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# Finding a sinusoidal function from its graph

## Video transcript

We're asked to determine
a function of the form y equals a sine of bx or y
is equal to a cosine bx represented by the graph below. So we need to figure out
maybe what the a's are, what the b's are,
and whether this is a sine or a cosine function. So let's see what
clues there are. So the first thing
that I notice is that whatever this function
is, when x is equal to 0, it does not equal 0. It is equal to negative 2. So based on that,
based on what we know about sine and
cosine functions, do you think that this is
going to be of the form y is equal to a sine
of bx or of the form y is equal to a cosine of bx? Without even knowing
what the a's and b's are, do you think this is going to
be a sine function or cosine function? Well, let's think about
what sine of 0 is. If you take sine
of 0, we already know that that is equal to 0. What is cosine of 0? Cosine of 0 is equal to 1. So it would be very hard--
and especially in this form, it would be impossible--
if sine of 0 is 0, to multiply 0 by
something to get to negative 2. So it can't be a sine
function of this form. You might say, well,
the cosine of 0 is 1. But here, it's negative 2. But at least if
you have a 1, you can then multiply
it by something to get to a negative 2. So what we now know is that
we are at least of this form. But now we have to figure
out what the a's and b's are going to be equal to. We know that this function is
y is equal to a cosine of bx. So the next question I ask
you is what is a going to be. Well, let's think about it. We already saw if we
just had cosine of bx, when x is equal to 0,
cosine of b times 0 would just be cosine of 0. And it would get us to 1. But we're not 1. We're at negative 2. It looks like [? it ?]
took a cosine function, and at least, when
x is equal to 0, we multiplied it by negative 2. So this should be negative 2. So now we have a little
bit more filled in of what we actually have. We know that it's y is equal
to negative 2 cosine of bx. And this gels with what we see
right over here, the amplitude here. You see that the difference
between the maximum value and the minimum value, or
the minimum and the maximum is 4, 1/2 of that is 2. Or another way you
think about it, we're varying 2 from
this center point. And over here, if you
think about the amplitude-- the amplitude is
the absolute value of this number right over here. The amplitude is equal
to the absolute value of this negative 2,
which is indeed equal 2. So it's consistent so far. Now let's think
about what b here is. And maybe we can
use our knowledge of what the period of
a periodic function is to think about
what b might be. Well, let's look over here. What is the period of
this periodic function? Well, let's draw one period. So if we use this as our
starting point-- or one cycle, I should say. Let's draw one cycle. If you view that as our
starting point, at 2 pi over 3, we have completed that cycle. And then we could
start the next cycle. We repeat the
pattern over again. Then you start the next cycle. So based on that,
what is the period? Well, it's the
length that you need to go in x to
complete one cycle. So that length right over
there, you see, is 2 pi over 3. So the period is 2 pi over 3. And given that the period
here is 2 pi over 3, can you figure out
what b is going to be? Well, the period
of this is going to be equal to 2 pi over
the absolute value of b. And you can solve
this multiple ways. You can multiply both sides by
3 and the absolute value of b. And you would be left
with the absolute value of b is equal to 3,
which means that b could be equal to positive
or negative 3. And so you might say,
well, Sal, what do I use? Does b equal positive
3 or negative 3? And so the next
question I'll ask you is, for a
cosine function, do you get different
values if you were to make this a cosine of 3x
or a cosine of negative 3x? Do you get different values? Well, if you play with the
unit circle a little bit. So I'm going to draw
a little rough unit circle right over here. Remember, cosine
is the x-coordinate where we intersect
the unit circle. And if we go in the
positive angle direction, if we go in that direction, our
x-coordinate-- it starts at 1, and then it gets a
little bit shorter. If we go in the negative
direction, it starts at 1, and then it gets a
little bit shorter. And so you can experiment
this with a good bit. But you'll see
that cosine of 3x-- and this is only
the case for cosine, not for sine-- cosine of 3x is
equal to cosine of negative 3x. So you can actually pick either
positive 3 or negative 3. But for simplicity in
this case right over here, I'll just go with
the positive 3. So this could be the
graph of-- and now we get our drum roll-- y is
equal to negative 2 times the cosine of-- I said I
wouldn't do the negative-- the cosine of positive 3x. And we are done.