 Welcome back to our lecture series Math 1060, Trigonometry for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In lecture 10, we're going to start talking about transforming trigonometric graphs. In this lecture and the next one, we're going to focus exclusively on the graphs of sine and cosine. But in the one after those, we're going to talk about transforming the other trigonometric functions. The transformations are going to be basically the same, but sine and cosine are a good place to start on these transformations. So we want to talk a little bit about general transformations of functions and see specifically how they affect sine and cosine when we transform these things. And so the first type of transformation we're going to talk about in this video is the idea of stretching and compressing a graph. And there's two types of stretches we have to talk about, a vertical stretch and a horizontal stretch. So in general principles, imagine f is a function and we have some positive real numbers c. And for the sake of this example, let's say that c is greater than one. It's like two, three pi, something like that. If you have a function, you have your function y equals f of x. If you put a coefficient of c in front of the function, this has the effect of it vertically stretching the graph by a factor of c. So if I were to illustrate this real quick, what we mean is here's our x-axis right there. And so we have some function. You know, it's something like this maybe. If we vertically stretch it by a factor of c, what it means is that the parts above the x-axis will get elongated and by a factor of c. So it's gonna get steeper. And the stuff that's below the x-axis will also get elongated. It gets stretched out by again that factor of c. So we get that thing stretched out. So this graph right here is y equals c f of x. While the original one was just y equals f of x. You don't have the c in there. So things, it vertically stretched the graph. If you put that c in front. Now that's if you have a number larger than one. If the number in front of the function is less than one, that actually causes it to be a horizontal compression. That is, things maybe get flatter. Maybe something like this. They get squished. So you might get something like y equals one over c f of x because if the number is less than one, then it's reciprocal be greater than one. Oh, I couldn't hold it in. I'm sorry. We'll start again in three, two, one. So a coefficient in front of the function causes it to either be stretched or compressed vertically. What if you put a number, I might not do that one here. All right, we'll come back to that one. So start again in three, two, one. So let's see how this would affect your typical sinusoidal wave. That is, if we were to look at sine or cosine. So if we look at the graph y equals two sine of x, let's sketch its graph just on a typical period from zero to two pi. So you see illustrated on the screen in yellow, just a typical sine wave, just this friend right here. And we're just doing the standard period as we go from x equals zero to x equals pi right here. Remember that the x intercepts of sine are gonna be zero pi and two pi, the maximum's obtained at pi halves and the minimum's obtained at three pi halves. These five points coincide with the quadrantal angles that is thinking of the unit circle. These are the four places, well, five places because we repeat the first and last one. These are the five places where the x and y axis intersect the unit circle at zero, at pi halves, at pi, at three pi halves, at two pi. So when we graph a trigonometric function, we transform it, we are very interested in what happens to these five points. Well, if you consider the graph of y equals two sine of x, the fact that there's a coefficient of two in front of it means that the graph has been vertically stretched by a factor of two. All of the y-coordinates are now double. Well, if you take an x intercept, if you double that, so for an x intercept, the y-coordinate is zero, if you double that, it's still zero. So x intercepts are not affected by vertical stretching, but this maximum value is what was y equals one will get doubled to y equals two. And what was y, it was equal to negative one, doubles to be y equals negative one. And if we connect those dots in the usual sine wave fashion, you get this graph right here. So y equals two sine of x is gonna be just a sine wave where the amplitude has now been doubled. The amplitude of this function is now equal to two. So we see that the humps here and the trough of the sine wave are now larger because the amplitude was increased. Another example, let's sketch the graph of y equals one half cosine of x. And again, we'll do this on the standard period, zero to two pi. So you see here illustrated in yellow the standard cosine wave. Remember the main difference between sine and cosine is that cosine starts at its maximum while sine would start at the origin. So we have our standard cosine wave. Look at the five critical points there, the five quadrental points. You have zero, pi halves, pi, three pi halves and two pi. For the standard cosine function, you get one, zero, negative one, zero and then one again. What we're gonna do is we're gonna cut each and every one of these in half. So the y value of one drops down to one half. Zero stays zero because the x intercepts are not affected by stretching. Negative one is gonna then drop down to be negative one half. Zero stays put and then the one goes back down to one half. And if we connect those dots, whoops, if you connect those dots like so, we then see that our cosine wave looks like it's been compressed because its amplitude has been changed. We get that the amplitude of this cosine wave is gonna be one half and we see that by looking at the coefficient in front of the cosine wave. And this principle is very much in general. If we have a coefficient in front of sine or cosine that's positive, that coefficient is the amplitude. So y equals a sine of x or y equals a cosine of x, a is the amplitude in that situation, meaning that the range of the sine or cosine wave will be negative a to a. But that's if a is positive. What if a is negative? What if you have a negative coefficient in front? Well, it's just a simple reminder that if you take y equals negative f of x, putting a negative sine in front of the function has the effect of reflecting the graph across the x-axis. So you take your graph and you reflect it across the x-axis. That's what that negative sign does. That's not gonna affect the amplitude because the amplitude is always a positive number. So if you were to look at the function y equals negative cosine of x, the fact they have a negative sign in there means you're going to reflect, you're gonna reflect across the x-axis. That's how that should be interpreted and it doesn't affect the amplitude. The amplitude in the situation would still be one. You take the absolute value of that coefficient. This time, let's actually graph three periods of the graph. We go from negative two pi to four pi. Let's think of the standard cosine, which you can see right here again. Just like on our last example, this is standard cosine. Well, if we reflect this across the x-axis, we should end up with this picture right here. Right? Still going. So instead of one, we're now at negative one. Zero's unaffected by stretching or reflection. One, y equals one that becomes, excuse me, y equals negative one then becomes a positive one right there. Zero's unaffected and then a positive one switches over to negative one. You have those five points in consideration right there. So let's look at them again. Negative one, zero, one, zero, negative one. It's now the reflection of what you see before. And so that's one period of the graph. Just copy it for the other side. So as we go from zero to two pi, we can copy that to go from two pi to four pi. And we can copy that as we go from negative two pi to zero, like so. And we can draw this picture as large or as small as it needs to be.