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CCSS Math: HSF.BF.B.3

We're asked to determine the
amplitude and the period of y equals negative
1/2 cosine of 3x. So the first thing
we have to ask ourselves is, what does
amplitude even refer to? Well the amplitude of
a periodic function is just half the difference
between the minimum and maximum values it takes on. So if I were to draw a
periodic function like this, and it would just go back
and forth between two-- let me draw it a
little bit neater-- it goes back and forth
between two values like that. So between that
value and that value. You take the difference between
the two, and half of that is the amplitude. Another way of thinking
about the amplitude is how much does it sway
from its middle position. Right over here,
we have y equals negative 1/2 cosine of 3x. So what is going to be
the amplitude of this? Well, the easy way
to think about it is just what is multiplying
the cosine function. And you could do the same thing
if it was a sine function. We have negative
1/2 multiplying it. So the amplitude
in this situation is going to be the absolute
value of negative 1/2, which is equal to 1/2. And you might say, well, why
do I not care about the sign? Why do I take the
absolute value of it? Well, the negative just
flips the function around. It's not going to
change how much it sways between its minimum
and maximum position. The other thing is,
well, how is it just simply the absolute
value of this thing? And to realize the
y, you just have to remember that a cosine
function or a sine function varies between positive
1 and negative 1, if it's just a simple function. So this is just multiplying
that positive 1 or negative 1. And so if normally
the amplitude, if you didn't have
any coefficient here, if the coefficient was
positive or negative 1, the amplitude would just be 1. Now, you're changing
it or you're multiplying it by this amount. So the amplitude is 1/2. Now let's think
about the period. So the first thing
I want to ask you is, what does the period
of a cyclical function-- or periodic function,
I should say-- what does the period of
a periodic function even refer to? Well let me draw some axes on
this function right over here. Let's say that this right
over here is the y-axis. That's the y-axis. And let's just say, for
the sake of argument, this is the x-axis
right over here. So the period of a
periodic function is the length of the
smallest interval that contains exactly one copy
of the repeating pattern of that periodic function. So what do they mean here? Well, what's repeating? So we go down and then
up just like that. Then we go down
and then we go up. So in this case, the length
of the smallest interval that contains exactly one
copy of the repeating pattern. This could be one of the
smallest repeating patterns. And so this length between here
and here would be one period. Then we could go between here
and here is another period. And there's multiple--
this isn't the only pattern that you could pick. You could say, well, I'm going
to define my pattern starting here going up and then
going down like that. So you could say that's
my smallest length. And then you would
see that, OK, well, if you go in the
negative direction, the next repeating
version of that pattern is right over there. But either way you're going
to get the same length that it takes to
repeat that pattern. So given that,
what is the period of this function
right over here? Well, to figure out the
period, we just take 2 pi and divide it by
the absolute value of the coefficient
right over here. So we divide it by the absolute
value of 3, which is just 3. So we get 2 pi over 3. Now we need to think
about why does this work? Well, if you think about
just a traditional cosine function, a traditional cosine
function or a traditional sine function, it has
a period of 2 pi. If you think about
the unit circle, 2 pi, if you start at
0, 2 pi radians later, you're back to
where you started. 2 pi radians,
another 2 pi, you're back to where you started. If you go in the
negative direction, you go negative 2 pi, you're
back to where you started. For any angle here,
if you go 2 pi, you're back to where
you were before. You go negative 2 pi, you're
back to where you were before. So the periods for
these are all 2 pi. And the reason why
this makes sense is that this coefficient
makes you get to 2 pi or negative-- in
this case 2 pi-- it's going to make you get
to 2 pi all that much faster. And so it gets-- your period
is going to be a lower number. It takes less length. You're going to get to 2
pi three times as fast. Now you might say,
well, why are you taking the absolute value here? Well, if this was
a negative number, it would get you to negative
2 pi all that much faster. But either way, you're going
to be completing one cycle. So with that out
of the way, let's visualize these two things. Let's actually draw
negative 1/2 cosine of 3x. So let me draw my axes here. My best attempt. So this is my y-axis. This is my x-axis. And then let me draw some--
So this is 0 right over here. x is equal to 0. And let me draw x is
equal to positive 1/2. I'll draw it right over here. So x is equal to positive 1/2. And we haven't shifted this
function up or down any. Then, if we wanted to, we
could add a constant out here, outside of the cosine function. But this is positive 1/2, or we
could just write that as 1/2. And then down here, let's say
that this is negative 1/2. And so let me draw that bound. I'm just drawing
these dotted lines so it'll become
easy for me to draw. And what happens when this is 0? Well cosine of 0 is 1. But we're going to multiply
it by negative 1/2. So it's going to be negative
1/2 right over here. And then it's going
to start going up. It can only go in
that direction. It's bounded. It's going to start going
up, then it'll come back down and then it will get back
to that original point right over here. And the question is,
what is this distance? What is this length? What is this length going to be? Well, we know what
its period is. It's 2 pi over 3. It's going to get to
this point three times as fast as a traditional
cosine function. So this is going
to be 2 pi over 3. And then if you give
it another 2 pi over 3, it's going to get back
to that same point again. So if you go another 2 pi
over 3, so in this case, you've now gone 4 pi over 3,
you've completed another cycle. So that length right
over there is a period. And then you could
also do the same thing in the negative direction. So this right over here would be
negative, negative 2 pi over 3. And to visualize the amplitude,
you see that it can go 1/2. Well, there's two ways
to think about it. The difference between the
maximum and the minimum point is 1. Half of that is 1/2. Or you could say that it's
going 1/2 in magnitude, or it's swaying 1/2 away
from its middle position in the positive or the
negative direction.