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# Finding a sinusoidal function from its graph

## Video transcript

We're asked to determine a function of the form y equals a sine of bx or y is equal to a cosine bx represented by the graph below. So we need to figure out maybe what the a's are, what the b's are, and whether this is a sine or a cosine function. So let's see what clues there are. So the first thing that I notice is that whatever this function is, when x is equal to 0, it does not equal 0. It is equal to negative 2. So based on that, based on what we know about sine and cosine functions, do you think that this is going to be of the form y is equal to a sine of bx or of the form y is equal to a cosine of bx? Without even knowing what the a's and b's are, do you think this is going to be a sine function or cosine function? Well, let's think about what sine of 0 is. If you take sine of 0, we already know that that is equal to 0. What is cosine of 0? Cosine of 0 is equal to 1. So it would be very hard-- and especially in this form, it would be impossible-- if sine of 0 is 0, to multiply 0 by something to get to negative 2. So it can't be a sine function of this form. You might say, well, the cosine of 0 is 1. But here, it's negative 2. But at least if you have a 1, you can then multiply it by something to get to a negative 2. So what we now know is that we are at least of this form. But now we have to figure out what the a's and b's are going to be equal to. We know that this function is y is equal to a cosine of bx. So the next question I ask you is what is a going to be. Well, let's think about it. We already saw if we just had cosine of bx, when x is equal to 0, cosine of b times 0 would just be cosine of 0. And it would get us to 1. But we're not 1. We're at negative 2. It looks like [? it ?] took a cosine function, and at least, when x is equal to 0, we multiplied it by negative 2. So this should be negative 2. So now we have a little bit more filled in of what we actually have. We know that it's y is equal to negative 2 cosine of bx. And this gels with what we see right over here, the amplitude here. You see that the difference between the maximum value and the minimum value, or the minimum and the maximum is 4, 1/2 of that is 2. Or another way you think about it, we're varying 2 from this center point. And over here, if you think about the amplitude-- the amplitude is the absolute value of this number right over here. The amplitude is equal to the absolute value of this negative 2, which is indeed equal 2. So it's consistent so far. Now let's think about what b here is. And maybe we can use our knowledge of what the period of a periodic function is to think about what b might be. Well, let's look over here. What is the period of this periodic function? Well, let's draw one period. So if we use this as our starting point-- or one cycle, I should say. Let's draw one cycle. If you view that as our starting point, at 2 pi over 3, we have completed that cycle. And then we could start the next cycle. We repeat the pattern over again. Then you start the next cycle. So based on that, what is the period? Well, it's the length that you need to go in x to complete one cycle. So that length right over there, you see, is 2 pi over 3. So the period is 2 pi over 3. And given that the period here is 2 pi over 3, can you figure out what b is going to be? Well, the period of this is going to be equal to 2 pi over the absolute value of b. And you can solve this multiple ways. You can multiply both sides by 3 and the absolute value of b. And you would be left with the absolute value of b is equal to 3, which means that b could be equal to positive or negative 3. And so you might say, well, Sal, what do I use? Does b equal positive 3 or negative 3? And so the next question I'll ask you is, for a cosine function, do you get different values if you were to make this a cosine of 3x or a cosine of negative 3x? Do you get different values? Well, if you play with the unit circle a little bit. So I'm going to draw a little rough unit circle right over here. Remember, cosine is the x-coordinate where we intersect the unit circle. And if we go in the positive angle direction, if we go in that direction, our x-coordinate-- it starts at 1, and then it gets a little bit shorter. If we go in the negative direction, it starts at 1, and then it gets a little bit shorter. And so you can experiment this with a good bit. But you'll see that cosine of 3x-- and this is only the case for cosine, not for sine-- cosine of 3x is equal to cosine of negative 3x. So you can actually pick either positive 3 or negative 3. But for simplicity in this case right over here, I'll just go with the positive 3. So this could be the graph of-- and now we get our drum roll-- y is equal to negative 2 times the cosine of-- I said I wouldn't do the negative-- the cosine of positive 3x. And we are done.