 In this video, we're gonna graph the function y equals four cosine squared of x over two, and we're gonna do this on the domain zero to four pi, as all trigonometric functions are periodic to some period. We don't necessarily have to graph the entire thing. If we graph, you know, a sufficient number of periods, then that's okay. Zero to four pi is gonna be sufficient for us here, which we're gonna see in retrospect, why that's perfectly fine. Now, when you try to graph this thing, and I should mention we're graphing this without a graphing calculator, without any technology whatsoever, we're just gonna do this by hand, graphing a quadratic function gets a little bit more complicated for trigonometry. We're good at graphing sine and cosine and transformations of that, but how do you deal with a cosine squared? Well, the idea is to use some type of trigonometry to trigonometric identity, I should say, to simplify this thing. And actually, the thing that cues us on what to do is the fact that we have cosine of, we get cosine squared of x over two, right? This x over two makes me think of a half-angle identity. We've seen before that, I should say we've seen before that cosine of x over two is equal to one half, excuse me, the square root of one plus cosine of x over two, plus or minus. This gets a lot better when you use cosine squared of x over two, which is exactly the setting we're in right now. If you have cosine squared of x over two, the half-angle, you're just gonna give one half of one plus cosine of x. For which, when you look at this over here, in terms of transformations, that's a lot easier to graph that thing. You have a shift up by some, you have some type of vertical stretch. So we're gonna make this substitution in for cosine, right there. We get y equals four times one half times one plus cosine of x, like so. One half of four is equal to two. So we get two times one plus cosine of x. I'm gonna distribute that two through, and now we just have to graph the function y equals two plus two cosine of x. How bad is that? We see that in terms of graphing this thing, the period has been changed, not the period, the amplitude's been changed, the amplitude's gonna be two. And this plus two right here means we're gonna have a shift up by a factor of two. So the midline's been moved up by two. So with that in mind, we're gonna move the midline to y equals two because of the vertical shift, y equals two. The amplitude is gonna now be, the amplitude is gonna be two. So it's gonna go to above the midline, to below the midline, to above the midline. And since it's a cosine function, we should start at its maximum, no reflection going on here. So your maximum here is going to be at four. The minimum will be at zero. There is no change to the period, so the minimum should happen at pi. The maximum should happen at zero and two pi. And then at pi halves and three pi halves, we should get the quote unquote X intercepts, because we've shifted things. We will cross the midline at pi halves and at three pi halves. So we get a picture like this. And then we're gonna repeat this picture for the second period, because again it wants us to go to four pi. So because there was no period change, this just turns out to be two cycles, completed right here. And so now here's the graph of y equals four cosine squared of x over two. It seems intimidating, but with the right half angle identity, this function is equivalent to two plus two cosine of x, which is fairly simple to graph comparatively.