Main content

## Topic B: Lesson 11: Graphing sinusoidal functions

Current time:0:00Total duration:3:10

# Example: Graphing y=-cos(π⋅x)+1.5

CCSS.Math: ,

## Video transcript

- [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.