ","End":"03:48.425","Text":"minus the square root of minus x plus 3."},{"Start":"03:48.425 ","End":"03:51.180","Text":"We\u0027re done with this exercise."}],"ID":5132},{"Watched":false,"Name":"Exercise 1","Duration":"1m 50s","ChapterTopicVideoID":5140,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5140.jpeg","UploadDate":"2016-02-29T13:50:38.7870000","DurationForVideoObject":"PT1M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.380 ","End":"00:04.080","Text":"In this exercise, we\u0027re given a graph of a function,"},{"Start":"00:04.080 ","End":"00:07.860","Text":"and we also are given two other graphs of the same function"},{"Start":"00:07.860 ","End":"00:11.745","Text":"shifted to new positions and we have to find the new equations."},{"Start":"00:11.745 ","End":"00:14.046","Text":"Specifically, the original one,"},{"Start":"00:14.046 ","End":"00:15.765","Text":"that\u0027s the one in black,"},{"Start":"00:15.765 ","End":"00:20.560","Text":"this one is y= -2x^2."},{"Start":"00:20.930 ","End":"00:23.220","Text":"This one in green,"},{"Start":"00:23.220 ","End":"00:29.805","Text":"we can see that it\u0027s simply the original one shifted to the right by"},{"Start":"00:29.805 ","End":"00:39.755","Text":"4 units and the red one is shifted to the left by 7 units."},{"Start":"00:39.755 ","End":"00:42.244","Text":"Now, how do we shift?"},{"Start":"00:42.244 ","End":"00:45.070","Text":"When we shift to the right,"},{"Start":"00:45.070 ","End":"00:48.650","Text":"we have to replace x with x-4."},{"Start":"00:48.650 ","End":"00:53.060","Text":"So in other words, here we have to replace x with x-4."},{"Start":"00:53.060 ","End":"00:59.123","Text":"When it\u0027s left, we\u0027ll replace x with x plus,"},{"Start":"00:59.123 ","End":"01:01.775","Text":"wherever the shift is, in this case, 7."},{"Start":"01:01.775 ","End":"01:04.910","Text":"We start with the original, that\u0027s the one in black,"},{"Start":"01:04.910 ","End":"01:11.935","Text":"y= -2x^2 and then to compute the green one,"},{"Start":"01:11.935 ","End":"01:14.355","Text":"and I\u0027ll switch to green."},{"Start":"01:14.355 ","End":"01:21.795","Text":"y= -2 instead of x, (x-4)^2."},{"Start":"01:21.795 ","End":"01:24.225","Text":"We could expand the brackets,"},{"Start":"01:24.225 ","End":"01:27.760","Text":"but this should be good enough."},{"Start":"01:28.810 ","End":"01:31.640","Text":"Next, the red one."},{"Start":"01:31.640 ","End":"01:33.815","Text":"The red one, I\u0027ll just switch to red."},{"Start":"01:33.815 ","End":"01:38.240","Text":"We get that y= -2 and this time,"},{"Start":"01:38.240 ","End":"01:44.320","Text":"we replace x with x+7, (x+7)^2."},{"Start":"01:44.320 ","End":"01:46.420","Text":"Again, I won\u0027t bother to expand the brackets,"},{"Start":"01:46.420 ","End":"01:50.040","Text":"we\u0027ll just leave the answer like this and we\u0027re done."}],"ID":5133},{"Watched":false,"Name":"Exercise 2","Duration":"1m 33s","ChapterTopicVideoID":5141,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5141.jpeg","UploadDate":"2016-02-29T13:50:47.0670000","DurationForVideoObject":"PT1M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.470 ","End":"00:05.685","Text":"In this exercise, we\u0027re given the graph of a function,"},{"Start":"00:05.685 ","End":"00:08.700","Text":"and we\u0027re also given 2 new graphs,"},{"Start":"00:08.700 ","End":"00:11.630","Text":"which is the original graph shifted into new positions,"},{"Start":"00:11.630 ","End":"00:14.565","Text":"and we have to write the equation for those new graphs."},{"Start":"00:14.565 ","End":"00:17.160","Text":"The original one is the one in black,"},{"Start":"00:17.160 ","End":"00:20.100","Text":"and that\u0027s the one that\u0027s given by this equation here."},{"Start":"00:20.100 ","End":"00:25.620","Text":"This is the 1 which is y equals minus 2x squared."},{"Start":"00:25.620 ","End":"00:35.265","Text":"The red one is gotten by the black one by shifting it up by 2 units,"},{"Start":"00:35.265 ","End":"00:41.105","Text":"whereas the olive brown one is the same original graph,"},{"Start":"00:41.105 ","End":"00:44.345","Text":"but shifted down 5 units."},{"Start":"00:44.345 ","End":"00:47.705","Text":"Now, when we shift something up,"},{"Start":"00:47.705 ","End":"00:54.380","Text":"we simply take the equation of the function and add plus 2 to it at the end,"},{"Start":"00:54.380 ","End":"00:57.230","Text":"and when we move something down,"},{"Start":"00:57.230 ","End":"01:02.615","Text":"we simply subtract 5 from the original function."},{"Start":"01:02.615 ","End":"01:10.745","Text":"In other words, if the original function is y equals minus 2x squared,"},{"Start":"01:10.745 ","End":"01:19.970","Text":"then the red one would be y equals minus 2x squared plus 2."},{"Start":"01:19.970 ","End":"01:25.630","Text":"The olive brown is"},{"Start":"01:25.630 ","End":"01:30.795","Text":"y equals minus 2x squared minus 5."},{"Start":"01:30.795 ","End":"01:33.280","Text":"That\u0027s all there is to it, we\u0027re done."}],"ID":5134},{"Watched":false,"Name":"Exercise 3","Duration":"1m 31s","ChapterTopicVideoID":5142,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5142.jpeg","UploadDate":"2016-02-29T13:50:55.7430000","DurationForVideoObject":"PT1M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:07.275","Text":"In this exercise, we\u0027re given 4 equations and 4 graphs."},{"Start":"00:07.275 ","End":"00:11.860","Text":"We have to match which graph belongs to which equation."},{"Start":"00:12.860 ","End":"00:15.750","Text":"Start with the first one."},{"Start":"00:15.750 ","End":"00:19.290","Text":"It looks like it\u0027s based on y=x^2,"},{"Start":"00:19.290 ","End":"00:22.775","Text":"but x is replaced by x-4 and there is a 2 on the end."},{"Start":"00:22.775 ","End":"00:26.935","Text":"What this means is that the graph of"},{"Start":"00:26.935 ","End":"00:31.605","Text":"x^2 is moved 4 units to the right and 2 units upwards."},{"Start":"00:31.605 ","End":"00:34.290","Text":"Now, 4 to the right and 2 upwards,"},{"Start":"00:34.290 ","End":"00:39.947","Text":"that\u0027s going to be the blue one. I\u0027ll just mark this."},{"Start":"00:39.947 ","End":"00:44.580","Text":"This belongs to number 1, that\u0027s blue one."},{"Start":"00:44.580 ","End":"00:47.595","Text":"In two, similarly,"},{"Start":"00:47.595 ","End":"00:50.985","Text":"we have 1 to the right and 3 down,"},{"Start":"00:50.985 ","End":"00:52.590","Text":"1 to the right and 3 down,"},{"Start":"00:52.590 ","End":"00:55.335","Text":"a quick inspection shows that it\u0027s this green one,"},{"Start":"00:55.335 ","End":"00:58.335","Text":"so that would be number 2."},{"Start":"00:58.335 ","End":"01:06.315","Text":"In three, the +5 means it\u0027s shifted to the left 5 units and 2 upwards,"},{"Start":"01:06.315 ","End":"01:10.745","Text":"so left 5 and up 2 is this one,"},{"Start":"01:10.745 ","End":"01:13.810","Text":"that would be number 3."},{"Start":"01:14.360 ","End":"01:24.675","Text":"Finally, the +4 here and the -5 means that it was shifted to the left 4 and down 5."},{"Start":"01:24.675 ","End":"01:30.970","Text":"That\u0027s got to be number 4 and we\u0027re done."}],"ID":5135},{"Watched":false,"Name":"Exercise 4","Duration":"7m 42s","ChapterTopicVideoID":5143,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5143.jpeg","UploadDate":"2016-02-29T13:51:39.4470000","DurationForVideoObject":"PT7M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"This exercise has eight parts."},{"Start":"00:02.220 ","End":"00:05.310","Text":"In each part, we\u0027re given an equation."},{"Start":"00:05.310 ","End":"00:07.980","Text":"For example, y=x^3,"},{"Start":"00:07.980 ","End":"00:12.435","Text":"and then we\u0027re given instructions on how to shift its graph."},{"Start":"00:12.435 ","End":"00:15.750","Text":"For example, move it left 2 and down 1."},{"Start":"00:15.750 ","End":"00:20.550","Text":"What we have to do is to write the equation of the new shifted graph and"},{"Start":"00:20.550 ","End":"00:25.530","Text":"then to sketch both the original and the shifted ones on the axes."},{"Start":"00:25.530 ","End":"00:26.730","Text":"Let\u0027s start."},{"Start":"00:26.730 ","End":"00:28.450","Text":"We\u0027ll start with the first one."},{"Start":"00:28.450 ","End":"00:33.365","Text":"The first one, the original is y=x^3."},{"Start":"00:33.365 ","End":"00:38.120","Text":"Now, a shift of left means replace x by x plus 2,"},{"Start":"00:38.120 ","End":"00:42.095","Text":"and down simply means that we have to subtract 1 at the end."},{"Start":"00:42.095 ","End":"00:47.070","Text":"What we get is y = x plus"},{"Start":"00:47.070 ","End":"00:52.810","Text":"2 because left is plus and down 1 is minus 1."},{"Start":"00:52.810 ","End":"00:55.910","Text":"Now, we have to just draw the graphs of each of these."},{"Start":"00:55.910 ","End":"00:57.710","Text":"Let\u0027s see, the original one,"},{"Start":"00:57.710 ","End":"01:02.370","Text":"y=x^3 is something like this,"},{"Start":"01:02.370 ","End":"01:06.790","Text":"and if we have to shift it left 2 and down 1,"},{"Start":"01:06.790 ","End":"01:09.080","Text":"it would look something like this."},{"Start":"01:09.080 ","End":"01:10.850","Text":"That\u0027s it for the first one."},{"Start":"01:10.850 ","End":"01:18.550","Text":"Next, we come to part 2, y=x^2/3."},{"Start":"01:18.550 ","End":"01:22.670","Text":"Let\u0027s first sketch the graph roughly something like this,"},{"Start":"01:22.670 ","End":"01:27.140","Text":"and now let\u0027s write the equation of the shifted graph."},{"Start":"01:27.140 ","End":"01:31.415","Text":"We have to shift it right 1 and down 3."},{"Start":"01:31.415 ","End":"01:36.045","Text":"That means that y is equal."},{"Start":"01:36.045 ","End":"01:38.830","Text":"Now, because of the shift right of 1,"},{"Start":"01:38.830 ","End":"01:41.585","Text":"we have to replace x by x minus 1."},{"Start":"01:41.585 ","End":"01:46.500","Text":"We get x minus 1 to the 2/3,"},{"Start":"01:46.500 ","End":"01:51.730","Text":"and then the down 3 means that we subtract 3."},{"Start":"01:51.730 ","End":"01:55.060","Text":"The graph should look something like this,"},{"Start":"01:55.060 ","End":"01:56.640","Text":"and that\u0027s roughly it."},{"Start":"01:56.640 ","End":"01:59.740","Text":"Next, we have 3,"},{"Start":"01:59.740 ","End":"02:02.915","Text":"y equals the square root of x."},{"Start":"02:02.915 ","End":"02:05.180","Text":"We have to move it left 1,"},{"Start":"02:05.180 ","End":"02:06.605","Text":"and to move left 1,"},{"Start":"02:06.605 ","End":"02:09.760","Text":"we simply replace x by x plus 1."},{"Start":"02:09.760 ","End":"02:17.570","Text":"We get y equals the square root of x plus 1 as the equation of the shifted graph."},{"Start":"02:17.570 ","End":"02:21.540","Text":"The original graph looks something like this roughly,"},{"Start":"02:21.540 ","End":"02:24.020","Text":"and if we move it to the left by 1,"},{"Start":"02:24.020 ","End":"02:25.835","Text":"it will look something like this."},{"Start":"02:25.835 ","End":"02:30.530","Text":"Again, roughly, and that\u0027s the graph of square root of x plus 1, and that\u0027s it."},{"Start":"02:30.530 ","End":"02:32.255","Text":"Now, part 4,"},{"Start":"02:32.255 ","End":"02:35.125","Text":"y equals minus the square root of x,"},{"Start":"02:35.125 ","End":"02:37.930","Text":"and if we shifted right 4,"},{"Start":"02:37.930 ","End":"02:42.080","Text":"it means that we replace x by x minus 4."},{"Start":"02:42.080 ","End":"02:51.080","Text":"The new graph will be y equals minus the square root of x minus 4, a new equation."},{"Start":"02:51.080 ","End":"02:52.595","Text":"Now, for the graphs."},{"Start":"02:52.595 ","End":"02:56.390","Text":"Before I draw the graph of y equals minus the square root of x,"},{"Start":"02:56.390 ","End":"03:02.090","Text":"it might help if we drew the graph of y equals the square root of x."},{"Start":"03:02.090 ","End":"03:06.785","Text":"Because the minus just makes it a reflection about the x-axis."},{"Start":"03:06.785 ","End":"03:08.420","Text":"One at a time. First,"},{"Start":"03:08.420 ","End":"03:11.045","Text":"y equals square root of x."},{"Start":"03:11.045 ","End":"03:13.220","Text":"Something like this,"},{"Start":"03:13.220 ","End":"03:15.890","Text":"and now we want to reflect it about"},{"Start":"03:15.890 ","End":"03:19.190","Text":"the x-axis to get y equals minus the square root of x,"},{"Start":"03:19.190 ","End":"03:20.930","Text":"so we get something like this."},{"Start":"03:20.930 ","End":"03:25.790","Text":"In other words, we just reflected the square root of x about the x-axis and got this,"},{"Start":"03:25.790 ","End":"03:31.065","Text":"and finally, we want to shift it to the right by 4 units,"},{"Start":"03:31.065 ","End":"03:34.035","Text":"we\u0027ll get something like this, and that\u0027s the red one."},{"Start":"03:34.035 ","End":"03:39.440","Text":"y equals minus the square root of x minus 4."},{"Start":"03:39.440 ","End":"03:41.660","Text":"Now, we come to part 5,"},{"Start":"03:41.660 ","End":"03:47.610","Text":"which is y = 2x - 7."},{"Start":"03:47.610 ","End":"03:49.200","Text":"If we move it up 4,"},{"Start":"03:49.200 ","End":"03:51.854","Text":"we just simply add 4 to the function."},{"Start":"03:51.854 ","End":"03:58.980","Text":"So we get y = 2x - 7 + 4,"},{"Start":"03:58.980 ","End":"04:00.345","Text":"or in other words,"},{"Start":"04:00.345 ","End":"04:05.305","Text":"y = 2x - 3."},{"Start":"04:05.305 ","End":"04:08.990","Text":"The first graph, it\u0027s a graph of a straight line,"},{"Start":"04:08.990 ","End":"04:10.745","Text":"it\u0027s easy enough to draw."},{"Start":"04:10.745 ","End":"04:14.630","Text":"I just computed a few points and join them with a straight line."},{"Start":"04:14.630 ","End":"04:18.650","Text":"The next one, which is the one that\u0027s shifted to the 4,"},{"Start":"04:18.650 ","End":"04:21.890","Text":"should look something like this at the red one,"},{"Start":"04:21.890 ","End":"04:25.100","Text":"which is just like the black one but moved upwards 4 units."},{"Start":"04:25.100 ","End":"04:30.530","Text":"In number 6, we can use a little trick if we see the original equation,"},{"Start":"04:30.530 ","End":"04:33.553","Text":"y=1/2(x + 1) + 2,"},{"Start":"04:33.553 ","End":"04:38.360","Text":"we see it looks very much like a shift of y=1/2(x)."},{"Start":"04:38.360 ","End":"04:43.540","Text":"In fact, if we take y=1/2(x),"},{"Start":"04:43.540 ","End":"04:49.850","Text":"we can make this into y=1/2(x +"},{"Start":"04:49.850 ","End":"04:57.645","Text":"1) + 2 if we go up 2 and left 1."},{"Start":"04:57.645 ","End":"05:03.510","Text":"Now, if we add to that down 5 and right 2,"},{"Start":"05:03.510 ","End":"05:06.345","Text":"we can combine these and say,"},{"Start":"05:06.345 ","End":"05:08.226","Text":"up 2, down 5,"},{"Start":"05:08.226 ","End":"05:11.655","Text":"that means down by 3,"},{"Start":"05:11.655 ","End":"05:16.980","Text":"and left 1 and right 2 is the same as right 1."},{"Start":"05:16.980 ","End":"05:21.155","Text":"If we apply this transformation to this equation,"},{"Start":"05:21.155 ","End":"05:22.670","Text":"we should get the answer."},{"Start":"05:22.670 ","End":"05:27.680","Text":"Yes, I forgot the equation of the final one would be to take this and"},{"Start":"05:27.680 ","End":"05:33.842","Text":"write y=1/2 of down 3,"},{"Start":"05:33.842 ","End":"05:35.563","Text":"we\u0027ll do that at the end,"},{"Start":"05:35.563 ","End":"05:38.090","Text":"that\u0027s the minus 3, right 1, first of all,"},{"Start":"05:38.090 ","End":"05:40.480","Text":"makes it x - 1,"},{"Start":"05:40.480 ","End":"05:45.390","Text":"and the down 3 means minus 3."},{"Start":"05:45.390 ","End":"05:49.925","Text":"This is the answer to the equation part of the end."},{"Start":"05:49.925 ","End":"05:54.470","Text":"Now, let\u0027s draw. First, we have y=1/2x."},{"Start":"05:54.470 ","End":"06:00.685","Text":"Next, we\u0027re going to do the original y=1/2(x + 1) + 2."},{"Start":"06:00.685 ","End":"06:06.725","Text":"What we\u0027ll do is we\u0027ll take this and we\u0027ll move it up by 2 and left by 1."},{"Start":"06:06.725 ","End":"06:10.295","Text":"So we\u0027ll get something like this in black."},{"Start":"06:10.295 ","End":"06:20.855","Text":"Finally, the last one we get by taking y=1/2x and making it move down 3 and right 1."},{"Start":"06:20.855 ","End":"06:22.640","Text":"We\u0027ll get something like this."},{"Start":"06:22.640 ","End":"06:26.170","Text":"That\u0027s the red graph and we\u0027re done."},{"Start":"06:26.170 ","End":"06:27.900","Text":"Next is part 7,"},{"Start":"06:27.900 ","End":"06:30.980","Text":"and then I save time by pre-drawing the pictures."},{"Start":"06:30.980 ","End":"06:32.690","Text":"Hope I didn\u0027t spoil the surprise."},{"Start":"06:32.690 ","End":"06:38.540","Text":"Anyway, y=1/x is the black one and it\u0027s a familiar graph."},{"Start":"06:38.540 ","End":"06:41.195","Text":"That\u0027s the usual hyperbola."},{"Start":"06:41.195 ","End":"06:44.870","Text":"If we need to shift it right by 1,"},{"Start":"06:44.870 ","End":"06:47.440","Text":"we replace x with x - 1."},{"Start":"06:47.440 ","End":"06:52.365","Text":"That part gives us 1 over x - 1."},{"Start":"06:52.365 ","End":"06:55.225","Text":"The up 3 means we add 3."},{"Start":"06:55.225 ","End":"07:00.455","Text":"Graphically, we just shifted to the right by 1 and up 3,"},{"Start":"07:00.455 ","End":"07:04.010","Text":"for example, this point here goes to this point here and so on."},{"Start":"07:04.010 ","End":"07:06.110","Text":"Just shift, right and up."},{"Start":"07:06.110 ","End":"07:07.750","Text":"That\u0027s it for part 7."},{"Start":"07:07.750 ","End":"07:13.335","Text":"Finally, we come to part 8, y=1/x^2."},{"Start":"07:13.335 ","End":"07:17.070","Text":"This has the usual shape familiar."},{"Start":"07:17.070 ","End":"07:20.040","Text":"I have sketched it here already,"},{"Start":"07:20.040 ","End":"07:24.400","Text":"and if we want to move it left by 2 and down by 3,"},{"Start":"07:24.400 ","End":"07:28.791","Text":"it becomes y=1 over,"},{"Start":"07:28.791 ","End":"07:33.165","Text":"now, the left 2 means we replace x by x plus 2."},{"Start":"07:33.165 ","End":"07:36.225","Text":"The down 3 means we subtract 3,"},{"Start":"07:36.225 ","End":"07:39.500","Text":"and if we sketch this in red,"},{"Start":"07:39.500 ","End":"07:43.050","Text":"it will look something like this, and that\u0027s it."}],"ID":5136},{"Watched":false,"Name":"Exercise 5","Duration":"1m 46s","ChapterTopicVideoID":5144,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5144.jpeg","UploadDate":"2016-02-29T13:51:48.8900000","DurationForVideoObject":"PT1M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.470 ","End":"00:05.775","Text":"In this exercise, we\u0027re given 2 equations"},{"Start":"00:05.775 ","End":"00:12.520","Text":"and 2 graphs and we have to say which graph belongs to which equation."},{"Start":"00:13.430 ","End":"00:15.570","Text":"Now, if you look at them,"},{"Start":"00:15.570 ","End":"00:23.985","Text":"they\u0027re both based on y=-x^2,"},{"Start":"00:23.985 ","End":"00:26.520","Text":"and each of them is shifted left and right,"},{"Start":"00:26.520 ","End":"00:29.385","Text":"up or down in various ways."},{"Start":"00:29.385 ","End":"00:38.770","Text":"If we draw a quick sketch of y=-x^2,"},{"Start":"00:38.780 ","End":"00:44.170","Text":"it\u0027ll look something like this."},{"Start":"00:48.040 ","End":"00:53.420","Text":"In this case, the difference is that instead of x, we have x-1."},{"Start":"00:53.420 ","End":"01:03.300","Text":"This is the same graph that\u0027s shifted to the right by 1 of this."},{"Start":"01:03.300 ","End":"01:06.960","Text":"The second 1, we have instead of x, x+2."},{"Start":"01:06.960 ","End":"01:12.285","Text":"That means it\u0027s shifted to the left too,"},{"Start":"01:12.285 ","End":"01:14.130","Text":"and because of the plus 3,"},{"Start":"01:14.130 ","End":"01:17.860","Text":"then we have up 3."},{"Start":"01:17.900 ","End":"01:23.550","Text":"Now if we look and see from the black one,"},{"Start":"01:23.550 ","End":"01:27.480","Text":"which one is left 2 and up 3 and which one is right 1."},{"Start":"01:27.480 ","End":"01:30.320","Text":"Well, clearly this one is being shifted right by 1,"},{"Start":"01:30.320 ","End":"01:35.050","Text":"so that belongs to the exercise number 1."},{"Start":"01:35.050 ","End":"01:39.885","Text":"This one is the one that\u0027s shifted left 2 and up 3."},{"Start":"01:39.885 ","End":"01:43.185","Text":"This is the one that belongs to number 2."},{"Start":"01:43.185 ","End":"01:45.760","Text":"That\u0027s all there is to it."}],"ID":5137},{"Watched":false,"Name":"Exercise 6","Duration":"2m 3s","ChapterTopicVideoID":5145,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5145.jpeg","UploadDate":"2016-02-29T13:51:59.2930000","DurationForVideoObject":"PT2M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.440 ","End":"00:07.515","Text":"In this exercise, we\u0027re given two equations here and here,"},{"Start":"00:07.515 ","End":"00:10.290","Text":"and we are given two graphs here and here."},{"Start":"00:10.290 ","End":"00:13.860","Text":"We have to say which one belongs to which,"},{"Start":"00:13.860 ","End":"00:18.252","Text":"which equation goes with which graph."},{"Start":"00:18.252 ","End":"00:23.490","Text":"Both of these look"},{"Start":"00:23.490 ","End":"00:30.210","Text":"like they\u0027re based on y equals the square root of minus x."},{"Start":"00:30.210 ","End":"00:36.675","Text":"They\u0027re similar except with some shifts to the left and right or up and down."},{"Start":"00:36.675 ","End":"00:38.940","Text":"It\u0027s easy to see this in"},{"Start":"00:38.940 ","End":"00:43.670","Text":"the first one because it\u0027s exactly the same except for the plus 2,"},{"Start":"00:43.670 ","End":"00:49.856","Text":"which means that we take y equals square root of minus x and move it up 2."},{"Start":"00:49.856 ","End":"00:52.865","Text":"Not immediately clear in the second one."},{"Start":"00:52.865 ","End":"00:54.065","Text":"But if you write"},{"Start":"00:54.065 ","End":"00:56.789","Text":"the 4 minus x"},{"Start":"00:58.120 ","End":"01:05.280","Text":"in terms of something algebraic equivalent minus x minus 4,"},{"Start":"01:05.780 ","End":"01:12.845","Text":"then we see that this comes from this by replacing x with x minus 4."},{"Start":"01:12.845 ","End":"01:15.570","Text":"First of all, which means,"},{"Start":"01:15.610 ","End":"01:25.658","Text":"right 4 and the minus 3 means down 3."},{"Start":"01:25.658 ","End":"01:34.830","Text":"The original y equals square root of x would look something like this roughly."},{"Start":"01:34.830 ","End":"01:41.800","Text":"Just to help make things easier highlight the endpoints on all 3 of them."},{"Start":"01:41.980 ","End":"01:48.310","Text":"It\u0027s quite easy to see that this one is the one which is up 2."},{"Start":"01:48.310 ","End":"01:52.320","Text":"That one is number 1."},{"Start":"01:52.490 ","End":"01:57.585","Text":"The one that\u0027s right 4 and down 3 is the lower one."},{"Start":"01:57.585 ","End":"02:03.010","Text":"This goes with number 2 and we\u0027re done."}],"ID":5138},{"Watched":false,"Name":"Exercise 7 - Parts 1-4","Duration":"6m 21s","ChapterTopicVideoID":5146,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5146.jpeg","UploadDate":"2016-02-29T13:52:30.0670000","DurationForVideoObject":"PT6M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.470","Text":"Begin with number 1,"},{"Start":"00:01.470 ","End":"00:04.710","Text":"we have y equals x squared minus 4 x plus 6,"},{"Start":"00:04.710 ","End":"00:06.525","Text":"and we need to draw its graph."},{"Start":"00:06.525 ","End":"00:11.670","Text":"What I\u0027m going to do is use the method of shifting left and right up and down,"},{"Start":"00:11.670 ","End":"00:15.240","Text":"and this one looks like it\u0027s a variation of x^2."},{"Start":"00:15.240 ","End":"00:17.775","Text":"We\u0027ll use the method of completing the square."},{"Start":"00:17.775 ","End":"00:22.350","Text":"So y equals x^2 minus 4x."},{"Start":"00:22.350 ","End":"00:24.120","Text":"Now we have to write something here,"},{"Start":"00:24.120 ","End":"00:25.800","Text":"which makes this a perfect square."},{"Start":"00:25.800 ","End":"00:27.225","Text":"Well, this is very familiar."},{"Start":"00:27.225 ","End":"00:31.485","Text":"It\u0027s x minus 2^2 is x^2 minus 4x plus 4."},{"Start":"00:31.485 ","End":"00:32.879","Text":"But there was a 6 here,"},{"Start":"00:32.879 ","End":"00:35.480","Text":"so I have to compensate by adding 2,"},{"Start":"00:35.480 ","End":"00:40.925","Text":"this equals now x minus 2^2 plus 2,"},{"Start":"00:40.925 ","End":"00:42.875","Text":"and the minus 2,"},{"Start":"00:42.875 ","End":"00:44.930","Text":"when you replace x with x minus 2,"},{"Start":"00:44.930 ","End":"00:46.535","Text":"this is a shift right."},{"Start":"00:46.535 ","End":"00:48.650","Text":"So I\u0027ll write right2,"},{"Start":"00:48.650 ","End":"00:52.505","Text":"and the plus 2 at the end means up 2,"},{"Start":"00:52.505 ","End":"00:55.295","Text":"and all this is based on the x^2 function."},{"Start":"00:55.295 ","End":"00:56.840","Text":"Let\u0027s start with the original,"},{"Start":"00:56.840 ","End":"00:59.270","Text":"we\u0027ll sketch that then we\u0027ll shift it right"},{"Start":"00:59.270 ","End":"01:01.955","Text":"2 and then we\u0027ll shift it up 2 and see where we are."},{"Start":"01:01.955 ","End":"01:04.640","Text":"First of all, y equals x^2."},{"Start":"01:04.640 ","End":"01:07.250","Text":"This is the familiar function."},{"Start":"01:07.250 ","End":"01:09.170","Text":"So I just drew a few points,"},{"Start":"01:09.170 ","End":"01:10.700","Text":"roughly sketch the line through it."},{"Start":"01:10.700 ","End":"01:11.825","Text":"This is very familiar."},{"Start":"01:11.825 ","End":"01:17.945","Text":"Next thing to do is to shift it right by 2 units,"},{"Start":"01:17.945 ","End":"01:21.050","Text":"and so we\u0027ll get something like this,"},{"Start":"01:21.050 ","End":"01:22.355","Text":"just a rough sketch."},{"Start":"01:22.355 ","End":"01:24.110","Text":"It should be the same shape as this."},{"Start":"01:24.110 ","End":"01:26.165","Text":"Just move 2 to the right."},{"Start":"01:26.165 ","End":"01:29.450","Text":"Next thing is to move it up 2,"},{"Start":"01:29.450 ","End":"01:35.810","Text":"want to see what it looks like when I move it up 2."},{"Start":"01:35.810 ","End":"01:40.100","Text":"We\u0027ll get something like this and we\u0027re done for Part 1."},{"Start":"01:40.100 ","End":"01:44.540","Text":"Part 2, we have to sketch the graph of y equals the square root of 4 minus"},{"Start":"01:44.540 ","End":"01:49.610","Text":"x. I\u0027ve written it below so we can scroll down a bit."},{"Start":"01:49.610 ","End":"01:52.670","Text":"Y equals the square root of 4 minus x."},{"Start":"01:52.670 ","End":"01:58.760","Text":"Now the idea is to find a more basic function such that this is just shifting up,"},{"Start":"01:58.760 ","End":"02:00.560","Text":"down left, right of it."},{"Start":"02:00.560 ","End":"02:07.265","Text":"A function that comes to mind is y equals the square root of minus x."},{"Start":"02:07.265 ","End":"02:09.200","Text":"This we know how to draw,"},{"Start":"02:09.200 ","End":"02:10.505","Text":"and I\u0027ll draw it in a minute."},{"Start":"02:10.505 ","End":"02:16.940","Text":"What we do here is slightly rewrite what\u0027s under the square root sign."},{"Start":"02:16.940 ","End":"02:19.670","Text":"So we can see a minus something here."},{"Start":"02:19.670 ","End":"02:21.740","Text":"I want a minus in front."},{"Start":"02:21.740 ","End":"02:26.510","Text":"Use the usual trick of putting a minus outside and reversing the order of the terms."},{"Start":"02:26.510 ","End":"02:29.555","Text":"So it\u0027s minus x minus 4,"},{"Start":"02:29.555 ","End":"02:31.655","Text":"and that way if I know the graph of this,"},{"Start":"02:31.655 ","End":"02:35.780","Text":"I\u0027ll be able to get the graph of this by shifting it 4 units to the right."},{"Start":"02:35.780 ","End":"02:38.570","Text":"So this is our basic function."},{"Start":"02:38.570 ","End":"02:40.580","Text":"This is the basic one,"},{"Start":"02:40.580 ","End":"02:45.740","Text":"and this one is gotten by shifting it right 4, and that\u0027s it."},{"Start":"02:45.740 ","End":"02:47.075","Text":"There is no up down."},{"Start":"02:47.075 ","End":"02:49.760","Text":"So the original one,"},{"Start":"02:49.760 ","End":"02:53.240","Text":"if you remember, it\u0027s just like y equals the square root of x."},{"Start":"02:53.240 ","End":"02:54.440","Text":"So because of the minus,"},{"Start":"02:54.440 ","End":"02:56.510","Text":"it\u0027s reflected to the left."},{"Start":"02:56.510 ","End":"03:03.425","Text":"In fact, it wouldn\u0027t hurt if I remind you of y equals the square root of x,"},{"Start":"03:03.425 ","End":"03:05.960","Text":"and then from here to here,"},{"Start":"03:05.960 ","End":"03:08.735","Text":"I get a reflection."},{"Start":"03:08.735 ","End":"03:12.545","Text":"So we start off with the y equals the square root of x."},{"Start":"03:12.545 ","End":"03:15.365","Text":"Here\u0027s a rough sketch, familiar function."},{"Start":"03:15.365 ","End":"03:19.760","Text":"Next, the square root of minus x reflecting about the,"},{"Start":"03:19.760 ","End":"03:21.815","Text":"I should have mentioned the y-axis."},{"Start":"03:21.815 ","End":"03:23.675","Text":"So we\u0027ll get something like this."},{"Start":"03:23.675 ","End":"03:25.085","Text":"That\u0027s the reflection,"},{"Start":"03:25.085 ","End":"03:28.010","Text":"and now we have to move it right by 4."},{"Start":"03:28.010 ","End":"03:32.969","Text":"So from here to here I\u0027ll shift it 4 to the right and we\u0027ll get something like this."},{"Start":"03:34.270 ","End":"03:38.060","Text":"We\u0027re done with Part 2."},{"Start":"03:38.060 ","End":"03:41.360","Text":"Now we want to sketch the graph in Part 3,"},{"Start":"03:41.360 ","End":"03:44.510","Text":"y equals absolute value of x minus 4."},{"Start":"03:44.510 ","End":"03:48.230","Text":"To give ourselves some room I\u0027ve written down below. Here it is."},{"Start":"03:48.230 ","End":"03:50.705","Text":"We look at it, we look for a basic function"},{"Start":"03:50.705 ","End":"03:53.450","Text":"such that this is a shift of that basic function,"},{"Start":"03:53.450 ","End":"03:58.579","Text":"and the obvious candidate is y equals absolute value of x."},{"Start":"03:58.579 ","End":"04:02.795","Text":"What we have here is x is replaced by x minus 4."},{"Start":"04:02.795 ","End":"04:08.360","Text":"Which means that if we take this and shift it right 4,"},{"Start":"04:08.360 ","End":"04:09.860","Text":"that will give us this one."},{"Start":"04:09.860 ","End":"04:15.800","Text":"Now, this is a very familiar one and its graph looks something like this."},{"Start":"04:15.800 ","End":"04:21.590","Text":"That\u0027s this one, that\u0027s y equals absolute value of x."},{"Start":"04:21.590 ","End":"04:24.140","Text":"If we shift it to the right 4,"},{"Start":"04:24.140 ","End":"04:26.900","Text":"simply take the point 4 here,"},{"Start":"04:26.900 ","End":"04:32.690","Text":"and same thing here and here, and that\u0027s it."},{"Start":"04:32.690 ","End":"04:34.790","Text":"We\u0027re done. Now for Part 4,"},{"Start":"04:34.790 ","End":"04:36.305","Text":"we have to sketch the graph."},{"Start":"04:36.305 ","End":"04:37.640","Text":"To give ourselves a bit of space,"},{"Start":"04:37.640 ","End":"04:40.316","Text":"I\u0027ve rewritten it below."},{"Start":"04:40.316 ","End":"04:43.280","Text":"Then we can scroll up a bit."},{"Start":"04:43.280 ","End":"04:47.075","Text":"This is a variation of one of the basic functions,"},{"Start":"04:47.075 ","End":"04:48.950","Text":"and we want to shift it left,"},{"Start":"04:48.950 ","End":"04:50.810","Text":"right, up, down, etc."},{"Start":"04:50.810 ","End":"04:56.600","Text":"The basic function, the most obvious one is y equals absolute value of x."},{"Start":"04:56.600 ","End":"04:59.345","Text":"But there\u0027s a slight difficulty here apparently,"},{"Start":"04:59.345 ","End":"05:03.830","Text":"because what we would like is x minus 2 and not 2 minus x."},{"Start":"05:03.830 ","End":"05:07.775","Text":"But if you remember, in general,"},{"Start":"05:07.775 ","End":"05:16.340","Text":"we have that the absolute value of a is equal to the absolute value of minus a."},{"Start":"05:16.340 ","End":"05:18.215","Text":"Since this is so,"},{"Start":"05:18.215 ","End":"05:25.145","Text":"we can just rewrite this as y equals absolute value of x minus 2."},{"Start":"05:25.145 ","End":"05:26.420","Text":"I\u0027m just reversing the order,"},{"Start":"05:26.420 ","End":"05:29.555","Text":"which is like putting minus, minus 3."},{"Start":"05:29.555 ","End":"05:34.955","Text":"Now if we look at this basic function and we look at this,"},{"Start":"05:34.955 ","End":"05:37.010","Text":"which is our function,"},{"Start":"05:37.010 ","End":"05:41.510","Text":"we see that x is replaced by x minus 2,"},{"Start":"05:41.510 ","End":"05:47.375","Text":"which means that the function is shifted right 2,"},{"Start":"05:47.375 ","End":"05:49.460","Text":"and the minus 3 at the end,"},{"Start":"05:49.460 ","End":"05:53.690","Text":"means that we then move it down 3."},{"Start":"05:53.690 ","End":"05:59.351","Text":"So let\u0027s start with y equals absolute value of x,"},{"Start":"05:59.351 ","End":"06:02.690","Text":"then we\u0027ll shift it right 2."},{"Start":"06:02.690 ","End":"06:06.305","Text":"I\u0027ll just mark off 2 units."},{"Start":"06:06.305 ","End":"06:12.650","Text":"Again, same thing here and here."},{"Start":"06:12.650 ","End":"06:14.825","Text":"Then we move it down 3."},{"Start":"06:14.825 ","End":"06:17.585","Text":"So this same apex."},{"Start":"06:17.585 ","End":"06:21.990","Text":"That\u0027s basically it for this part, we\u0027re done."}],"ID":5139},{"Watched":false,"Name":"Exercise 8 - Parts 1-2","Duration":"14m 11s","ChapterTopicVideoID":26402,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/26402.jpeg","UploadDate":"2021-11-21T04:15:41.2970000","DurationForVideoObject":"PT14M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.985","Text":"This exercise has 4 parts,"},{"Start":"00:02.985 ","End":"00:04.830","Text":"and in each part we have to find"},{"Start":"00:04.830 ","End":"00:09.645","Text":"the first and second derivatives of y using implicit differentiation."},{"Start":"00:09.645 ","End":"00:11.625","Text":"Start with Part 1,"},{"Start":"00:11.625 ","End":"00:14.850","Text":"x squared plus y cubed equals 1."},{"Start":"00:14.850 ","End":"00:16.980","Text":"We just go ahead and differentiate."},{"Start":"00:16.980 ","End":"00:19.320","Text":"Derivative of x squared is 2x."},{"Start":"00:19.320 ","End":"00:22.950","Text":"Derivative of y cubed is not just 3y squared,"},{"Start":"00:22.950 ","End":"00:27.885","Text":"but we have to add y prime and derivative of 1 is 0."},{"Start":"00:27.885 ","End":"00:31.680","Text":"We collect terms and do it in 1 step."},{"Start":"00:31.680 ","End":"00:33.930","Text":"Bring the 2x over to the other side,"},{"Start":"00:33.930 ","End":"00:38.765","Text":"make it minus 2x and then divide by the 3y squared,"},{"Start":"00:38.765 ","End":"00:41.110","Text":"so this is what y prime is."},{"Start":"00:41.110 ","End":"00:43.610","Text":"Now, we differentiate a second time,"},{"Start":"00:43.610 ","End":"00:49.505","Text":"and remember the quotient rule for derivatives."},{"Start":"00:49.505 ","End":"00:53.630","Text":"This is what it is and f is going to be minus 2x,"},{"Start":"00:53.630 ","End":"00:56.120","Text":"g will be 3y squared."},{"Start":"00:56.120 ","End":"01:05.630","Text":"The derivative of f will be minus 2 and g is 3y squared and minus minus,"},{"Start":"01:05.630 ","End":"01:10.315","Text":"so it\u0027s plus 2x times 6y."},{"Start":"01:10.315 ","End":"01:14.580","Text":"But that\u0027s not all because it\u0027s y prime."},{"Start":"01:14.580 ","End":"01:16.605","Text":"The implicit derivative."},{"Start":"01:16.605 ","End":"01:22.515","Text":"On the denominator (3y squared squared is 9y^4."},{"Start":"01:22.515 ","End":"01:27.050","Text":"Now we can replace y prime,"},{"Start":"01:27.050 ","End":"01:29.420","Text":"what we have here,"},{"Start":"01:29.420 ","End":"01:32.780","Text":"with the y prime that we found here."},{"Start":"01:32.780 ","End":"01:35.225","Text":"Instead of y prime,"},{"Start":"01:35.225 ","End":"01:39.350","Text":"we put minus 2x over 3y squared."},{"Start":"01:39.350 ","End":"01:42.080","Text":"Now, we can multiply top and bottom."},{"Start":"01:42.080 ","End":"01:46.225","Text":"We don\u0027t need 3y squared because we have a y here, we just need 3y."},{"Start":"01:46.225 ","End":"01:54.785","Text":"What we get is 3y times minus 2 times 3y squared will be minus 18y cubed."},{"Start":"01:54.785 ","End":"01:59.610","Text":"Then here,3y squared cancels with y and 3y,"},{"Start":"01:59.610 ","End":"02:02.880","Text":"so we have 2x times 6 times minus 2x,"},{"Start":"02:02.880 ","End":"02:04.860","Text":"it\u0027s minus 24x squared,"},{"Start":"02:04.860 ","End":"02:09.950","Text":"and 3y times 9y^4 is 27y^5."},{"Start":"02:09.950 ","End":"02:12.260","Text":"That\u0027s basically the answer,"},{"Start":"02:12.260 ","End":"02:16.370","Text":"but I would cancel by 3 everything."},{"Start":"02:16.370 ","End":"02:19.175","Text":"I\u0027ll make you pull the minus out in front optionally."},{"Start":"02:19.175 ","End":"02:20.675","Text":"So 18 becomes 6,"},{"Start":"02:20.675 ","End":"02:23.135","Text":"24 becomes 8, 27 becomes 9."},{"Start":"02:23.135 ","End":"02:26.015","Text":"Slightly neater. Anyway, this is the answer."},{"Start":"02:26.015 ","End":"02:30.980","Text":"Now, let\u0027s move on to Part 2. x squared"},{"Start":"02:30.980 ","End":"02:37.535","Text":"y cubed equals x plus y."},{"Start":"02:37.535 ","End":"02:40.249","Text":"Same tricks as usual."},{"Start":"02:40.249 ","End":"02:42.695","Text":"This time, we have a product."},{"Start":"02:42.695 ","End":"02:45.065","Text":"Let\u0027s write down the product rule."},{"Start":"02:45.065 ","End":"02:47.360","Text":"Other people just forget it."},{"Start":"02:47.360 ","End":"02:50.715","Text":"Just write it each time,"},{"Start":"02:50.715 ","End":"02:52.580","Text":"it\u0027s not very long to do."},{"Start":"02:52.580 ","End":"02:57.935","Text":"f prime g plus fg prime."},{"Start":"02:57.935 ","End":"03:01.040","Text":"Of course, f is the x squared bit,"},{"Start":"03:01.040 ","End":"03:03.905","Text":"and g is the y cubed bit."},{"Start":"03:03.905 ","End":"03:06.110","Text":"Just following it,"},{"Start":"03:06.110 ","End":"03:11.325","Text":"2x and then this 1 is untouched, y cubed."},{"Start":"03:11.325 ","End":"03:13.895","Text":"Then plus, that\u0027s this plus f,"},{"Start":"03:13.895 ","End":"03:17.810","Text":"that\u0027s the untouched part x squared and the g prime,"},{"Start":"03:17.810 ","End":"03:22.160","Text":"very tempting to just write 3y squared and leave it at that,"},{"Start":"03:22.160 ","End":"03:24.605","Text":"but it\u0027s an expression with y,"},{"Start":"03:24.605 ","End":"03:27.230","Text":"so we have to add the y prime."},{"Start":"03:27.230 ","End":"03:30.245","Text":"That\u0027s the left-hand side. Right-hand side."},{"Start":"03:30.245 ","End":"03:32.240","Text":"Derivative of x is 1,"},{"Start":"03:32.240 ","End":"03:34.190","Text":"derivative of y is not 1,"},{"Start":"03:34.190 ","End":"03:36.845","Text":"it\u0027s 1 times y prime."},{"Start":"03:36.845 ","End":"03:40.940","Text":"Or you could just say when you derive anything you put prime over it."},{"Start":"03:40.940 ","End":"03:44.705","Text":"What we want is y prime."},{"Start":"03:44.705 ","End":"03:50.660","Text":"Let me just highlight that we have a y prime here,"},{"Start":"03:50.660 ","End":"03:53.125","Text":"and we have a y prime here,"},{"Start":"03:53.125 ","End":"03:55.850","Text":"and we want to isolate it;"},{"Start":"03:55.850 ","End":"03:58.415","Text":"see what it is on its own."},{"Start":"03:58.415 ","End":"04:01.445","Text":"Let\u0027s see how shall we do that?"},{"Start":"04:01.445 ","End":"04:11.630","Text":"Well, let\u0027s say we throw the 1 over,"},{"Start":"04:11.630 ","End":"04:13.835","Text":"it doesn\u0027t really matter which way."},{"Start":"04:13.835 ","End":"04:17.300","Text":"1 of them we have to throw on 1 side and 1 on the other side."},{"Start":"04:17.300 ","End":"04:22.280","Text":"Let\u0027s say I throw y prime over to,"},{"Start":"04:22.280 ","End":"04:26.075","Text":"this 1 will go over to this side of the equation,"},{"Start":"04:26.075 ","End":"04:29.120","Text":"and this bit will go over to that side."},{"Start":"04:29.120 ","End":"04:32.640","Text":"Then we\u0027ll get all the y primes on the left."},{"Start":"04:32.710 ","End":"04:40.083","Text":"But on the other side, it\u0027s minus,"},{"Start":"04:40.083 ","End":"04:48.290","Text":"so we have x squared times 3y."},{"Start":"04:48.290 ","End":"04:50.910","Text":"I don\u0027t know what\u0027s going on today."},{"Start":"04:51.530 ","End":"05:01.055","Text":"Yes, x squared times 3y squared, on this side."},{"Start":"05:01.055 ","End":"05:04.820","Text":"I\u0027m leaving out the y prime just for a moment, you\u0027ll see."},{"Start":"05:04.820 ","End":"05:08.690","Text":"Minus y prime, and again,"},{"Start":"05:08.690 ","End":"05:13.880","Text":"basically what I\u0027m saying is I\u0027m taking y prime outside the brackets,"},{"Start":"05:13.880 ","End":"05:15.800","Text":"so the y prime is just 1,"},{"Start":"05:15.800 ","End":"05:18.560","Text":"and there it goes from there."},{"Start":"05:18.560 ","End":"05:20.750","Text":"The rest of it is,"},{"Start":"05:20.750 ","End":"05:22.385","Text":"leave the 1 here,"},{"Start":"05:22.385 ","End":"05:24.020","Text":"subtract the other bit,"},{"Start":"05:24.020 ","End":"05:26.790","Text":"which is the 2xy cubed."},{"Start":"05:28.600 ","End":"05:31.640","Text":"We still want to isolate y prime,"},{"Start":"05:31.640 ","End":"05:38.150","Text":"so all we do now is divide by this part and get that y prime is equal to"},{"Start":"05:38.150 ","End":"05:46.470","Text":"1 minus 2xy cubed over x squared."},{"Start":"05:46.470 ","End":"05:47.720","Text":"We\u0027ll put the 3 in front,"},{"Start":"05:47.720 ","End":"05:49.820","Text":"usually put the constant in front."},{"Start":"05:49.820 ","End":"05:58.235","Text":"3x squared, y squared minus 1. That\u0027s y prime."},{"Start":"05:58.235 ","End":"06:00.635","Text":"That\u0027s the first derivative."},{"Start":"06:00.635 ","End":"06:04.180","Text":"But we still need the second derivative."},{"Start":"06:04.180 ","End":"06:08.615","Text":"I probably like to also have that highlighted there,"},{"Start":"06:08.615 ","End":"06:10.910","Text":"so that\u0027s our y prime."},{"Start":"06:10.910 ","End":"06:16.170","Text":"It\u0027s consistent, this pack,"},{"Start":"06:16.170 ","End":"06:18.910","Text":"all our y primes."},{"Start":"06:19.180 ","End":"06:21.815","Text":"It\u0027s not nice with all those colors."},{"Start":"06:21.815 ","End":"06:26.255","Text":"Continuing, we need y double prime."},{"Start":"06:26.255 ","End":"06:29.810","Text":"This time it\u0027s not the product rule we need,"},{"Start":"06:29.810 ","End":"06:32.450","Text":"it\u0027s the quotient rule because we have a fraction."},{"Start":"06:32.450 ","End":"06:36.125","Text":"I\u0027m going to write what it is again."},{"Start":"06:36.125 ","End":"06:40.685","Text":"f over g prime is"},{"Start":"06:40.685 ","End":"06:50.885","Text":"f prime g minus fg prime all over what was the original denominator squared."},{"Start":"06:50.885 ","End":"06:55.100","Text":"If we do that, then we get,"},{"Start":"06:55.100 ","End":"06:57.335","Text":"well obviously, I\u0027m taking this as f,"},{"Start":"06:57.335 ","End":"07:00.650","Text":"but this as g. So we get f prime."},{"Start":"07:00.650 ","End":"07:02.615","Text":"Now, what is f prime?"},{"Start":"07:02.615 ","End":"07:05.037","Text":"1 is 0."},{"Start":"07:05.037 ","End":"07:12.505","Text":"This primed is, you can take a constant outside the bracket, it\u0027s minus twice."},{"Start":"07:12.505 ","End":"07:14.425","Text":"We have a product here."},{"Start":"07:14.425 ","End":"07:16.775","Text":"That just mean not differentiate,"},{"Start":"07:16.775 ","End":"07:23.290","Text":"but just that we have to differentiate xy cubed prime."},{"Start":"07:23.290 ","End":"07:27.145","Text":"That\u0027s the f prime and we still have to throw in the g,"},{"Start":"07:27.145 ","End":"07:29.105","Text":"which is 3x squared,"},{"Start":"07:29.105 ","End":"07:33.970","Text":"y squared minus 1,"},{"Start":"07:33.970 ","End":"07:35.215","Text":"that\u0027s the g. Now,"},{"Start":"07:35.215 ","End":"07:37.825","Text":"we\u0027re up to this minus here,"},{"Start":"07:37.825 ","End":"07:42.225","Text":"and this minus, and then f,"},{"Start":"07:42.225 ","End":"07:48.725","Text":"1 minus 2xy cubed."},{"Start":"07:48.725 ","End":"07:50.170","Text":"Then g prime,"},{"Start":"07:50.170 ","End":"07:52.400","Text":"now similar to before,"},{"Start":"07:52.400 ","End":"07:57.305","Text":"we\u0027ve got to differentiate each piece separately."},{"Start":"07:57.305 ","End":"08:00.115","Text":"The minus 1 is not going to give us anything, but this will."},{"Start":"08:00.115 ","End":"08:02.770","Text":"3 comes out of the brackets."},{"Start":"08:02.770 ","End":"08:09.290","Text":"Because constants can come out of the derivative or the prime and we\u0027re left"},{"Start":"08:09.990 ","End":"08:18.910","Text":"with x squared, y squared prime."},{"Start":"08:18.910 ","End":"08:25.225","Text":"Not still but a freshly over the original denominator squared,"},{"Start":"08:25.225 ","End":"08:27.623","Text":"g squared, there it is,"},{"Start":"08:27.623 ","End":"08:32.330","Text":"3x squared, y squared minus 1 all squared."},{"Start":"08:33.480 ","End":"08:38.980","Text":"We still haven\u0027t finished differentiating because we have a prime here and a prime here."},{"Start":"08:38.980 ","End":"08:41.690","Text":"What we need to do,"},{"Start":"08:41.690 ","End":"08:45.310","Text":"still could see the product rule,"},{"Start":"08:45.310 ","End":"08:46.690","Text":"so I\u0027ll just follow that."},{"Start":"08:46.690 ","End":"08:52.270","Text":"I\u0027m not going to go into every tiny detail at this stage is sophisticated enough."},{"Start":"08:52.270 ","End":"08:54.725","Text":"Minus 2, now,"},{"Start":"08:54.725 ","End":"09:00.660","Text":"I\u0027m just going to write all this in a square bracket derived is"},{"Start":"09:00.660 ","End":"09:06.588","Text":"1 and underived y"},{"Start":"09:06.588 ","End":"09:12.485","Text":"cubed plus underived x"},{"Start":"09:12.485 ","End":"09:15.860","Text":"underived 3y squared."},{"Start":"09:17.040 ","End":"09:20.290","Text":"Almost forgot to do it, but there it is."},{"Start":"09:20.290 ","End":"09:21.610","Text":"You need to add the y prime."},{"Start":"09:21.610 ","End":"09:23.405","Text":"Do not forget that,"},{"Start":"09:23.405 ","End":"09:26.920","Text":"very important, ruined the whole exercise."},{"Start":"09:28.250 ","End":"09:35.165","Text":"That\u0027s this part, minus 2 the derived parties is here."},{"Start":"09:35.165 ","End":"09:37.840","Text":"Then we come to this bit,"},{"Start":"09:37.840 ","End":"09:39.420","Text":"which is 3x squared,"},{"Start":"09:39.420 ","End":"09:41.540","Text":"y squared minus 1."},{"Start":"09:43.640 ","End":"09:46.700","Text":"Turning to look messy,"},{"Start":"09:46.700 ","End":"09:57.715","Text":"minus 1 minus 2xy cubed times 3."},{"Start":"09:57.715 ","End":"10:01.450","Text":"Now, make sure this is a 3."},{"Start":"10:01.450 ","End":"10:03.500","Text":"Now, we come, again,"},{"Start":"10:03.500 ","End":"10:08.350","Text":"to the product rule only this time this is f and this is"},{"Start":"10:08.350 ","End":"10:16.390","Text":"g. Derived is 2x and derived y squared and vice versa,"},{"Start":"10:16.390 ","End":"10:20.080","Text":"x squared as is, y squared derived."},{"Start":"10:20.080 ","End":"10:22.390","Text":"Y squared derived is simply 2y, isn\u0027t it?"},{"Start":"10:22.390 ","End":"10:28.770","Text":"Oh no, we need to add the y prime."},{"Start":"10:28.770 ","End":"10:30.675","Text":"All this is over."},{"Start":"10:30.675 ","End":"10:34.641","Text":"This thing, keep dragging along with us,"},{"Start":"10:34.641 ","End":"10:40.600","Text":"3x squared, y squared minus 1 all squared."},{"Start":"10:40.600 ","End":"10:42.295","Text":"Now other than simplification,"},{"Start":"10:42.295 ","End":"10:45.800","Text":"why don\u0027t I accept this really as an answer?"},{"Start":"10:45.800 ","End":"10:48.190","Text":"Because when I write y prime,"},{"Start":"10:48.190 ","End":"10:51.785","Text":"I really expect to have it just in terms of x and y."},{"Start":"10:51.785 ","End":"10:54.665","Text":"It\u0027s bad enough that it\u0027s not just x, it\u0027s x and y."},{"Start":"10:54.665 ","End":"10:56.730","Text":"Why do I need the y prime in here?"},{"Start":"10:56.730 ","End":"10:59.620","Text":"I\u0027ll just highlight it."},{"Start":"10:59.620 ","End":"11:04.310","Text":"Make it stand out as a y prime here, are there anymore?"},{"Start":"11:04.310 ","End":"11:06.325","Text":"There\u0027s another 1 here."},{"Start":"11:06.325 ","End":"11:08.740","Text":"I don\u0027t want the y prime in there,"},{"Start":"11:08.740 ","End":"11:12.985","Text":"but I do have what y prime is in terms of this."},{"Start":"11:12.985 ","End":"11:14.995","Text":"I\u0027m going to substitute,"},{"Start":"11:14.995 ","End":"11:16.510","Text":"y prime is this."},{"Start":"11:16.510 ","End":"11:19.000","Text":"But after that I\u0027m going to leave the thing alone and not"},{"Start":"11:19.000 ","End":"11:22.565","Text":"simply going to simplify it because it\u0027s a whole mess in algebra."},{"Start":"11:22.565 ","End":"11:26.530","Text":"Let\u0027s just write it first and you\u0027ll see what a mess it is."},{"Start":"11:26.820 ","End":"11:37.390","Text":"What I have here is that y double prime equals minus"},{"Start":"11:37.390 ","End":"11:40.750","Text":"2 times"},{"Start":"11:40.880 ","End":"11:47.500","Text":"y cubed plus 3xy squared."},{"Start":"11:47.500 ","End":"11:50.510","Text":"Put the 3 in front,"},{"Start":"11:51.240 ","End":"11:54.190","Text":"3xy squared, then y prime,"},{"Start":"11:54.190 ","End":"11:58.640","Text":"which is 1 minus 2xy cubed"},{"Start":"11:59.750 ","End":"12:08.320","Text":"over 3x squared y squared minus 1."},{"Start":"12:10.110 ","End":"12:16.780","Text":"That was just putting the y prime in and still times 3x squared,"},{"Start":"12:16.780 ","End":"12:18.860","Text":"y squared minus 1."},{"Start":"12:22.940 ","End":"12:26.450","Text":"I\u0027ll just put the 3 in front there."},{"Start":"12:26.450 ","End":"12:29.140","Text":"Actually, while I\u0027m at it,"},{"Start":"12:29.140 ","End":"12:32.123","Text":"I might as well take the 2 out of here also."},{"Start":"12:32.123 ","End":"12:33.720","Text":"The 2 and the 3 make it 6."},{"Start":"12:33.720 ","End":"12:35.480","Text":"If you multiply them,"},{"Start":"12:35.480 ","End":"12:41.810","Text":"minus 6 of 1 minus 2xy cubed"},{"Start":"12:44.370 ","End":"12:48.055","Text":"times this thing,"},{"Start":"12:48.055 ","End":"12:50.170","Text":"now with 2,"},{"Start":"12:50.170 ","End":"12:52.490","Text":"we\u0027ve taken out already."},{"Start":"12:55.910 ","End":"13:06.916","Text":"xy squared plus 2x squared y,"},{"Start":"13:06.916 ","End":"13:09.245","Text":"again, times y prime,"},{"Start":"13:09.245 ","End":"13:17.010","Text":"which is 1 minus 2xy cubed,"},{"Start":"13:17.010 ","End":"13:22.893","Text":"over 3x squared y"},{"Start":"13:22.893 ","End":"13:31.610","Text":"squared minus 1."},{"Start":"13:32.490 ","End":"13:36.565","Text":"That\u0027s it, y prime in terms of x and y."},{"Start":"13:36.565 ","End":"13:40.315","Text":"Now, it is possible to simplify this algebraically."},{"Start":"13:40.315 ","End":"13:42.215","Text":"I don\u0027t feel like doing it."},{"Start":"13:42.215 ","End":"13:44.320","Text":"I mean, what would you get out of it?"},{"Start":"13:44.320 ","End":"13:48.190","Text":"Another algebra exercise, we\u0027re doing calculus."},{"Start":"13:48.190 ","End":"13:53.210","Text":"Yeah, I\u0027m sneaking out of doing this,"},{"Start":"13:53.210 ","End":"13:57.270","Text":"but I don\u0027t really don\u0027t think it has any value which equals,"},{"Start":"13:57.270 ","End":"13:59.260","Text":"this is the prime, double prime."},{"Start":"13:59.260 ","End":"14:02.525","Text":"It looks confusing. Which equals blah,"},{"Start":"14:02.525 ","End":"14:03.620","Text":"blah, blah, blah, blah."},{"Start":"14:03.620 ","End":"14:05.240","Text":"But that\u0027s the answer."},{"Start":"14:05.240 ","End":"14:09.100","Text":"We\u0027re done with the number 2."}],"ID":27336},{"Watched":false,"Name":"Exercise 8 - Parts 3-6","Duration":"4m 11s","ChapterTopicVideoID":5148,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5148.jpeg","UploadDate":"2016-02-29T13:53:21.3430000","DurationForVideoObject":"PT4M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.590","Text":"The next four exercises,"},{"Start":"00:01.590 ","End":"00:03.090","Text":"3, 4, 5, and 6,"},{"Start":"00:03.090 ","End":"00:06.120","Text":"are all based on the same prototype of 1/x."},{"Start":"00:06.120 ","End":"00:08.085","Text":"We begin with number 3."},{"Start":"00:08.085 ","End":"00:13.230","Text":"I\u0027ve copied the function and the prototypes down below."},{"Start":"00:13.230 ","End":"00:16.904","Text":"This is the prototype and this is the function."},{"Start":"00:16.904 ","End":"00:24.840","Text":"The way to get from here to here is simply to replace x with x minus 6,"},{"Start":"00:24.840 ","End":"00:25.995","Text":"that\u0027s the only difference."},{"Start":"00:25.995 ","End":"00:28.560","Text":"Whenever we do a substitution like this,"},{"Start":"00:28.560 ","End":"00:34.685","Text":"the minus 6 indicates that this is a shift right by 6 units."},{"Start":"00:34.685 ","End":"00:37.250","Text":"Let\u0027s first of all sketch the basic,"},{"Start":"00:37.250 ","End":"00:41.585","Text":"the prototype y=1/x over here."},{"Start":"00:41.585 ","End":"00:43.325","Text":"This is roughly what it looks like."},{"Start":"00:43.325 ","End":"00:44.990","Text":"It should be familiar to you."},{"Start":"00:44.990 ","End":"00:50.670","Text":"It\u0027s the hyperbola and goes through 1,"},{"Start":"00:50.670 ","End":"00:53.125","Text":"1 and minus 1, minus 1."},{"Start":"00:53.125 ","End":"00:57.215","Text":"I\u0027ll keep this picture for the next few exercises."},{"Start":"00:57.215 ","End":"01:02.265","Text":"Now, back to the original function, y = 1 over x minus 6."},{"Start":"01:02.265 ","End":"01:06.800","Text":"Like I said, we just have to take this one and shift it 6 units to the right,"},{"Start":"01:06.800 ","End":"01:08.735","Text":"so we\u0027ll get something like this."},{"Start":"01:08.735 ","End":"01:10.400","Text":"This is meant to be the same as this,"},{"Start":"01:10.400 ","End":"01:12.515","Text":"just moved 6 units to the right."},{"Start":"01:12.515 ","End":"01:14.075","Text":"That\u0027s it. We\u0027re done."},{"Start":"01:14.075 ","End":"01:17.240","Text":"Now let\u0027s sketch the graph of the function in part 4,"},{"Start":"01:17.240 ","End":"01:20.360","Text":"which is also based on y=1/x."},{"Start":"01:20.360 ","End":"01:22.675","Text":"I\u0027ll copy it down below."},{"Start":"01:22.675 ","End":"01:24.320","Text":"Here it is."},{"Start":"01:24.320 ","End":"01:28.234","Text":"I kept the graph of y=1/x from before."},{"Start":"01:28.234 ","End":"01:33.790","Text":"The way to get from this to this is simply by subtracting 6."},{"Start":"01:33.790 ","End":"01:36.065","Text":"This is one of those elementary things,"},{"Start":"01:36.065 ","End":"01:42.725","Text":"which just means that we take the graph of this and move down 6 units."},{"Start":"01:42.725 ","End":"01:46.745","Text":"Start from here, then copy it here,"},{"Start":"01:46.745 ","End":"01:48.800","Text":"but shift it 6 units down."},{"Start":"01:48.800 ","End":"01:50.130","Text":"We\u0027ll get something like this."},{"Start":"01:50.130 ","End":"01:51.575","Text":"Just to make things easier,"},{"Start":"01:51.575 ","End":"01:57.460","Text":"I drew the horizontal asymptote through minus 6 just as a guideline."},{"Start":"01:57.460 ","End":"01:59.475","Text":"That\u0027s this part done."},{"Start":"01:59.475 ","End":"02:00.830","Text":"Next is number 5."},{"Start":"02:00.830 ","End":"02:03.320","Text":"We have to draw the sketch of the graph."},{"Start":"02:03.320 ","End":"02:06.350","Text":"It\u0027s also based on y=1/x, and in fact,"},{"Start":"02:06.350 ","End":"02:09.079","Text":"is remarkably similar to the previous exercise,"},{"Start":"02:09.079 ","End":"02:11.870","Text":"only with minus x replaced by plus 5."},{"Start":"02:11.870 ","End":"02:17.225","Text":"I\u0027ve copied this exercise down below to give us more space. Here it is."},{"Start":"02:17.225 ","End":"02:19.910","Text":"I kept the original prototype,"},{"Start":"02:19.910 ","End":"02:22.175","Text":"1/x, together with its graph."},{"Start":"02:22.175 ","End":"02:27.560","Text":"The only difference between this and this is that here we\u0027ve added the plus 5."},{"Start":"02:27.560 ","End":"02:35.750","Text":"Adding 5 geometrically means to take this graph and move up by 5 units."},{"Start":"02:35.750 ","End":"02:42.405","Text":"All you have to do is get a copy of this here and move it up 5 units."},{"Start":"02:42.405 ","End":"02:44.555","Text":"We\u0027ll get something like this."},{"Start":"02:44.555 ","End":"02:47.905","Text":"Same as this, move it up 5, and that\u0027s it."},{"Start":"02:47.905 ","End":"02:52.040","Text":"Part 6, we have to again draw the graph of the function."},{"Start":"02:52.040 ","End":"02:56.405","Text":"This also is based on y=1/x."},{"Start":"02:56.405 ","End":"03:01.110","Text":"I\u0027ll copy this down below for our convenience. Here it is."},{"Start":"03:01.110 ","End":"03:05.375","Text":"I kept the original y=1/x from the previous exercise."},{"Start":"03:05.375 ","End":"03:08.194","Text":"Now, to get from here to here,"},{"Start":"03:08.194 ","End":"03:10.625","Text":"there are two stages."},{"Start":"03:10.625 ","End":"03:16.780","Text":"One is replacing x by x plus 5."},{"Start":"03:16.780 ","End":"03:23.150","Text":"Geometrically, this has the effect of shifting the graph left by 5 units."},{"Start":"03:23.150 ","End":"03:26.435","Text":"The next thing we did was just subtract 2 from y,"},{"Start":"03:26.435 ","End":"03:35.975","Text":"and this minus 2 from here has the effect of making the graph go down 2 units."},{"Start":"03:35.975 ","End":"03:38.705","Text":"What we have to do is start from this graph,"},{"Start":"03:38.705 ","End":"03:40.534","Text":"draw an intermediate stage,"},{"Start":"03:40.534 ","End":"03:43.340","Text":"which is moved left 5 units,"},{"Start":"03:43.340 ","End":"03:45.140","Text":"and then the result of that,"},{"Start":"03:45.140 ","End":"03:46.925","Text":"we take down 2 units."},{"Start":"03:46.925 ","End":"03:48.595","Text":"The first one we have,"},{"Start":"03:48.595 ","End":"03:51.980","Text":"now moving this one left 5 units here,"},{"Start":"03:51.980 ","End":"03:53.480","Text":"we get something like this."},{"Start":"03:53.480 ","End":"03:58.625","Text":"I used the minus 5 as a guideline to help me draw the sketch."},{"Start":"03:58.625 ","End":"04:02.925","Text":"Next thing to do is to take this and move it down 2."},{"Start":"04:02.925 ","End":"04:04.010","Text":"We draw that over here,"},{"Start":"04:04.010 ","End":"04:05.570","Text":"it will look something like this."},{"Start":"04:05.570 ","End":"04:06.680","Text":"Just a rough sketch."},{"Start":"04:06.680 ","End":"04:09.425","Text":"Again, I used the guideline at minus 2,"},{"Start":"04:09.425 ","End":"04:11.910","Text":"and that\u0027s roughly it."}],"ID":5141},{"Watched":false,"Name":"Exercise 8 - Parts 7-10","Duration":"4m 44s","ChapterTopicVideoID":5149,"CourseChapterTopicPlaylistID":56144,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5149.jpeg","UploadDate":"2016-02-29T13:53:44.7470000","DurationForVideoObject":"PT4M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.539","Text":"In Exercise 7, we have to draw the graph of this function."},{"Start":"00:03.539 ","End":"00:08.220","Text":"We already decided it\u0027s based on 1 over x^2, that\u0027s our prototype."},{"Start":"00:08.220 ","End":"00:12.975","Text":"In fact, the following four exercises are all based on 1 over x^2."},{"Start":"00:12.975 ","End":"00:17.940","Text":"Let\u0027s begin with 7 and I copied it down below to give us some more space."},{"Start":"00:17.940 ","End":"00:19.170","Text":"Here we are below,"},{"Start":"00:19.170 ","End":"00:20.820","Text":"I\u0027ve copied the function."},{"Start":"00:20.820 ","End":"00:24.945","Text":"This is the basic functional prototype it\u0027s based on."},{"Start":"00:24.945 ","End":"00:29.130","Text":"I took the liberty of free drawing the graph of 1 over x^2,"},{"Start":"00:29.130 ","End":"00:30.554","Text":"which should be familiar."},{"Start":"00:30.554 ","End":"00:35.565","Text":"If not, you can just substitute a few points and see what it is."},{"Start":"00:35.565 ","End":"00:40.790","Text":"We have the graph of this and what\u0027s the difference between this and our function?"},{"Start":"00:40.790 ","End":"00:47.975","Text":"The difference is that we\u0027ve replaced x by x minus 2."},{"Start":"00:47.975 ","End":"00:52.085","Text":"When we do this, it has the effect graphically of"},{"Start":"00:52.085 ","End":"00:57.110","Text":"moving the graph to the right by 2 units."},{"Start":"00:57.110 ","End":"01:01.715","Text":"All we have to do is take this one and shift it 2 units to the right."},{"Start":"01:01.715 ","End":"01:03.410","Text":"We\u0027ll draw it over here."},{"Start":"01:03.410 ","End":"01:05.405","Text":"We\u0027ll get something like this."},{"Start":"01:05.405 ","End":"01:10.430","Text":"This is supposed to be the same as this except move 2 units to the right."},{"Start":"01:10.430 ","End":"01:13.685","Text":"That\u0027s it for part 7."},{"Start":"01:13.685 ","End":"01:17.975","Text":"Part 8, we have to draw the graph of this function."},{"Start":"01:17.975 ","End":"01:22.220","Text":"Once again, it\u0027s based on y equals 1 over x^2."},{"Start":"01:22.220 ","End":"01:25.910","Text":"I\u0027ll copy it below for our convenience."},{"Start":"01:25.910 ","End":"01:31.130","Text":"Here it is. Here\u0027s the basic function prototype it was based on."},{"Start":"01:31.130 ","End":"01:34.625","Text":"The difference is that in our function,"},{"Start":"01:34.625 ","End":"01:36.560","Text":"we have a minus 2,"},{"Start":"01:36.560 ","End":"01:42.665","Text":"which has the effect of lowering the whole function or its graph by 2 units."},{"Start":"01:42.665 ","End":"01:48.930","Text":"Basically, we start from here and we just go down 2."},{"Start":"01:48.930 ","End":"01:53.660","Text":"Other words, I have to take this picture and pull it down 2 units,"},{"Start":"01:53.660 ","End":"01:55.445","Text":"and I\u0027ll draw it over here."},{"Start":"01:55.445 ","End":"01:57.170","Text":"I\u0027ll get something like this,"},{"Start":"01:57.170 ","End":"02:00.214","Text":"a rough sketch and we\u0027re done."},{"Start":"02:00.214 ","End":"02:02.060","Text":"In number 9, as usual,"},{"Start":"02:02.060 ","End":"02:04.205","Text":"we have to draw the graph of the function."},{"Start":"02:04.205 ","End":"02:11.610","Text":"It\u0027s also based on y equals 1 over x^2 and is remarkably similar to number 8. Here it is."},{"Start":"02:11.610 ","End":"02:15.380","Text":"Here\u0027s the original y equals 1 over x^2,"},{"Start":"02:15.380 ","End":"02:17.330","Text":"which is based on the prototype."},{"Start":"02:17.330 ","End":"02:18.830","Text":"I\u0027ve kept the picture,"},{"Start":"02:18.830 ","End":"02:20.980","Text":"the sketch of that one too."},{"Start":"02:20.980 ","End":"02:24.950","Text":"As before, the only difference between this one and"},{"Start":"02:24.950 ","End":"02:30.245","Text":"this one is that we take this and we add a 4 to it."},{"Start":"02:30.245 ","End":"02:38.645","Text":"Adding 4 to a function has the effect graphically of moving it up 4 units."},{"Start":"02:38.645 ","End":"02:43.249","Text":"We have to do is take this and move it up 4 units."},{"Start":"02:43.249 ","End":"02:44.630","Text":"We\u0027ll draw that over here."},{"Start":"02:44.630 ","End":"02:46.895","Text":"It\u0027ll look something like this."},{"Start":"02:46.895 ","End":"02:50.819","Text":"We just needed a rough sketch and we\u0027re done."},{"Start":"02:50.819 ","End":"02:52.910","Text":"Finally, we come to number 10."},{"Start":"02:52.910 ","End":"02:55.969","Text":"We have to sketch the graph of this function."},{"Start":"02:55.969 ","End":"03:01.520","Text":"Once again, it\u0027s based on the y equals 1 over x^2 prototype."},{"Start":"03:01.520 ","End":"03:06.050","Text":"I\u0027ve copied the exercise down below for our convenience."},{"Start":"03:06.050 ","End":"03:10.490","Text":"Here it is. The prototype 1 over x^2,"},{"Start":"03:10.490 ","End":"03:14.525","Text":"which has the sketch here I kept from before."},{"Start":"03:14.525 ","End":"03:20.465","Text":"Now how do we get from 1 over x^2 to 1 over x plus 4^2 plus 2?"},{"Start":"03:20.465 ","End":"03:22.879","Text":"Well, there\u0027s two steps."},{"Start":"03:22.879 ","End":"03:31.310","Text":"One is changing x into x plus 4, I mean substituting."},{"Start":"03:31.310 ","End":"03:40.605","Text":"This has the effect of moving the graph left by 4 units."},{"Start":"03:40.605 ","End":"03:45.045","Text":"The other bit is the plus 2 on the end."},{"Start":"03:45.045 ","End":"03:54.784","Text":"This plus 2 has the effect of making the graph go up 2 units."},{"Start":"03:54.784 ","End":"03:59.255","Text":"What we\u0027re going to do is start from this 1 over x^2."},{"Start":"03:59.255 ","End":"04:05.195","Text":"We\u0027re going to move it left on the x-axis."},{"Start":"04:05.195 ","End":"04:07.790","Text":"Then down here, we\u0027ll take the result of this and move it up 2."},{"Start":"04:07.790 ","End":"04:10.730","Text":"Here it goes,"},{"Start":"04:10.730 ","End":"04:13.590","Text":"first of all, left 4."},{"Start":"04:13.730 ","End":"04:19.545","Text":"Something like this, I used minus 4 vertical line as a guide,"},{"Start":"04:19.545 ","End":"04:23.380","Text":"that\u0027s the asymptote, to help the sketch."},{"Start":"04:23.380 ","End":"04:25.805","Text":"Next thing is to take this and move it up 2,"},{"Start":"04:25.805 ","End":"04:30.140","Text":"and drawing it over here,"},{"Start":"04:30.140 ","End":"04:33.635","Text":"we get roughly something like this."},{"Start":"04:33.635 ","End":"04:41.615","Text":"Once again, I use the horizontal 2 asymptote as a guideline to help me draw the graph."},{"Start":"04:41.615 ","End":"04:44.970","Text":"That\u0027s it. We\u0027re done."}],"ID":5142}],"Thumbnail":null,"ID":56144}]

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