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# Transforming sinusoidal graphs: vertical & horizontal stretches

Sal graphs y=-2.5*cos(1/3*x) by considering it as a vertical stretch and reflection, and a horizontal stretch, of y=cos(x). Created by Sal Khan.

Video transcript

We are asked to
graph the function y is equal to negative
2.5 cosine of 1/3 x on the interval, 0 to 6
pi, including the endpoints. So let me do my best
attempt at graphing that. And to start off, I'm going
to graph with the simplest function, or the
simplest version of this, or the root of this,
which is just cosine of x. So let me just graph,
and eventually you can kind of-- let me
just graph cosine of x. So that's my y-axis. And I want to have some space
here so I can eventually graph this entire thing. So let's say that this is
negative 1, this is negative 2. This is positive 1,
this is positive 2. And let's say that this
right over here is 2 pi. And then of course that
could be pi right over there. Now, the first thing
I'm going to do-- let me copy this because
I could use it later to graph the whole thing. So let's start off. So I'm just going to graph
y is equal to cosine of x. So when x is equal
to 0-- and I'm just going to do it between
the interval 0 and 2 pi. Obvious it's a
periodic function, it'll keep going in the negative
and the positive directions. So what happens when
x is equal to 0? What is cosine of x? Well cosine of 0 is 1. What about when
x is equal to pi? What is cosine of pi? Well cosine of pi is negative 1. And then what's cosine of 2 pi? Well that's 1 again. We get back-- we've
completed a period, or we've completed
an entire cycle. And 2 pi is the
period of cosine of x. So this is one cycle
right over here. I could keep going if I wanted
to, but the whole point, I just wanted to graph this
one cycle between 0 and 2 pi. Now what I want to think about
is, what happens to this graph? Instead of graphing
y equals cosine of x-- let me draw
some graph paper again. Instead of drawing y is
equal to cosine of x, I'm going to draw y is
equal to cosine of 1/3 x. So the only difference
between that and that is now I'm multiplying
the x by 1/3. What's going to happen
to the graph over here? How is this going to change
instead of being an x, if it's a 1/3 x? What's going to
happen over here? And now I'm going to do it
over the entire interval between zero and six pi. So let me just make sure
I have enough space. So that's 3 pi, 4
pi, 5 pi, and 6 pi. What's going to
happen to this graph? Well, there's a couple of
ways to think about it. The easiest might
just be to say, well to complete an entire cycle,
we're going to go 1/3 as fast. Or we're going to go
three times slower. Or if you just want to
think about the period here, what's the period
of cosine of 1/3 x? Well the period is
going to be 2 pi divided by the absolute value of this
coefficient right over here. So it's the absolute value
of 1/3, which is just 1/3. So the period is 2
pi over 1/3, which is the same thing as 2 pi
times 3, which is 6 pi. Which gels with the intuition. That's going to take
three times as much time to get whatever we input
into the cosine function to get back to 2 pi. Because whatever we take
x, we're taking 1/3 of it. So to get to 2 pi, you can't
just have x equals 2 pi. x now has to equal 6
pi to get 2 pi inputted into the cosine function. So the period is now 6 pi. At x is equal to 0, 1/3 times 0
is 0, and the cosine of 0 is 1. When x is equal to 6 pi, you
have 6 pi divided by 3 is 2 pi. Cosine of 2 pi is equal to 1. And if you want to go
in between, over here to go in between, we tried pi. But over here, we
could try 3 pi. When x is 3 pi, you
have cosine of 1/3 of 3 pi, that's cosine of pi. Cosine of pi is negative 1. So when x is equal to 3 pi,
we have cosine of 1/3 times 3 pi is negative 1. So it's going to look
something like this. Trying my best attempt. to draw it. So it's going to look
something like this. So you see, to go from y
equals cosine of x to y equals cosine of 1/3 x, it
essentially stretched out to this function
by a factor of 3. You can see this period
is three times longer. The period here was 2 pi. All right, well there's
only one more transformation we need in order to
get to the function that they're asking us about. We just have to, instead of
having a cosine of 1/3 x, we just have to negative
2.5 cosine of 1/3 x. So let's try to draw that. So let me put my
axis here again. And let me label it. So that's 2 pi, 3 pi,
4 pi, 5 pi, and 6 pi. And our goal now is to draw
the graph of y is equal to-- and we're just doing it over
between 0 and 6 pi here. We only did it between
0 and 2 pi here. Obviously they're all periodic,
they all keep going on and on. But now we want to graph y is
equal to negative 2.5 times cosine of 1/3 x. So given this change, we're now
multiplying by negative 2.5, what is going to
be-- well actually, let's think about a few things. What was the amplitude
in the first two graphs right over here? Well there's two ways
to think about it. You could say the amplitude
is half the difference between the minimum
and the maximum points. In either of these case,
the minimum is negative 1, maximum is 1. The difference is 2,
half of that is 1. Or you could just say
it's the absolute value of the coefficient here,
which is implicitly a 1. And the absolute value
of 1 is, once again, 1. What's going to be the amplitude
for this thing right over here? Well the amplitude is going to
be the absolute value of what's multiplying the cosine function. So the amplitude in this
case, do it in green, the amplitude is going to be
equal to the absolute value of negative 2.5,
which is equal to 2.5. So given that, how is
multiplying by negative 2.5 going to transform this
graph right over here? Well let's think about it. If it was multiplying
by just a positive 2.5, you would stretch it out. At each point it would go
up by a factor of 2 and 1/2. But it's a negative
2.5, so at each point, you're going to
stretch it out and then you're going to flip
it over the x-axis. So let's do that. So when x was 0, you
got 1 in this case. But now we're going to
multiply that by negative 2.5, which means you're going
to get to negative 2.5. So let me draw negative
2.5 right over there. So that's negative 2.5. That'd be negative--
let me make it clear. This would be negative
3 right over here, this would be positive 3. So that number right over
there is negative 2.5. And let me draw a
dotted line there. It could serve to be useful. Now when cosine
of 1/3 x is 0, it doesn't matter what
you multiply it by, you're still going to
get 0 right over here. Now, when cosine of
1/3 x was negative 1, which was the case when x
is equal to 3 pi, what's going to happen over here? Well cosine of 1/3 x,
we see, is negative 1. Negative 1 times negative
2.5 is positive 2.5. So we're going to get to
positive 2.5, which is right-- let me draw a dotted
line over here. We're going to get to positive
2.5, which is right over there. And then when cosine
of 1/3 x is equal to 0, doesn't matter what we
multiply it by, we get to 0. And then finally, when x
is at 6 pi, cosine of 1/3 x is equal to 1. What's that going to be when
you multiply it by negative 2.5? Well it's going to
be negative 2.5. So we're going to
get back over here. So we're ready to
draw our graph. It looks something-- let
me do that in magenta color since that's what the
color I wrote this in. It will look like this. I can draw it as a solid line. So it will look like that. So you saw what happened. By putting this 1/3 here,
it stretched out the graph. It increased the period
by a factor of 3. And then multiplying
it by negative 2.5-- if you just multiply
it by 2.5, you would just multiply
that out a little bit. But now it's a
negative, so not only do you increase the amplitude,
but you flip it over. So it is, indeed, the case
that the amplitude here is 2.5. We vary 2.5 from
our middle position. Or you could say that the
difference between the minimum and the maximum is 5,
so half of that is 2.5. But it isn't just multiplying
this graph by 2.5. If you multiply
this graph by 2.5, you'd get something-- let
me be a little neater. You would get something that
looked something like that. But because we
had a negative, we had to flip it over the x-axis. And we got this here. So this amplitude
is 2.5, but it's a flipped over
version of this graph.