WEBVTT 00:00:06.850 --> 00:00:14.105 Let's take a look at graphing functions of the form y equals sine (x minus C) and y equals cosine (x minus C). 00:00:14.107 --> 00:00:21.535 In these forms, C shifts your graphs horizontally to the right or the left. Let's look at our basic cosine curve. 00:00:21.537 --> 00:00:29.036 We know what it looks like. The period is 0 to 2 pi, so if C is positive it's going to take our graph and shift it to the 00:00:29.038 --> 00:00:34.630 right C units. So the first x-coordinate of the quarter point in this case would be C, and then the last x-coordinate 00:00:34.632 --> 00:00:43.629 would be C plus 2 pi. If C is negative, we're going to shift to the left and so your first x-coordinate of your quarter 00:00:43.631 --> 00:00:51.631 point will be C, and again the period's 2 pi, so the last x-coordinate will be C plus 2 pi. Let's look at an example. 00:00:51.633 --> 00:01:00.127 Let's look at y equals sine (x minus pi over 2). Well we know what our basic sine curve looks like. I've listed the 00:01:00.129 --> 00:01:08.878 quarter points for y equals sine x. So here, remember the amplitude is going to be the absolute value of A which is 00:01:08.880 --> 00:01:20.678 1 in this case, so the amplitude is 1. And the period is 2 pi over B, and notice in this case B is 1, so the period is 00:01:20.680 --> 00:01:29.269 still 2 pi. But the graph is going to be shifted pi over 2 to the right. So that means that the first x-coordinate of the 00:01:29.271 --> 00:01:39.018 quarter point is going to be pi over 2. So let's figure out all the x-coordinates. So since the period is 2 pi, one fourth 00:01:39.020 --> 00:01:46.954 of that period is pi over 2. That's going to represent the interval between each x-coordinate. So what we're going 00:01:46.956 --> 00:01:53.545 to do is, we're going to add pi over 2 to each of these x-coordinates to get our next x-coordinate. Pi over 2 and 00:01:53.547 --> 00:02:09.230 pi over 2 is pi. Pi plus pi over 2 is 3 pi over 2. 3 pi over 2 plus pi over 2 is 2 pi, and 2 pi plus pi over 2 would be 00:02:09.232 --> 00:02:21.320 5 pi over 2. So again, all we're doing is we're adding one fourth of the period to get these x-coordinates. Now the 00:02:21.322 --> 00:02:29.573 y-coordinates haven't changed at all. So we're going to have (pi over 2, 0), (pi, 1), (3 pi over 2, 0), (2 pi, -1), 00:02:29.575 --> 00:02:45.846 (5 pi over 2, 0), so the graph is going to look like this. 00:02:45.848 --> 00:02:59.097 So let's see (pi over 2, 0), (pi, 1), (3 pi over 2, 0), (2 pi, -1), (5 pi over 2, back to 0). Ok, here we've drawn one 00:02:59.099 --> 00:03:07.595 cycle or period for this function. Of course the domain is all real numbers, so it's going to continue in both directions. 00:03:09.840 --> 00:03:18.618 Let's look at sketching graphs of functions of the form y equals A sine (Bx minus C) and y equals A cosine (Bx minus C). 00:03:18.620 --> 00:03:25.551 I've listed the characteristics of the graphs here in this box, and I want to show you why the period is 2 pi over B 00:03:25.553 --> 00:03:33.896 and the phase shift is C over B. The first thing I want to talk about is B. You need to make sure that B, the coefficient 00:03:33.898 --> 00:03:41.646 of x, is positive first. If it's not, you're going to rewrite the function using even/odd properties to make B positive. 00:03:41.648 --> 00:03:51.077 So let's also remember, if you recall, when you're graphing y equals A sine x and y equals A cosine x, recall that 00:03:51.079 --> 00:03:57.583 the period for those graphs is 2 pi. So that means they complete one cycle when x is between 0 and 2 pi. 00:03:57.585 --> 00:04:04.762 Well graphs in either one of these forms will complete one cycle when Bx minus C is between 0 and 2 pi. So let's 00:04:04.764 --> 00:04:14.766 set that up. So we have Bx minus C between 0 and 2 pi, and we've made B positive remember by even/odd 00:04:14.768 --> 00:04:35.539 properties if we need to. Now we're going to solve this for x. So let's add C to both sides and divide by B. 00:04:35.541 --> 00:04:43.038 So what we can see here is that graphs in either one of those forms are going to complete one cycle when x is on 00:04:43.040 --> 00:04:50.039 the interval C over B to 2 pi over B plus C over B. That's when it will complete one cycle. Well the length of that cycle 00:04:50.041 --> 00:05:01.563 is the period. So the period is going to be when you take 2 pi over B plus C over B, if you subtract C over B 00:05:01.565 --> 00:05:11.315 we can see that we get 2 pi over B. So the period is 2 pi over B, and we can see here that this expression right here 00:05:11.317 --> 00:05:18.744 is the phase shift. Remember that the first quarter point of these functions used to start at 0. Now it's going to start 00:05:18.746 --> 00:05:26.248 at C over B, so we call C over B the phase shift. So the phase shift is C over B, period is 2 pi over B, 00:05:26.250 --> 00:05:33.180 amplitude is absolute value of A, and the range is negative absolute value of A to the positive absolute value of A. 00:05:33.182 --> 00:05:44.677 Let's look at an example. Here I want to sketch the graph of y equals 4 cosine (2x plus pi). Now, I need to figure out 00:05:44.679 --> 00:05:58.200 what's the amplitude. Well remember the amplitude is the absolute value of A which in this case is 4. The range then 00:05:58.203 --> 00:06:14.139 is going to be negative 4 to 4. The period is going to be 2 pi over B. Ok, let's look in this example. What's B? 00:06:14.141 --> 00:06:31.775 Notice that B is positive. B is 2. 2 pi over 2, so the period for this graph is pi. And what's the phase shift? 00:06:31.777 --> 00:06:41.662 Well the phase shift is going to be C over B. Now notice that I see a plus sign. If you like you can actually 00:06:41.664 --> 00:06:54.754 factor out the 2. If I factor out the 2 I'm going to have x minus a pi over 2. Oops I need two minuses, minus a negative. 00:06:54.756 --> 00:07:07.532 There we go. So I'm just changing the plus sign to subtracting, so this is x minus. I can see here this is my phase shift. 00:07:07.534 --> 00:07:15.032 Some students like to factor out the B. Some students just like to use the formulas. Whichever works for you. 00:07:15.034 --> 00:07:25.463 So I know that the period is pi, and the phase shift is, we said C was a negative pi, over 2. 00:07:25.465 --> 00:07:32.553 So the phase shift is negative pi over 2. Ok, so now we need to figure out the quarter points for this graph. 00:07:32.555 --> 00:07:40.800 Well I've listed the quarter points for y equals cosine x. B and C are going to change the x-coordinates of those 00:07:40.802 --> 00:07:46.398 quarter points. So remember it used to start at 0. Well now it's going to start at the phase shift which is 00:07:46.400 --> 00:07:54.237 negative pi over 2. Now, how do I get the next x-coordinate? Remember how to do that? You take one fourth of the 00:07:54.239 --> 00:08:04.921 period. Well one fourth of the period is just pi over 4, so I want to be adding pi over 4 to each of these x-coordinates. 00:08:04.923 --> 00:08:14.672 Well negative pi over 2 is negative 2 pi over 4. If you add pi over 4, you're going to get negative pi over 4. And again 00:08:14.674 --> 00:08:25.355 add pi over 4 to negative pi over 4, you get 0. Add pi over 4 to 0 you get pi over 4, and lastly, add pi over 4 to 00:08:25.357 --> 00:08:32.853 pi over 4. You get 2 pi over 4 which is pi over 2. So now we've found all of the x-coordinates. How do we find 00:08:32.855 --> 00:08:39.602 the y-coordinates? Well the y-coordinates are found by taking the y-coordinates of these quarter points and 00:08:39.604 --> 00:08:51.356 multiplying by A which in this case is 4. So let's see. 1 times 4 is 4. 0 times 4 is 0. Negative 1 times 4, negative 4. 00:08:51.358 --> 00:09:02.469 0 times 4 is still 0, and then 1 times 4 is 4. So now we've found our quarter points. We can plot those points. 00:09:02.471 --> 00:09:22.750 So let's see. Negative pi over 2, and then I need 4 and negative 4. And so we have (-pi over 2, 4), (0, -4), (pi over 4, 0), 00:09:22.752 --> 00:09:34.090 back up to 4. Alright, so here's one cycle for this particular function, and of course the domain is all real numbers, 00:09:34.092 --> 00:09:42.585 so it's going to keep going in either direction. Let's look at another example. 00:09:44.200 --> 00:09:56.363 In the previous example, we had y equals 4 cosine (2x plus pi). We can rewrite this as 2x minus a negative pi 00:09:56.365 --> 00:10:06.703 so that we can see the A is 4, the B is 2, and the C is negative pi. Well in our next example, y equals 2 sine 00:10:06.705 --> 00:10:15.269 (negative 4x plus pi). We need to rewrite it for a different reason. Let's write down this function. We have 2 sine 00:10:15.271 --> 00:10:23.953 (negative 4x plus pi). Well I see a plus sign here and I see a plus sign here, but it turns out the phase shift is not 00:10:23.955 --> 00:10:32.456 negative because more importantly I have B that's negative, and whenever B is negative your first step is to rewrite 00:10:32.458 --> 00:10:39.980 it using even/odd identities. So if you recall, the sine is odd. Which means what? It means the sine of negative x 00:10:39.982 --> 00:10:52.727 is negative sine x. So what I can do is I can factor that negative out, and then I actually have 4x minus pi, and then 00:10:52.729 --> 00:11:04.981 using even/odd identitites this is equal to negative 2 sine of (4x minus pi). So now I can see that my A is negative 2, 00:11:04.983 --> 00:11:16.436 my B is 4, and my C is pi. So now let's figure out the amplitude. Well the amplitude is going to be the absolute value 00:11:16.438 --> 00:11:35.961 of A which absolute value of negative 2 is 2. The range is going to be negative 2 to 2. Period is going to be 2 pi over B. 00:11:35.963 --> 00:11:56.891 My B remember is 4, so that's pi over 2. And my phase shift is C over B, so it's positive this time. Ok, so we have 00:11:56.893 --> 00:12:01.988 the quarter points for y equals sine x. We know what those are. You should know those very well. So how do we get 00:12:01.990 --> 00:12:09.825 the quarter points here? Well, the x-coordinates, the first x-coordinate starts at your phase shift, so pi over 4 is the 00:12:09.827 --> 00:12:14.328 first x-coordinate. Now how do we get the next one? Do you remember? Exactly. 00:12:14.330 --> 00:12:23.671 You have to take one fourth of the period which is pi over 8. So this time I want to add pi over 8 to each of the 00:12:23.673 --> 00:12:33.105 x-coordinates. So let's see. Pi over 4 is the same thing as 2 pi over 8 plus pi over 8 gives you 3 pi over 8. 00:12:33.107 --> 00:12:46.194 3 pi over 8 plus pi over 8 is 4 pi over 8 which reduces to pi over 2. 4 pi over 8 plus pi over 8 is 5 pi over 8, and 00:12:46.196 --> 00:12:54.539 last x-coordinate, 5 pi over 8 plus pi over 8 gives you 6 pi over 8 which reduces to 3 pi over 4. 00:12:54.541 --> 00:13:01.627 Ok, now how do we get the y-coordinates? Remember, to get the y-coordinates you multiply these y-coordinates by A. 00:13:01.629 --> 00:13:12.219 Be careful, remember A is negative 2, so 0 times negative 2 is still 0, 1 times negative 2, 0 times negative 2, 00:13:12.221 --> 00:13:25.158 negative 1 times negative 2, and 0 times negative 2. So now we can plot these points. Let's see. 00:13:25.160 --> 00:13:43.838 Let's make this pi over 8, so pi over 4 here, 3 pi over 8, pi over 2, and 3 pi over 4. Then we have...(keeps writing) 00:13:43.840 --> 00:13:57.769 Ok so our first point (pi over 4, 0), and then it's decreasing first (3 pi over 8, negative 2), back to 0, 2, back to 0, 00:13:57.771 --> 00:14:04.500 and of course since we have one cycle we can continue this graph in either direction. Just continue the graph. 00:14:04.502 --> 00:14:11.025 So remember, very important when B is negative that you rewrite it first to get B positive. 00:14:12.750 --> 00:14:20.022 Let's look at sketching graphs of functions of the form y equals A sine (Bx minus C) plus D and y equals A cosine 00:14:20.024 --> 00:14:26.772 (Bx minus C) plus D. You may recall from algebra that when you add a constant to a function it shifts the graph 00:14:26.774 --> 00:14:36.019 vertically. The same thing is going to happen here. So remember your quarter points for A sine (Bx minus C). 00:14:36.021 --> 00:14:41.730 Well all you're going to do is you're going to add D to the y-coordinates of all the quarter points of this function, 00:14:41.732 --> 00:14:48.226 or of course if it's a cosine you're going to add D to all the quarter points of A cosine (Bx minus C). 00:14:48.228 --> 00:14:53.570 Now, remember we're shifting vertically so that's actually going to change the range. Remember the range when we 00:14:53.572 --> 00:14:59.071 don't have a D, when D is zero, is negative absolute value of A to [absolute value of] A. So to get the range 00:14:59.073 --> 00:15:06.001 for functions in this form we have to add D to get the range. We have to add negative absolute value of A plus D, 00:15:06.003 --> 00:15:13.093 so you're adding D to all the y-coordinates, to absolute value of A plus D. Period and phase shift do not change. 00:15:13.095 --> 00:15:25.091 Let's look at an example. I have y equals 3 cosine (6x minus pi) plus 2. So here I can see that A is 3, so I know my 00:15:25.093 --> 00:15:40.872 amplitude is 3. Absolute value of 3 is 3. But what's the range? Well, remember the range is going to be negative 3 and 00:15:40.874 --> 00:15:57.553 then we have to add 2 to positive 3 plus 2, so the range is negative 1 to 5. The period is going to be 2 pi over B. 00:15:57.555 --> 00:16:14.235 Ok so my B is 6, my C is pi, and my D is 2. So let's see. 2 pi over 6, my period is going to be pi over 3. And my 00:16:14.237 --> 00:16:26.984 phase shift is going to be C over B, so pi over 6. Now, I've listed the quarter points for y equals cosine x. Let's get 00:16:26.986 --> 00:16:37.265 the quarter points for this function. The first x-coordinate is your phase shift, so pi over 6. Now, remember how to 00:16:37.267 --> 00:16:46.412 what you add to each of these quarter points. We need to find one fourth of the period, one fourth of pi over 3 is 00:16:46.414 --> 00:16:55.603 pi over 12, so what I want to add to each of these x-coordinates is pi over 12. Think of pi over 6 as 2 pi over 12. 00:16:55.605 --> 00:17:05.538 When you add pi over 12 to 2 pi over 12 you get 3 pi over 12 which reduces to pi over 4. 3 pi over 12 plus pi over 12 00:17:05.540 --> 00:17:18.219 is 4 pi over 12 which reduces to pi over 3. 4 pi over 12 plus pi over 12 is 5 pi over 12, and then 5 pi over 12 plus 00:17:18.221 --> 00:17:27.812 pi over 12 gives me 6 pi over 12 which reduces to pi over 2. So notice that my x-coordinates are affected by B and C. 00:17:27.814 --> 00:17:32.312 Now we need to get the y-coordinates. Well remember when we didn't have a D we got the y-coordinates by 00:17:32.314 --> 00:17:39.902 multiplying our y-coordinates by A. Well now we multiply our y-coordinates by A, but then we add 2. So let's go. 00:17:39.904 --> 00:17:52.334 It's 1, A is 3, so we're going to do 1 times 3 is 3 plus 2 gives me 5. 0 times 3 is 0 plus 2 is 2. 00:17:52.336 --> 00:18:05.837 Negative 1 times 3 is negative 3 plus 2 is negative 1. 0 times 3 is 0 plus 2 is 2. And finally, 1 times 3 is 3 plus 2 is 5. 00:18:05.839 --> 00:18:20.114 So let's plot these quarter points. 00:18:20.116 --> 00:18:32.793 And let's see. I need to go up to 5 and down to negative 1. Ok, so we have pi over 6, we're up at 5. 00:18:32.795 --> 00:18:49.293 Pi over 4 we're at 2. Pi over 3, we're down at negative 1, back to 2 and back to 5. Alright. 00:18:49.295 --> 00:18:57.044 So here's one cycle for this function. Of course our domain is all real numbers so it doesn't stop it continues in both 00:18:57.046 --> 00:19:05.662 directions forever. So I want to reiterate. To get your x-coordinates you find the phase shift. Then you add one 00:19:05.664 --> 00:19:14.913 fourth of the period. To get your y-coordinates you multiply the old y-coordinates by A, and then you add D. 00:19:17.000 --> 00:19:24.754 Suppose we want to determine the equation of a function given the graph. Look at this graph. Suppose we were asked 00:19:24.756 --> 00:19:33.442 to choose an equation of a function from this list below that matches this graph. Take a minute and you'll realize that 00:19:33.444 --> 00:19:41.188 all five of these equations match this graph. So we're going to have to know more information in order to determine 00:19:41.190 --> 00:19:49.688 a unique function. Let's look at this example here. We're given the graph, and we want to determine the equation of a 00:19:49.690 --> 00:19:56.939 function of the form, and we're going to be told it's a cosine curve, and we're going to be told B is positive. 00:19:56.941 --> 00:20:04.188 And also we're going to associate these five labeled quarter points with the five quarter points for y equals cosine x 00:20:04.190 --> 00:20:11.989 over the interval 0 to 2 pi. Well let's write those quarter points down. We should know what they are. For y equals 00:20:11.991 --> 00:20:28.943 cosine x those quarter points are (0, 1), (pi over 2, 0), (pi, -1), (3 pi over 2, 0), (2 pi, 1), and we can see the quarter 00:20:28.945 --> 00:20:51.270 points for this one (pi over 6, -4), (pi over 3, 0), (pi over 2, 4), (2 pi over 3, 0), and (5 pi over 6, -4). So this 00:20:51.272 --> 00:20:58.511 quarter point is associated with this one and so on. So what this tells me, I can see that the phase shift is pi over 6. 00:20:58.513 --> 00:21:06.443 Also, we can figure out the period because the first quarter point is pi over 6, or the first x-coordinate of the quarter 00:21:06.445 --> 00:21:11.943 point is pi over 6, and the x-coordinate of the last quarter point is 5 pi over 6. The difference between those two 00:21:11.945 --> 00:21:21.788 x-coordinates gives us the period. So we can figure out the period. We can take 5 pi over 6 and subtract pi over 6 00:21:21.790 --> 00:21:33.221 to get 4 pi over 6 and reduce that. That's 2 pi over 3, so I know what my period is. Now, do you remember how to 00:21:33.223 --> 00:21:42.472 find the period? Exactly. So to get the period you take 2 pi and divide by B. Remember we're assuming B is positive 00:21:42.474 --> 00:21:55.496 this time, so we could easily see that in this case B is 3. Ok so now we know B. We want to come up with C and A. 00:21:55.498 --> 00:22:02.089 Well remember we know the phase shift is pi over 6, and remember to get the phase shift you take C over B. 00:22:02.091 --> 00:22:14.358 So I know C over B is pi over 6, and we just found that B was 3, so we know C over 3 is pi over 6. We can 00:22:14.360 --> 00:22:28.827 solve this for C, so C is going to be 3 pi divided by 6 which is pi over 2. So we know B, we know C, we need to find A. 00:22:28.829 --> 00:22:37.024 Well looking at this graph, we can easily determine the amplitude is 4. But be careful. A is not positive 4. Notice 00:22:37.026 --> 00:22:46.018 that this cosine is increasing first which tells me that A is negative 4. So now we can write our function. 00:22:46.020 --> 00:23:01.480 A is negative 4, B was 3, and C was pi over 2. So we can determine a unique function if we make the assumptions. 00:23:01.482 --> 00:23:07.749 We're going to be told whether it's a sine or a cosine, in this case it's a cosine. We're going to be told B is positive, 00:23:07.751 --> 00:23:14.073 and we're going to given five quarter points of one cycle that associates or corresponds to the quarter points, 00:23:14.075 --> 00:23:18.664 in this case for y equals cosine x for x between 0 and 2 pi. 00:23:18.666 --> 00:23:24.919 Practice working these examples until you've mastered the skills and concepts involved.