 In this video, we're gonna graph the function y equals three minus six sine squared of x. We're gonna do this without any technology whatsoever. We're just gonna graph this by hand. Now, because it's a trigonometric function, you have a sine x there. Sine x is two pi periodic. It'll repeat itself any length along the x axis of two pi. So we don't have to graph the whole domain. If we can graph from zero to two pi, then that'd be sufficient. We know the basic shape and we can repeat it over and over again if we needed larger than that. So how do you graph something like y equals three minus six sine squared? Well, graphing a quadratic trigonometric function can be somewhat of a challenge, but with trigonometric identities, we can actually simplify it into just a linear trigonometric function that is just sines or cosines. And just graphing a sinusoidal wave is much easier to do. So how do we do that? So the first thing I'm gonna do is I notice that the coefficients three and six are both multiples of three. So I'm gonna factor out the three and see what we have there. You're gonna get one minus two sine squared of x. And this is where life gets a little bit easier for us here. One minus two sine squared of x, this makes my trigonometric identity sense start tingling. One minus two sine squared of x is just half, I can apply the double angle identity for cosine right here. One minus two sine squared of x, this is just cosine of two x. So we can simplify that very much so. And so then you look at those numbers here, three. Three here gives us the amplitude of our sinusoidal wave. It's gonna be three, great. And then what about this two? This two affects the period for which the period of the cosine wave is gonna be two pi over two, which simplifies just to be pi. So we're actually gonna graph two cycles of this thing. So one cycle would be from zero to pi. The other cycle would be from zero to two pi like so. And since the amplitude is three, we have to graph all the way up to three and all the way down to negative three. So let's get started here. Cosine starts at its highest point, which here is gonna be three. Then it's also gonna end at the high note, which at the end of the period, which would be pi. And then in the exact middle, the midpoint here, it's gonna get its lowest point right here. In this case, it would be pi halves. So therefore then between the max and the minimum, you're gonna have an x intercept. So cosine will have an intercept at pi force. It'll have one at three pi force. And so if we graph this using the typical sinusoidal graph shape, we get a single cycle of graph. You're gonna get something like this. And then we're gonna repeat this because we agreed we were gonna do this towards, we were gonna do this towards the two pi, the whole two pi there. So we get these pictures like this and then graph it one more time like so. And so then we get the graph of y equals three minus six sine squared, which again, when you use the right trigonometric identity, it doesn't seem so challenging. And that's the whole point of trigonometric identities. They can simplify what seems to be hard trigonometric expressions, a much easier one, but the hard part is knowing what's the right identity to use in a given spot.