[ROCK MUSIC] INSTRUCTOR: Welcome to a video on how to determine the equation of a sine and cosine function. The goal of the video is to write the equation of a sine and cosine function given a graph. 

So we've been working with transformations of the sine and cosine functions. So we should be familiar with these two equations and what these variables A, B, C, and D tell us about the function. And one of the issues with finding the equation of a sine and cosine graph is that it's not unique. We could use either sine or cosine. So really, we have a choice on which of these equations, we want to use. 

And the way we're going to do this is, first, we're going to pick a piece of the graph. And we're going to highlight one period of it. So for this graph, I'm going to isolate this piece of the graph. And you can see right away it resembles the graph of sine theta. So we're actually going to use this form to find the equation of this graph. 

The next thing I recommend is drawing a horizontal line through the center of the graph, like this. What this does is make finding the amplitude easier. And we can see, if this is a center line and this is the maximum, this distance is 1/2. Therefore, our amplitude is equal to 1/2. And so we know that A is equal to 1/2. 

And the next thing it shows us is that, typically, the sine function hugs the x-axis-- or the x-axis is the center of the graph. And we can see now it's been shifted up one. So now we know that C is equal to 1. 

So now what we have to do is find the period and the phase shift or horizontal shift. And what we can also tell that from the basic sine function, this graph has been shifted pi/4 units to the right. So that tells us that D is going to be pi/4. 

And lastly, we need to find the value of B. And we can figure that out from the period. Notice that from pi/4 to 5 pi/4, we have one complete cycle of the sine function. So 5 pi/4 minus 1 pi/4 is 4 pi over 4 or pi radians. So the period is pi radians. 

Remember, the period is equal to 2 pi divided by B. And again, we're saying that's equal to pi radians. So we can do cross products here. We have B pi is equal to 2 pi. Dividing by pi, we have B equal to 2. 

This is all the information we need. We have A, B, C, and D. So let's write our equation. y equals 1/2 sine of the quantity B times the quantity x minus D. D is pi/4 plus C, and our C value is 1. 

So let's check this out. Our amplitude is 1/2. Our period, 2 pi divided by 2, is pi radians. The phase shift, remember, if it's minus pi/4, that means right pi/4 and then up one. So there we go. 

Let's try another. First, let's take a look at the graph and determine which piece of the graph we want to focus on. We want to highlight one period of what looks like either the graph of sine or cosine. And this is why the equation that we find will not be unique. 

So when I look at this, I see the graph of a sine function starting at pi/4 going up, down, and then back up. So we'll use this piece to find the equation of this function. And again, you can see we're going to use the sine function. 

Next, we'll draw a line through the center. Looks like it's going to be at negative 1. So right away, when we draw the center line through y equals negative 1, We know that our vertical shift C will be negative 1. 

Next, the amplitude from negative 1 to positive 2 would be three units. So A is equal to 3. The phase shift again is right pi/4 units. So D is equal to pi/4. And now we have to determine the period so we can find the value of B. 

Remember, 2 pi divided by B is equal to our period. So we have 3 pi over 4 minus 1 pi over 4 would be 2 pi over 4 or pi over 2. And again, we'll perform cross products here. B times pi must equal 4 pi. Dividing both sides by pi, we have B equals 4. 

And that's all we need. Again, we're focusing on the sine function. So we have y equals 3 sine of the quantity B, which is 4. x minus D, so x minus pi/4. And the shift was down one. So C equals negative 1. 

OK, let's take a look at one more. Let's identify the piece we want to focus on. And for this one, I'm going to focus on cosine. So we'll start here at this point and end at this point. Because from there on, it starts to repeat. 

Draw a line through the center. Looks like it's at y equals negative 2. So right away, we know that C is equal to negative 2. Next, we can see the distance from our center to a maximum is two units. We're looking at this graph on the interval from negative pi/4 over 4 to 7 pi/4. 

That horizontal distance is 2 pi. So our period is 2 pi, which makes B equal to 1. And lastly, we need to find D. The horizontal shift or phase shift is left pi over four units. So D is actually negative pi/4. 

Now, there's one other thing. Typically when we graph the cosine function on the interval from 0 to 2 pi, it looks something like this. And notice that we usually start at a maximum, but now we're starting at a minimum. 

So what happened was the cosine function was reflected across the x-axis before it was shifted down two units. The result is A is equal to negative 2. So we have all the information we need now. We have y equals negative 2 cosine. Our B value is 1. So we'll leave that off. And then we have x minus D. Well, minus a negative pi/4 is plus pi/4, and then minus 2. 

Now, on this problem, I did pick a more challenging piece of the graph to find the equation to this time. Remember, these equations are not unique. So we could have chosen a different piece of this graph to find an equivalent equation in a different form. OK, I hope you found this video helpful. Thank you. 

[ROCK MUSIC]