 In this video, we want to consider the graph of the function y equals sine of x. Now, some things we know about sine is that its domain is all real numbers, so its graph is going to cover the entire x-axis, but the function is also periodic. It repeats itself every two pi units along the x-axis. If I can graph a single period, like if I can graph it from zero to two pi, then I can just rubber stamp that and just stamp, stamp, stamp over and over again to get the entire graph. We also know that the range of sine is from negative one to one, that the graph will never go past negative one and one. The amplitude of the function is just going to be one. So if we graph some points, we know about sine. So when the angle is zero, sine will be zero. If the angle is pi four, then sine will be square root of two over two, which is about 0.7. So you can see that right here. The y-corner will be 0.5 when angle is pi six there. At pi halves, sine is equal to one, like so. At three pi fours, it's again square root of two over two, and at pi, it's returned to zero. So if we assume sine is a nice, smooth, continuous function, we can connect the dots and get a picture that looks like this, this little bump on the screen here. Then if we continue on at x equals five pi fours, sine is actually going to be negative square root of two over two, about negative point seven in that situation. So we get a point about right here. At three pi halves, sine is going to equal negative one, and then at seven pi fours, we also get negative root two over two again, and that two pi, it's going to repeat itself. So kind of connecting the dots, we get something that looks like the following. And so this here gives us a picture of a single period of a sine function. Let's switch over to a graphing category to get a more precise picture here. If we open up Desmos, we can see the following points that we put on the screen a moment ago. So you'll see here zero zero, pi fours root two over two, pi halves one, three pi fours root two over two, here is pi zero. This point right here is going to be five pi fours negative root two over two, three pi halves negative one. We have seven pi fours negative root two over two and two pi common zero. So like we saw earlier, if we try to connect the dots, we're going to get a picture that looks something like the following. This is a single period of our function. So we would anticipate that if we were to continue this thing on, we can kind of just replicate this picture over and over again, over and over again. We're going to get this sinusoidal wave that looks something like what we just drawn on the screen just now. Let's take a look at what the computer thinks it should look like. We're going to get this picture right here. And that sure enough gives us what we were expecting. If we zoom out, the sinusoidal wave will continue to repeat this pattern over and over and over again. We never exceed the value one. We never get below negative one like we would expect there. And again, this graph will just repeat itself over and over and over again.