WEBVTT
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Let's take a look at graphing functions of the form y equals sine (x minus C) and y equals cosine (x minus C).
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In these forms, C shifts your graphs horizontally to the right or the left. Let's look at our basic cosine curve.
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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
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right C units. So the first x-coordinate of the quarter point in this case would be C, and then the last x-coordinate
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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
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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.
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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
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quarter points for y equals sine x. So here, remember the amplitude is going to be the absolute value of A which is
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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
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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
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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
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of that period is pi over 2. That's going to represent the interval between each x-coordinate. So what we're going
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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
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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
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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
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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),
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(5 pi over 2, 0), so the graph is going to look like this.
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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
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cycle or period for this function. Of course the domain is all real numbers, so it's going to continue in both directions.
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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).
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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
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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
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of x, is positive first. If it's not, you're going to rewrite the function using even/odd properties to make B positive.
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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
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the period for those graphs is 2 pi. So that means they complete one cycle when x is between 0 and 2 pi.
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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
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set that up. So we have Bx minus C between 0 and 2 pi, and we've made B positive remember by even/odd
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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.
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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
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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
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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
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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
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is the phase shift. Remember that the first quarter point of these functions used to start at 0. Now it's going to start
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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,
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amplitude is absolute value of A, and the range is negative absolute value of A to the positive absolute value of A.
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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
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what's the amplitude. Well remember the amplitude is the absolute value of A which in this case is 4. The range then
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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?
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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?
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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
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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.
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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.
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Some students like to factor out the B. Some students just like to use the formulas. Whichever works for you.
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So I know that the period is pi, and the phase shift is, we said C was a negative pi, over 2.
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So the phase shift is negative pi over 2. Ok, so now we need to figure out the quarter points for this graph.
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Well I've listed the quarter points for y equals cosine x. B and C are going to change the x-coordinates of those
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quarter points. So remember it used to start at 0. Well now it's going to start at the phase shift which is
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negative pi over 2. Now, how do I get the next x-coordinate? Remember how to do that? You take one fourth of the
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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.
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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
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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
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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
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the y-coordinates? Well the y-coordinates are found by taking the y-coordinates of these quarter points and
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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.
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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.
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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),
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back up to 4. Alright, so here's one cycle for this particular function, and of course the domain is all real numbers,
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so it's going to keep going in either direction. Let's look at another example.
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In the previous example, we had y equals 4 cosine (2x plus pi). We can rewrite this as 2x minus a negative pi
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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
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(negative 4x plus pi). We need to rewrite it for a different reason. Let's write down this function. We have 2 sine
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(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
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negative because more importantly I have B that's negative, and whenever B is negative your first step is to rewrite
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it using even/odd identities. So if you recall, the sine is odd. Which means what? It means the sine of negative x
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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
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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,
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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
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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.
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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
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the quarter points for y equals sine x. We know what those are. You should know those very well. So how do we get
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the quarter points here? Well, the x-coordinates, the first x-coordinate starts at your phase shift, so pi over 4 is the
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first x-coordinate. Now how do we get the next one? Do you remember? Exactly.
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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
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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.
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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
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last x-coordinate, 5 pi over 8 plus pi over 8 gives you 6 pi over 8 which reduces to 3 pi over 4.
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Ok, now how do we get the y-coordinates? Remember, to get the y-coordinates you multiply these y-coordinates by A.
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Be careful, remember A is negative 2, so 0 times negative 2 is still 0, 1 times negative 2, 0 times negative 2,
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negative 1 times negative 2, and 0 times negative 2. So now we can plot these points. Let's see.
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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)
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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,
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and of course since we have one cycle we can continue this graph in either direction. Just continue the graph.
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So remember, very important when B is negative that you rewrite it first to get B positive.
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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
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(Bx minus C) plus D. You may recall from algebra that when you add a constant to a function it shifts the graph
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vertically. The same thing is going to happen here. So remember your quarter points for A sine (Bx minus C).
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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,
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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).
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Now, remember we're shifting vertically so that's actually going to change the range. Remember the range when we
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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
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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,
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so you're adding D to all the y-coordinates, to absolute value of A plus D. Period and phase shift do not change.
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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
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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
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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.
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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
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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
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the quarter points for this function. The first x-coordinate is your phase shift, so pi over 6. Now, remember how to
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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
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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.
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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
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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
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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.
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Now we need to get the y-coordinates. Well remember when we didn't have a D we got the y-coordinates by
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multiplying our y-coordinates by A. Well now we multiply our y-coordinates by A, but then we add 2. So let's go.
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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.
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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.
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So let's plot these quarter points.
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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.
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Pi over 4 we're at 2. Pi over 3, we're down at negative 1, back to 2 and back to 5. Alright.
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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
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directions forever. So I want to reiterate. To get your x-coordinates you find the phase shift. Then you add one
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fourth of the period. To get your y-coordinates you multiply the old y-coordinates by A, and then you add D.
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Suppose we want to determine the equation of a function given the graph. Look at this graph. Suppose we were asked
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to choose an equation of a function from this list below that matches this graph. Take a minute and you'll realize that
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all five of these equations match this graph. So we're going to have to know more information in order to determine
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a unique function. Let's look at this example here. We're given the graph, and we want to determine the equation of a
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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.
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And also we're going to associate these five labeled quarter points with the five quarter points for y equals cosine x
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over the interval 0 to 2 pi. Well let's write those quarter points down. We should know what they are. For y equals
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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
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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
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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.
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Also, we can figure out the period because the first quarter point is pi over 6, or the first x-coordinate of the quarter
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point is pi over 6, and the x-coordinate of the last quarter point is 5 pi over 6. The difference between those two
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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
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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
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find the period? Exactly. So to get the period you take 2 pi and divide by B. Remember we're assuming B is positive
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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.
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Well remember we know the phase shift is pi over 6, and remember to get the phase shift you take C over B.
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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
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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.
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Well looking at this graph, we can easily determine the amplitude is 4. But be careful. A is not positive 4. Notice
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that this cosine is increasing first which tells me that A is negative 4. So now we can write our function.
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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.
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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,
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and we're going to given five quarter points of one cycle that associates or corresponds to the quarter points,
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in this case for y equals cosine x for x between 0 and 2 pi.
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Practice working these examples until you've mastered the skills and concepts involved.