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# Features of sinusoidal functions

Sal introduces the main features of sinusoidal functions:

**midline, amplitude, & period**. He shows how these can be found from a sinusoidal function's graph. Created by Sal Khan.Video transcript

We have a periodic
function depicted here and what I want
you to do is think about what the midline
of this function is. The midline is a line,
a horizontal line, where half of the
function is above it, and half of the
function is below it. And then I want you to
think about the amplitude. How far does this function
vary from that midline-- either how far above does it go or
how far does it go below it? It should be the same amount
because the midline should be between the highest
and the lowest points. And then finally,
think about what the period of this function is. How much do you have
to have a change in x to get to the same
point in the cycle of this periodic function? So I encourage you to pause
the video now and think about those questions. So let's tackle
the midline first. So one way to think
about is, well, how high does this function go? Well, the highest y-value for
this function we see is 4. It keeps hitting 4 on
a fairly regular basis. And we'll talk about
how regular that is when we talk
about the period. And what's the lowest value
that this function gets to? Well, it gets to y
equals negative 2. So what's halfway
between 4 and negative 2? Well, you could eyeball
it, or you could count, or you could,
literally, just take the average between
4 and negative 2. So 4-- so the midline is going
to be the horizontal line-- y is equal to 4 plus
negative 2 over 2. Just literally the
mean, the arithmetic mean, between 4 and negative 2. The average of 4 and
negative 2, which is just going to
be equal to one. So the line y equals
1 is the midline. So that's the midline
right over here. And you see that it's kind
of cutting the function where you have half of the
function is above it, and half of the
function is below it. So that's the midline. Now, let's think
about the amplitude. Well, the amplitude is
how much this function varies from the midline--
either above the midline or below the midline. And the midline
is in the middle, so it's going to be the
same amount whether you go above or below. One way to say it is, well,
at this maximum point, right over here, how far above
the midline is this? Well, to get from 1
to 4 you have to go-- you're 3 above the midline. Another way of thinking
about this maximum point is y equals 4 minus y equals 1. Well, your y can go as much
as 3 above the midline. Or you could say your
y-value could be as much as 3 below the midline. That's this point right over
here, 1 minus 3 is negative 1. So your amplitude right
over here is equal to 3. You could vary as much as
3, either above the midline or below the midline. Finally, the period. And when I think
about the period I try to look for a relatively
convenient spot on the curve. And I'm calling this
a convenient spot because it's a nice-- when
x is at negative 2, y is it one-- it's at a
nice integer value. And so what I want to do is
keep traveling along this curve until I get to the same y-value
but not just the same y-value but I get the same
y-value that I'm also traveling in the same direction. So for example, let's
travel along this curve. So essentially our
x is increasing. Our x keeps increasing. Now you might say, hey, have
I completed a cycle here because, once again,
y is equal to 1? You haven't completed a cycle
here because notice over here where our y is increasing
as x increases. Well here our y is
decreasing as x increases. Our slope is positive here. Our slope is negative here. So this isn't the same
point on the cycle. We need to get to the point
where y once again equals 1. Or we could say, especially in
this case, we're at the midline again, but our
slope is increasing. So let's just keep going. So that gets us to
right over there. So notice, now we have
completed one cycle. So the change in x needed
to complete one cycle. That is your period. So to go from negative 2
to 0, your period is 2. So your period here is 2. And you could do it again. So we're at that point. Let's see, we want to
get back to a point where we're at the
midline-- and I just happen to start right
over here at the midline. I could have started
really at any point. You want to get
to the same point but also where the
slope is the same. We're at the same point
in the cycle once again. So I could go-- so if I travel
1 I'm at the midline again but I'm now going down. So I have to go further. Now I am back at that
same point in the cycle. I'm at y equals 1 and
the slope is positive. And notice, I traveled. My change in x was the
length of the period. It was 2.