WEBVTT 00:00:06.850 --> 00:00:14.117 Let's explore the graph of the sine function, y equals sine x. Here I've made a table where x is the angle 00:00:14.119 --> 00:00:22.121 in standard position and y is the value of the sine function. Notice that all the angles in my table are special angles, 00:00:22.123 --> 00:00:29.622 so you should be able to determine all these values on your own. Then I've plotted all the ordered pairs, 00:00:29.624 --> 00:00:44.895 as you can see, so now let's see what the graph of sine x looks like. Let's connect our points. 00:00:44.897 --> 00:00:55.155 So we have a nice graph of the sine function. If I had chosen values bigger than 2 pi, say 2 pi plus pi over 6 00:00:55.157 --> 00:01:05.667 that would be 13 pi over 6. The value of the sine function of 13 pi over 6 would be what? 00:01:05.669 --> 00:01:13.425 One half, exactly. So you can see that this pattern is going to repeat itself, if I had chosen values bigger than 2 pi 00:01:13.427 --> 00:01:20.900 or less than zero. Now let's look at some characteristics of the sine function now that we know the graph. 00:01:22.673 --> 00:01:29.942 Here is the graph of y equals sine x. Let's look at some of the characteristics of the graph. The first thing we want to 00:01:29.944 --> 00:01:37.447 notice is what's the domain? Remember domain means the values of x. Well notice that the left side there's arrows, 00:01:37.449 --> 00:01:44.693 so it's going to negative infinity on the right, on the left, excuse me. And on the right it's going to positive infinity. 00:01:44.695 --> 00:01:58.444 So the domain would be all real numbers. Now let's look at the range of the function, the sine function. 00:01:58.446 --> 00:02:04.695 That's looking at the y-values. What do you notice about the y-values? Notice the smallest y-value is negative 1, 00:02:04.697 --> 00:02:16.987 the largest y-value is 1, and it takes on all values in between. So the range would be negative 1 to 1. 00:02:16.989 --> 00:02:24.236 Another thing we notice about this, there are three periods shown here. Here's one period, here's another period, 00:02:24.236 --> 00:02:31.737 and another period where the sine function is repeating itself three times. So that means the period of these intervals 00:02:31.739 --> 00:02:48.767 in between here, here's another period, and finally we have one more period drawn here. So the period for the sine 00:02:48.769 --> 00:02:59.268 function would be 2 pi. Now what that means is that if you're trying to find values, that means the sine of some value 00:02:59.270 --> 00:03:09.267 x is going to be the same thing as the sine of x plus 2 pi times, say n, where n is any integer. They'll be the same value 00:03:09.69 --> 00:03:18.564 because the sine function is periodic. Now, another thing we notice about the graph is that sine is an odd function. 00:03:18.566 --> 00:03:33.300 What does that mean? That means that the sine of negative x is the opposite of the sine of x. That also means 00:03:33.302 --> 00:03:42.051 the graph is symmetric about the origin. Another characteristic, let's look at the intercepts. What's the y-intercept? 00:03:42.053 --> 00:03:54.585 Where does the graph cross the y-axis? We can see that the y-intercept is 0. Now, what about the x-intercepts? 00:03:54.587 --> 00:04:00.340 Well, there's an infinite number of them, as we can see, because you remember that this goes on forever, but let's 00:04:00.342 --> 00:04:09.601 describe where they occur. Well, let's see. One x-intercept is at zero, and then pi, and then 2 pi, and so on. 00:04:09.603 --> 00:04:20.362 Do you see the pattern? So the x-intercepts are at all multiples of pi. So we can describe that by just saying n pi, 00:04:20.364 --> 00:04:35.613 again where n is an integer. Two more things we want to notice about the sine function. Notice that there's a maximum 00:04:35.615 --> 00:04:40.630 point and a minimum point, a relative max and a relative min, and we want to talk about where they occur. 00:04:40.632 --> 00:04:46.384 Well we know there's an infinite number of them because the sine function repeats itself. Let's look at the relative 00:04:46.386 --> 00:04:56.638 max and talk about where they occur. The relative max occurs at what x-values? Well, let's see. The first one occurs, 00:04:56.640 --> 00:05:03.405 well not the first one, but one of them occurs at pi over 2. Where is the next one? Well, it's going to be a period away, 00:05:03.407 --> 00:05:22.904 2 pi away. So I can describe where they occur by saying at x equals pi over 2 plus 2 pi n, where n is any integer. 00:05:22.906 --> 00:05:33.934 And then of course now we see there's a relative min. Where does it occur? The relative min occurs at what x-value? 00:05:33.936 --> 00:05:40.189 Well one of them is 3 pi over 2, and again to get to the next one we add multiples of the period, so we can say at 00:05:40.191 --> 00:05:57.454 3 pi over 2 plus 2 pi n, again where n is an integer. You should learn all these characteristics of the sine function. 00:05:57.456 --> 00:06:03.997 They're very important. Now we're going to focus on just one period of the sine function, and look at some of those 00:06:03.999 --> 00:06:06.693 important points that you're going to need to know. 00:06:08.700 --> 00:06:15.720 Here is the graph of y equals sine x with x between 0 and 2 pi or one period or cycle of the sine function, 00:06:15.722 --> 00:06:23.220 and we want to notice some important points on this cycle. Those important points occur at the x-intercepts 00:06:23.222 --> 00:06:30.234 and the y-intercept, which of course is the same in this case. So I've got the x-intercepts and then the points where 00:06:30.236 --> 00:06:36.738 the relative max and the relative min occur. Notice that these points occur at one fourth of the period, so we call them 00:06:36.740 --> 00:06:55.265 quarter points. Let's list the quarter points. The first one is the origin, (0,0). The next one is (pi over 2,1). 00:06:55.267 --> 00:07:09.026 Then (pi,0), (3 pi over 2,-1) and (2 pi,0). These are important points for you to know, so you should know these 00:07:09.028 --> 00:07:12.527 quarter points for the sine function so that you can graph it easily. 00:07:14.020 --> 00:07:23.026 Let's explore the graph of the cosine function, y equals cosine x. Here I've made a table of values where x is the 00:07:23.028 --> 00:07:31.523 angle in standard position in radians and y is the value of the cosine function. Notice that all the angles in my table 00:07:31.525 --> 00:07:38.778 are special angles, so you should be able to get all these values on your own. Then I plotted the ordered pairs, 00:07:38.780 --> 00:07:46.065 as you can see here, so we can see what the graph of the cosine looks like. Let's connect our points. So let's see. 00:07:46.067 --> 00:08:00.567 It's going to start up at 1, go down, and do this. So I've plotted the points from 0 to 2 pi. What would happen 00:08:00.569 --> 00:08:08.814 if I chose values bigger than 2 pi, or say, less than zero? Let's take an angle. For example, 2 pi plus pi over 6 00:08:08.816 --> 00:08:19.064 would be 13 pi over 6. What would the cosine of 13 pi over 6 be? Well let's see. That's in quadrant one, cosine's 00:08:19.066 --> 00:08:25.854 positive, so it's going to be the square root of 3 over 2. So we can see that this pattern will just repeat itself 00:08:25.856 --> 00:08:32.855 for values bigger than 2 pi or less than zero. Now that we know what the graph of the cosine looks like, 00:08:32.857 --> 00:08:36.000 let's look at some characteristics of the graph. 00:08:37.600 --> 00:08:44.127 Let's look at the graph of the cosine function. Here I've got three periods of the cosine function, and we want to look 00:08:44.129 --> 00:08:50.377 at some of the characteristics. Well the domain of the cosine function, the values of x, are all real numbers. 00:08:50.379 --> 00:09:01.380 We can see that the graph goes from negative infinity to positive infinity, so the domain is all real numbers. 00:09:01.382 --> 00:09:08.377 The range will be the y-values. Let's look at those. The y-values go from negative 1 up to 1 and take in all values 00:09:08.379 --> 00:09:21.411 in between, so the range would be negative 1 to 1. The period is 2 pi. We can see here's a period, between here, 00:09:21.413 --> 00:09:35.160 that it repeats itself again here, so here's another period of 2 pi, and then I see another period over here. 00:09:35.162 --> 00:09:46.436 So the period for the cosine function is 2 pi. Now let's look at another characteristic which would be I notice that 00:09:46.438 --> 00:09:55.935 this function is symmetric about the y-axis which means it's an even function. That means that the cosine of 00:09:55.937 --> 00:10:07.454 negative x is the same as the cosine of x. It's symmetric about the y-axis. Next let's look at the intercepts. Well, 00:10:07.456 --> 00:10:19.465 the y-intercept is what? The y-intercept is 1. It crosses the y-axis at 1. What about the x-intercepts? Let's look at those. 00:10:19.467 --> 00:10:31.313 Where do they occur? Well we can see we have infinitely many of them. We can see one, two, three, four, five, six 00:10:31.315 --> 00:10:40.052 here on these cycles here. But where do they occur? Well, let's see. pi over 2, then the next one's at 3 pi over 2, 00:10:40.054 --> 00:10:46.551 5 pi over 2, 7 pi over 2. Can you predict where the next one would be? 00:10:46.553 --> 00:10:54.348 9 pi over 2, exactly. So you can see that they occur at odd multiples of pi over 2. How do we write that? 00:10:54.350 --> 00:11:06.094 Well, odd number is any number 2n plus 1 gives you an odd number, and we multiply that odd number times 00:11:06.096 --> 00:11:15.344 pi over 2. That's where all the x-intercepts occur. The other thing we want to look at is the graph has relative max 00:11:15.346 --> 00:11:22.372 and relative min points, and we want to talk about where they occur. Let's look at the relative max first. Where does 00:11:22.374 --> 00:11:36.887 it occur? Well this one's of course at x equals 0, and then the next one occurs 2 pi away so we can add multiples 00:11:36.889 --> 00:11:53.153 of the period 2 pi, which is the same thing of course as saying 2 pi n where n is any integer. Now let's look at the 00:11:53.155 --> 00:12:00.903 relative min. Here's a relative min. Where does it occur? At pi. Where does the next one occur? A period away, this 00:12:00.905 --> 00:12:24.903 is an interval of 2 pi, so the relative min occurs at x equals, this would be pi plus 2 pi n again where n is an integer. 00:12:24.905 --> 00:12:30.650 So here are the characteristics of the cosine function that you should be very familiar with. 00:12:30.652 --> 00:12:36.903 Now I want to focus on just this one period and look at some important points. 00:12:37.690 --> 00:12:45.193 Here we have one cycle of the cosine curve, and we want to look at some important points. The important points 00:12:45.195 --> 00:12:53.693 are where the intercepts or the relative max, the x-intercepts, and the relative min. Those points occur at one fourth 00:12:53.695 --> 00:13:06.226 of the period, so we call them quarter points. Let's list the quarter points for the cosine function between 0 and 2 pi. 00:13:06.228 --> 00:13:23.479 Well the first one is at 0, (0,1). The next one occurs at (pi over 2, 0). The next point here is (pi, -1), (3 pi over 2, 0), 00:13:23.481 --> 00:13:33.256 and finally (2 pi, 1). So here are the five quarter points for cosine x between 0 and 2 pi. You need to know 00:13:33.258 --> 00:13:37.500 these quarter points for the cosine function so you can graph it. 00:13:39.500 --> 00:13:47.018 Here we want to graph functions of the form y equals A sine x and y equals A cosine x to see what effect A has 00:13:47.020 --> 00:13:56.780 on the graph. First thing we need to talk about is the amplitude of these graphs will be the absolute value of A. 00:13:56.782 --> 00:14:05.533 It's always positive. Let's look at an example. So I have this function y equals 3 sine x. What's going to happen is 00:14:05.535 --> 00:14:13.293 this 3 is going to be multiplied by all these y-coordinates of the quarter points. So you can see that I've got all 00:14:13.295 --> 00:14:21.301 those quarter points of sine x listed. So now let's figure out what the quarter points will be for 3 sine x. 00:14:21.303 --> 00:14:38.322 Well it doesn't change the x-coordinates, so they're still going to be 0, pi over 2, pi, 3 pi over 2, and 2 pi. 00:14:38.324 --> 00:14:48.072 So to get the y-coordinates for 3 sine x we're going to multiply these values by 3. So 0 times 3 is of course 0. 00:14:48.074 --> 00:15:01.092 1 times 3 is 3. 0 times 3 is 0. Negative 1 times 3, negative 3, and then again 0 times 3 is 0. So here will be 00:15:01.094 --> 00:15:20.112 the quarter points for y equals 3 sine x. Let's graph that function. 00:15:20.114 --> 00:15:35.613 So we have (0, 0), then we have (pi over 2, 3), (pi , 0), (3 pi over 2, -3), and (2 pi, back up to 0). So the amplitude is 00:15:35.615 --> 00:15:45.389 this distance, half the distance between the max and the min value. Let's write the amplitude for y equals 3 sine x. 00:15:45.391 --> 00:15:58.152 The amplitude will be the absolute value of 3 which is 3. Again, that's the distance, half the distance between the 00:15:58.154 --> 00:16:05.148 relative max and the relative min, which makes the range of this graph what? Exactly. 00:16:05.150 --> 00:16:16.000 The range is now going to be negative 3 to 3. Now let's look at another example. 00:16:17.665 --> 00:16:29.432 Let's look at the graph of y equals negative 2 cosine x. The amplitude will be the absolute value of negative 2, 00:16:29.434 --> 00:16:38.684 which of course is positive 2. Now, in order to graph this function, what we're going to do is multiply negative 2 00:16:38.686 --> 00:16:46.434 by our y-coordinates of our quarter points for y equals cosine x. So let's write the quarter points for y equals 00:16:46.436 --> 00:17:06.432 negative 2 cosine x. Well the A only affects the y-coordinates so the x-coordinates will not change. 00:17:06.434 --> 00:17:13.934 Ok, so let's look at the first one. So the y-coordinate of cosine x is 1, multiply by negative 2, and what do you get? 00:17:13.936 --> 00:17:20.230 Negative 2. It's important to note that you don't multiply your y-coordinates by the amplitude. You multiply your 00:17:20.232 --> 00:17:31.481 y-coordinates by A. Alright so let's keep going. 0 times negative 2 is still 0. Negative 1 times negative 2 is positive 2. 00:17:31.483 --> 00:17:38.477 0 times negative 2 is 0, and 1 times negative 2 is negative 2. So here are our quarter points 00:17:38.479 --> 00:17:55.979 for y equals negative 2 cosine x. Let's graph this function. 00:17:55.981 --> 00:18:10.479 So let's see. I have (0, -2), and (pi over 2, 0), (pi , 2), (3 pi over 2, 0), (2 pi , -2). So here's the graph 00:18:10.581 --> 00:18:20.538 of y equals negative 2 cosine x. What is the range for this graph? Well the range is going to be 00:18:20.540 --> 00:18:31.539 negative 2 to positive 2. So in general when you're graphing functions y equals A sine x or y equals A cosine x, 00:18:31.541 --> 00:18:38.560 the amplitude will always be the absolute value of A, 00:18:38.562 --> 00:18:49.050 and the range will always be negative absolute value of A to positive [absolute value of] A. 00:18:50.562 --> 00:18:58.321 Here we want to look at graphing functions of the form y equals sine Bx and y equals cosine Bx. For these functions, 00:18:58.323 --> 00:19:12.590 the period will be 2 pi divided by B. Now it's important to note that B must be positive here. Let's look at an example. 00:19:12.592 --> 00:19:25.098 y equals cosine 3x, so B is positive 3. So the period's going to be 2 pi divided by 3. So B is going to change 00:19:25.100 --> 00:19:33.352 the x-coordinates. Here I've listed the quarter points for y equals cosine x which of course has a period of 2 pi. 00:19:33.354 --> 00:19:41.349 For cosine 3x, the period is 2 pi over 3. The quarter points occur at one fourth of the period, so let's figure out, 00:19:41.351 --> 00:19:53.880 what is one fourth of this period? One fourth of 2 pi over 3 is pi over 6, so now our x-coordinates are going to be 00:19:53.882 --> 00:20:06.890 in increments of pi over 6. The y-coordinates will stay the same. So let's list the quarter points for y equals cosine 3x. 00:20:06.892 --> 00:20:18.640 Ok, so the first one still occurs at 0. The next quarter point is going to be pi over 6 increment, so you're always 00:20:18.642 --> 00:20:30.169 adding pi over 6 to each x-coordinate to get the next one. What is pi over 6 plus pi over 6? Well, it's 2 pi over 6, 00:20:30.171 --> 00:20:42.926 but of course you always reduce your answers. Ok, and then 2 pi over 6, or pi over 3, plus pi over 6 gives you 00:20:42.928 --> 00:20:58.444 3 pi over 6 which is pi over 2. And finally, 3 pi over 6 plus another pi over 6 is 4 pi over 6 or the period of 2 pi over 3. 00:20:58.446 --> 00:21:16.707 So here are the quarter points for y equals cosine 3x. Let's plot those points. So we have a period of 2 pi over 3. 00:21:16.709 --> 00:21:27.469 And let's see. Notice the y-coordinates are going to be 1 and negative 1, so let's see. The first point we want to 00:21:27.471 --> 00:21:43.483 plot is (0,1). The next point, quarter point, is (pi over 6, 0), (pi over 3, -1), (pi over 2, 0), and (2 pi over 3, positive 1). 00:21:43.483 --> 00:21:58.247 So here is one period for y equals cosine 3x. What is its amplitude? Well we can see that it's still 1. Notice that, 00:21:58.249 --> 00:22:10.240 in this case, A is 1, so the amplitude is 1. What is the range? Well it's negative 1 up to positive 1. 00:22:11.900 --> 00:22:20.236 Let's look at graphing functions of the form y equals A sine Bx and y equals A cosine Bx. The amplitude of these 00:22:20.238 --> 00:22:28.030 functions will be the absolute value of A. The period will be 2 pi over B, again for B positive. Let's look at this 00:22:28.032 --> 00:22:37.293 example y equals sine negative pi x. Right off the bat we notice that our B is negative. So what do we do? Well, we 00:22:37.295 --> 00:22:46.300 have to remember something about the sine function, and that is that the sine is odd. And that means that the 00:22:46.302 --> 00:22:58.304 sine of negative x is the opposite of the sine of x, so we can rewrite this function as negative sine of pi x. 00:22:58.306 --> 00:23:10.075 Now we can find our amplitude which will be the absolute value of negative 1, which is 1, and our period 00:23:10.077 --> 00:23:22.319 for this function will be 2 pi divided by pi, which is 2. Now we want to get the quarter points for this function. We need 00:23:22.321 --> 00:23:28.592 to get one fourth of the period to figure out the interval between the quarter points, which will be 1/2. So we're going 00:23:28.594 --> 00:23:40.600 to be adding 1/2 to each of our x-coordinates to get the x-coordinates for y equals negative sine pi x. The first 00:23:40.602 --> 00:23:49.854 x-coordinate will stay 0. Let's just do the x-coordinates first and then we'll come back and do the y-coordinates. 00:23:49.856 --> 00:24:03.630 So what we want to do is we want to add 1/2 to each of these x-coordinates. So 0 plus 1/2 is 1/2. 1/2 plus 1/2 is 1. 00:24:03.632 --> 00:24:18.890 1 plus 1/2 is 3/2. And then 3/2 plus another 1/2 gives you 2. So that's our x-coordinates for our quarter points. 00:24:18.892 --> 00:24:25.143 Now how do we get the y-coordinates? Remember the y-coordinates are affected by A, so we want to be multiplying 00:24:25.145 --> 00:24:34.142 these y-coordinates by A, which in this case is negative 1. So we're going to multiply each of these y-coordinates 00:24:34.144 --> 00:24:42.669 by negative 1. So let's see. We're going to get 0. 1 times negative 1 is negative 1. 0 times negative 1 is 0. 00:24:42.671 --> 00:24:50.913 Negative 1 times negative 1 is positive 1, and then 0 times negative 1 is 0. So here are our quarter points 00:24:50.915 --> 00:25:08.165 for y equals negative sine pi x. Let's graph that function. 00:25:08.167 --> 00:25:18.200 Ok, so let's see. We have (0, 0), and then we have ... oops, not pi over 2. It should have been 1/2. So all of these 00:25:18.202 --> 00:25:39.474 should be 1/2, 1, 3/2, and 2. So this point (1/2, -1) down here. (1/2, 0) I mean sorry, (1, 0). (3/2, 1) and then (2, 0). 00:25:39.476 --> 00:25:48.971 So here would be one period for the function y equals negative sine pi x. Notice that it's going down first. 00:25:48.973 --> 00:25:56.737 There's the negative which reflects it about the x-axis. What's the amplitude? We said that was 1. 00:25:56.739 --> 00:26:09.230 What's the range for this function? Let's write that down. The range would be negative 1 to positive 1. 00:26:10.000 --> 00:26:17.502 Let's take a look at graphing y equals negative 2 cosine of negative 4x. Again, the first thing I need to notice is 00:26:17.504 --> 00:26:24.753 when B is negative. To find the period you need a positive B. So we need to rewrite this function. Remember that 00:26:24.755 --> 00:26:39.003 the cosine is even. What does that mean? It means that the cosine of negative x is the same as the cosine of x, 00:26:39.005 --> 00:26:46.292 which means the cosine of negative 4x is the same as the cosine of 4x. So this can be rewritten as y equals 00:26:46.294 --> 00:26:57.789 negative 2 cosine of 4x. Now, what's the amplitude? Well the amplitude is going to be the absolute value 00:26:57.791 --> 00:27:08.535 of negative 2 which of course is positive 2. What's the period? Well we find the period by taking 2 pi dividing by B, 00:27:08.537 --> 00:27:19.288 which in this case is 4, so our period will be pi over 2. Now how do we get our x-coordinates? We need to know 00:27:19.290 --> 00:27:28.584 one fourth of the period. And what is one fourth of pi over 2? Well, that's pi over 8. So pi over 8 is going to be what 00:27:28.586 --> 00:27:33.589 we add to each x-coordinate to get our x-coordinates for our quarter points for this function. 00:27:33.591 --> 00:27:48.589 So let's write those down. So let's see, we have negative 2 cosine 4x. The first x-coordinate will stay 0. 00:27:48.591 --> 00:27:56.339 Now what do we add to 0 to get to the next one? You add pi eighths. So this next x-coordinate will be pi over 8. 00:27:56.341 --> 00:28:04.088 Again, you're adding pi over 8 to each x-coordinate. What's this x-coordinate going to be? Well, what is pi over 8 00:28:04.090 --> 00:28:16.629 plus pi over 8? Well, that's 2 pi over 8 which you reduce to pi over 4. And then keep adding pi over 8. 00:28:16.631 --> 00:28:27.882 That's 2 pi over 8 plus pi over 8 gives you 3 pi over 8. And then 3 pi over 8 plus another pi over 8 gives you 00:28:27.884 --> 00:28:36.142 4 pi over 8, which of course is pi over 2. So those are our x-coordinates for our quarter points. How do we get our 00:28:36.144 --> 00:28:48.662 y-coordinates? Well, we need to multiply each of these y-coordinates by A, which is negative 2. Make sure you 00:28:48.664 --> 00:28:56.161 don't multiply by 2. It's always A, not the amplitude. So let's see. What is 1 times negative 2? Negative 2. 00:28:56.163 --> 00:29:07.412 0 times negative 2 is 0. Negative 1 times negative 2 is a positive 2. 0 times negative 2, 0. And 1 times negative 2, 00:29:07.414 --> 00:29:33.412 negative 2. So here are our quarter points for y equals negative 2 cosine 4x. Let's look at that graph. 00:29:33.414 --> 00:29:50.409 So we have (0, -2), (pi over 8, 0), (pi over 4, 2), (3 pi over 8, 0), and (pi over 2, -2). So here would be one period for 00:29:50.411 --> 00:29:58.985 this function y equals negative 2 cosine 4x. We figured out the amplitude for this function was 2. 00:29:58.987 --> 00:30:07.233 What's the range for this function? Well the range is going to be negative 2 to 2. Remember it's always 00:30:07.235 --> 00:30:13.700 negative absolute value of A to positive absolute value of A. 00:30:14.999 --> 00:30:21.246 In this example, we're given a graph and we want to determine the equation of a function of the form 00:30:21.248 --> 00:30:29.260 y equals A sine Bx or y equals A cosine Bx. So we have to decide whether we're going to use a sine function or a 00:30:29.262 --> 00:30:35.768 cosine function. Well, how do we know? Let's look at the first quarter point. It's (0, 0). It goes through the origin, so 00:30:35.770 --> 00:30:45.764 we know we're going to use a sine function. So we're going to write y equals A sine Bx, so now we have to find 00:30:45.766 --> 00:30:51.533 B and A. Well remember B is with the period, so let's look at this graph. What's the period for this graph? 00:30:51.535 --> 00:31:01.045 We can see the period is 8 pi, so let's write that down. And how do we get the period? Do you remember? 00:31:01.047 --> 00:31:13.052 That's right. You take 2 pi and you divide by B to get the period. So we can solve for B. Multiply both sides by B. 00:31:13.054 --> 00:31:23.814 So we get 2 pi divided by 8 pi, which is 1/4. So now we know B is 1/4. Now we need to find A. Well I can see from 00:31:23.816 --> 00:31:32.070 the graph that the amplitude is 2, but that doesn't mean that A is 2. Notice that this graph is decreasing first. 00:31:32.072 --> 00:31:41.326 This y-coordinate of this quarter point is negative 2, so I know A is negative 2. So now what's our sine function? 00:31:41.328 --> 00:31:49.593 y equals negative 2 sine of 1/4 times x. 00:31:49.595 --> 00:31:57.500 You need to practice working these examples until you've mastered the skills and concepts involved.