 In this video, we provide the solution to question number 18 from the practice final exam for math 1060, in which case we're asked to graph and simplify the function y equals four minus eight sine squared of x and graph it on the domain zero to two pi. So since this is a sine squared, it's not immediately obvious how to graph this thing, but we can use some trigonometric ideas to help us out here. For example, if you factor out the four, you're left behind with a one minus two sine squared of x, for which this makes us think of the double angle identity for a cosine. One minus two sine squared of x is the same thing as just cosine of two x, like so. So this thing simplifies to be four times cosine of two x. That dramatically simplifies this calculation. We see that the amplitude of this cosine wave will be four and we see that this coefficient b is two, which means the period, which will be two pi over two is actually equal to pi. Like so. And so keeping that in mind here, our amplitude's gonna be four, so cosine starts at its maximum, there's no reflection going on here whatsoever. And then it's gonna finish a single period at pi, like so. Halfway in between is going to, of course, be pi halves. That's where it hits its minimum value right here. And then it'll be x intercepts between these max and men's. So if we use these five points to help us out here, we can draw our cosine wave. It'll look something like this. And then we're just going to rinse and repeat and do it again. Just copy the picture we have already, carry through like that. In which case then we have the graph of our function y equals four minus eight sine squared of x, which was equivalent to graphing y equals four cosine of two x.