Example: Graphing y=-cos(π⋅x)+1.5

- [Instructor] We're told to graph y is equal to negative cosine of pi times x plus 1.5 in the interactive widget. So, pause this video and think about how you would do that. And just to explain how this widget works if you're trying to do it on Khan Academy, this dot right over here helps define the midline. You can move that up and down. And then this one right over here is a neighboring extreme point. So either a minimum or a maximum point. So, there's a couple of ways that we could approach this. First of all, let's just think about what would cosine of pi x look like, and then we'll think about what the negative does and the plus 1.5. So, cosine of pi x. When x is equal zero, pi times zero, is just going to be zero, cosine of zero is equal to one. And if we're just talking about cosine of pi x, that's going to be a maximum point when you hit one. Just cosine of pi x would oscillate between one and negative one. And then what would its period be if we're talking about cosine of pi x? Well, you might remember, one way to think about the period is to take two pi and divide it by whatever the coefficient is on the x right over here. So two pi divided by pi would tell us that we have a period of two. And so how do we construct a period of two here? Well, that means that as we start here at x equals zero, we're at one, we want to get back to that maximum point by the time x is equal to two. So let me see how I can do that. If I were to squeeze it a little bit, that looks pretty good. And the reason why I worked on this midline point is I liked having this maximum point at one when x is equal to zero, because we said cosine of pi times zero should be equal to one. So that's why I'm just manipulating this other point in order to set the period right. But this looks right. We're going from this maximum point and we're going all the way down and then back to that maximum point, and it looks like our period is indeed two. So this is what the graph of cosine of pi x would look like. Now, what about this negative sign? Well, the negative would essentially flip it around. So, instead of whenever we're equaling one, we should be equal to negative one. And every time we're equal to negative one, we should be equal to one. So what I could is I could just take that and then bring it down here, and there you have it, I flipped it around. So this is the graph of y equals negative cosine of pi x. And then last but not least, we have this plus 1.5. So that's just going to shift everything up by 1.5. So I'm just going to shift everything up by, shift it up by 1.5 and shift it up by 1.5. And there you have it. That is the graph of negative cosine of pi x plus 1.5. And you can validate that that's our midline. We're still oscillating one above and one below. The negative sign, when cosine of pi time zero, that should be one, but then you take the negative that, we get to negative one. You add 1.5 to that, you get to positive .5. And so this is all looking quite good.