Welcome back, everyone. So in the last video, we got introduced to the sine and cosine functions as graphs. And we also saw ways that these functions could be transformed by a vertical shift. Now what we're going to be talking about in this video is new ways to transform the sine and cosine function by using the amplitude or reflection. Now this might sound a bit complicated, but don't sweat it because what we're going to learn in this video is that these types of transformations you can do to the sine and cosine are actually very similar to transformations we've already learned about for functions back in algebra. So you're going to find, I think, that a lot of these graphs are actually very straightforward. So without further ado, let's get right into this.

Now when it comes to the amplitude, the amplitude is a number, and it's a number that affects how tall the peaks of your graph are. Now I want you to recall that the sine and cosine function are both repeating waves. And if you find the distance from the midline to the peaks or valleys of that wave, that is going to tell you the amplitude. So let's say for example that we're looking at a sine graph, and we take a midline and we draw it right through the middle of this graph. Well, if we draw a midline horizontally through this graph, we can find the amplitude by looking at the distance from this midline to either a peak or a valley to this wave. And we can see that that distance is 1 unit, meaning the amplitude of this sine function is 1.

But what would happen if we had a different amplitude, or how could we have a different amplitude? Well, let's take a look at a different situation where rather than having a 1 in front of our sine function, we have 2. So to figure out what this looks like I can plot some points on this graph by taking all the outputs of the sine and multiplying them by 2. We would have 2×0, which is 0 meaning we would start here on our graph. Now we get π2 on the x-axis, we would have 2×1, which is 2. When we get π on the x-axis, we'd have 2×0, which is again just 0. Then we would have 2×-1, which is negative 2. Then we would have 2×0, which is 0.

So if you go ahead and draw this graph by connecting these points with a smooth curve, you're going to get a function that looks something like this. And I want you to notice what this function looks like for 2sin x. Notice it's very similar to what we had for the sine of x, except it appears that our graph has been vertically stretched in some kind of way, like it's taller. Well, that's the idea of changing the amplitude. It's literally just like a vertical stretch that we learned about back in algebra. And just like we had a number in front of the function, we now have a number in front of the sine. So if you want to change the amplitude of your graph, you just need to change this number in front of the sine or cosine function, and that's really the main idea of the amplitude. We can see that we multiply the sine function by 2, so our amplitude for this function was 2. And likewise, when we just had a 1 in front of the sine function, our amplitude was 1.

Now, another thing that you'll frequently see in this course is situations where you have a negative amplitude. And when the amplitude is negative, you're going to get a graph that is reflected over the x-axis. So to see what this looks like, well, let's plot another graph. So let's say that we have -sin x. Well, if we want to find what this looks like, we can take all the output for the sign of x and make them negative. Now negative 0 would just be 0, negative one would be look something like that. We then have negative 0, once again, which is just 0. And if we had a negative negative one, that's the same thing as canceling the negative signs giving you a positive one. And then you'd have negative one times 0, which is again just 0 on the graph, meaning that it's going to look something like this when you connect to these points with a smooth curve. Now notice for the negative sine graph, it literally looks like we took the sine function and just flipped it upside down over the x-axis, and that's exactly what happened. And you may recall back in algebra that a negative number in front of the function would flip the function upside down, and that's exactly what happens with the sine and cosine as well.

So this is really the main idea behind changing the amplitude or reflecting your graph over the x-axis. Now to make sure that we're really understanding this, let's actually try an example where we have both a change in the amplitude and a reflection. So here we're asked to graph the function y=-32cos x. Now this function right here could also be written as y=-1.5cos x. 1.5 is the same thing as 32. So basically, our amplitude is 1.5. And if I go to our graph, 1.5 would be right between 1 and 2 here, and then we would have negative 1.5, which is right down there between negative one and negative 2. Now recall for the cosine function, the cosine typically starts at a high value, then goes through π2, reaches a valley, then goes back up through 3π÷2, reaches a peak, and then comes back down through 5π÷2 and keeps waving like that. But recall that we have a negative sign in front of this function. So what that's going to do is take our little cosine graph, and it's going to flip it over the x-axis. So what we're actually going to do for the cosine is we're going to start at a low value, then we're going to go up through π2, and we're going to reach a peak at π. Then we're going to go down through 3π÷2, reach a valley at 2π. Then we're going to come back up here and touch 5π÷2 on the x-axis, and the graph is going to keep waving like that. So this is what the graph will look like, and that's the answer to this problem. So hopefully, this gives you a better understanding of how to deal with changes in the amplitude or a reflection for the sine or cosine functions. Hope you found this video helpful. Thanks for watching.