Transforming sinusoidal graphs: vertical & horizontal stretches

We are asked to graph the function y is equal to negative 2.5 cosine of 1/3 x on the interval, 0 to 6 pi, including the endpoints. So let me do my best attempt at graphing that. And to start off, I'm going to graph with the simplest function, or the simplest version of this, or the root of this, which is just cosine of x. So let me just graph, and eventually you can kind of-- let me just graph cosine of x. So that's my y-axis. And I want to have some space here so I can eventually graph this entire thing. So let's say that this is negative 1, this is negative 2. This is positive 1, this is positive 2. And let's say that this right over here is 2 pi. And then of course that could be pi right over there. Now, the first thing I'm going to do-- let me copy this because I could use it later to graph the whole thing. So let's start off. So I'm just going to graph y is equal to cosine of x. So when x is equal to 0-- and I'm just going to do it between the interval 0 and 2 pi. Obvious it's a periodic function, it'll keep going in the negative and the positive directions. So what happens when x is equal to 0? What is cosine of x? Well cosine of 0 is 1. What about when x is equal to pi? What is cosine of pi? Well cosine of pi is negative 1. And then what's cosine of 2 pi? Well that's 1 again. We get back-- we've completed a period, or we've completed an entire cycle. And 2 pi is the period of cosine of x. So this is one cycle right over here. I could keep going if I wanted to, but the whole point, I just wanted to graph this one cycle between 0 and 2 pi. Now what I want to think about is, what happens to this graph? Instead of graphing y equals cosine of x-- let me draw some graph paper again. Instead of drawing y is equal to cosine of x, I'm going to draw y is equal to cosine of 1/3 x. So the only difference between that and that is now I'm multiplying the x by 1/3. What's going to happen to the graph over here? How is this going to change instead of being an x, if it's a 1/3 x? What's going to happen over here? And now I'm going to do it over the entire interval between zero and six pi. So let me just make sure I have enough space. So that's 3 pi, 4 pi, 5 pi, and 6 pi. What's going to happen to this graph? Well, there's a couple of ways to think about it. The easiest might just be to say, well to complete an entire cycle, we're going to go 1/3 as fast. Or we're going to go three times slower. Or if you just want to think about the period here, what's the period of cosine of 1/3 x? Well the period is going to be 2 pi divided by the absolute value of this coefficient right over here. So it's the absolute value of 1/3, which is just 1/3. So the period is 2 pi over 1/3, which is the same thing as 2 pi times 3, which is 6 pi. Which gels with the intuition. That's going to take three times as much time to get whatever we input into the cosine function to get back to 2 pi. Because whatever we take x, we're taking 1/3 of it. So to get to 2 pi, you can't just have x equals 2 pi. x now has to equal 6 pi to get 2 pi inputted into the cosine function. So the period is now 6 pi. At x is equal to 0, 1/3 times 0 is 0, and the cosine of 0 is 1. When x is equal to 6 pi, you have 6 pi divided by 3 is 2 pi. Cosine of 2 pi is equal to 1. And if you want to go in between, over here to go in between, we tried pi. But over here, we could try 3 pi. When x is 3 pi, you have cosine of 1/3 of 3 pi, that's cosine of pi. Cosine of pi is negative 1. So when x is equal to 3 pi, we have cosine of 1/3 times 3 pi is negative 1. So it's going to look something like this. Trying my best attempt. to draw it. So it's going to look something like this. So you see, to go from y equals cosine of x to y equals cosine of 1/3 x, it essentially stretched out to this function by a factor of 3. You can see this period is three times longer. The period here was 2 pi. All right, well there's only one more transformation we need in order to get to the function that they're asking us about. We just have to, instead of having a cosine of 1/3 x, we just have to negative 2.5 cosine of 1/3 x. So let's try to draw that. So let me put my axis here again. And let me label it. So that's 2 pi, 3 pi, 4 pi, 5 pi, and 6 pi. And our goal now is to draw the graph of y is equal to-- and we're just doing it over between 0 and 6 pi here. We only did it between 0 and 2 pi here. Obviously they're all periodic, they all keep going on and on. But now we want to graph y is equal to negative 2.5 times cosine of 1/3 x. So given this change, we're now multiplying by negative 2.5, what is going to be-- well actually, let's think about a few things. What was the amplitude in the first two graphs right over here? Well there's two ways to think about it. You could say the amplitude is half the difference between the minimum and the maximum points. In either of these case, the minimum is negative 1, maximum is 1. The difference is 2, half of that is 1. Or you could just say it's the absolute value of the coefficient here, which is implicitly a 1. And the absolute value of 1 is, once again, 1. What's going to be the amplitude for this thing right over here? Well the amplitude is going to be the absolute value of what's multiplying the cosine function. So the amplitude in this case, do it in green, the amplitude is going to be equal to the absolute value of negative 2.5, which is equal to 2.5. So given that, how is multiplying by negative 2.5 going to transform this graph right over here? Well let's think about it. If it was multiplying by just a positive 2.5, you would stretch it out. At each point it would go up by a factor of 2 and 1/2. But it's a negative 2.5, so at each point, you're going to stretch it out and then you're going to flip it over the x-axis. So let's do that. So when x was 0, you got 1 in this case. But now we're going to multiply that by negative 2.5, which means you're going to get to negative 2.5. So let me draw negative 2.5 right over there. So that's negative 2.5. That'd be negative-- let me make it clear. This would be negative 3 right over here, this would be positive 3. So that number right over there is negative 2.5. And let me draw a dotted line there. It could serve to be useful. Now when cosine of 1/3 x is 0, it doesn't matter what you multiply it by, you're still going to get 0 right over here. Now, when cosine of 1/3 x was negative 1, which was the case when x is equal to 3 pi, what's going to happen over here? Well cosine of 1/3 x, we see, is negative 1. Negative 1 times negative 2.5 is positive 2.5. So we're going to get to positive 2.5, which is right-- let me draw a dotted line over here. We're going to get to positive 2.5, which is right over there. And then when cosine of 1/3 x is equal to 0, doesn't matter what we multiply it by, we get to 0. And then finally, when x is at 6 pi, cosine of 1/3 x is equal to 1. What's that going to be when you multiply it by negative 2.5? Well it's going to be negative 2.5. So we're going to get back over here. So we're ready to draw our graph. It looks something-- let me do that in magenta color since that's what the color I wrote this in. It will look like this. I can draw it as a solid line. So it will look like that. So you saw what happened. By putting this 1/3 here, it stretched out the graph. It increased the period by a factor of 3. And then multiplying it by negative 2.5-- if you just multiply it by 2.5, you would just multiply that out a little bit. But now it's a negative, so not only do you increase the amplitude, but you flip it over. So it is, indeed, the case that the amplitude here is 2.5. We vary 2.5 from our middle position. Or you could say that the difference between the minimum and the maximum is 5, so half of that is 2.5. But it isn't just multiplying this graph by 2.5. If you multiply this graph by 2.5, you'd get something-- let me be a little neater. You would get something that looked something like that. But because we had a negative, we had to flip it over the x-axis. And we got this here. So this amplitude is 2.5, but it's a flipped over version of this graph.