Write the equation of the

function f of x graphed below. And so we have this

clearly periodic function. So immediately you

might say, well, this is either going to be a sine

function or a cosine function. But its midline

and its amplitude are not just the plain vanilla

sine or cosine function. And we can see that

right over here. The midline is halfway

between the maximum point and the minimum point. The maximum point right over

here, it hits a value of y equals 1. At the minimum points,

it’s a value of y is equal to negative 5. So halfway between

those, the average of 1 and negative 5, 1 plus

negative 5 is negative 4. Divided by 2 is negative 2. So this right over

here is the midline. So this is y is

equal to negative 2. So it’s clearly shifted down. Actually, I’ll talk

in a second about what type of an expression

it might be. But now, also, let’s

think about its amplitude. Its amplitude–

that’s how far it might get away from the

midline– we see here. It went 3 above the midline. Going from negative 2 to 1,

it went 3 above the midline at the maximum point. And it can also go 3 below the

midline at the minimum point. So this thing clearly

has an amplitude of 3. So immediately, we

can say, well, look. This is going to have

a form something like f of x is equal to

the amplitude 3. We haven’t figured

out yet whether this is going to be a cosine

function or a sine function. So I’ll write “cosine” first. Cosine maybe some coefficient

times x plus the midline. The midline– we

already figured out– was minus 2 or negative 2. So it could take

that form or it could take f of x is equal to 3

times– it could be sine of x or sine of some

coefficient times x. Sine of kx minus 2 plus

the midline– so minus 2. So how do we figure

out which of these are? Well, let’s just think about

the behavior of this function when x is equal to 0. When x is equal to

0, if this is kx, then the input into the

cosine is going to be 0. Cosine of 0 is 1. Whether you’re talking about

degrees or radians, cosine of 0 is 1. While sine of 0– so

if x is 0, k times 0 is going to be

0– sine of 0 is 0. So what’s this thing doing

when x is equal to 0? Well, when x is equal to

0, we are at the midline. If we’re at the midline, that

means that all of this stuff right over here evaluated to 0. So since, when x equals 0, all

of this stuff evaluated to 0, we can rule out the

cosine function. When x equals 0 here, this

stuff doesn’t evaluate to 0. So we can rule out this

one right over there. And so we are left with this. And we just really

need to figure out– what could this

constant actually be? And to think about

that, let’s look at the period of this function. Let’s see. If we went from

this point– where we intersect the midline–

and we have a positive slope, the next point that we do

that is right over here. So our period is 8. So what coefficient

could we have here to make the period of

this thing be equal to 8? Well, let us just

remind ourselves what the period of sine of x is. So the period of

sine of x– so I’ll write “period” right

over here– is 2pi. You increase your angle by

2 pi radians or decrease it. you’re back at the same

point on the unit circle. So what would be the

period of sine of kx? Well, now, your x, your input

is increasing k times faster. So you’re going to get to the

same point k times faster. So your period is going

to be 1/k’th as long. So now your period is

going to be 2 pi over k. Notice, as x increases, your

argument into the sine function is increasing k times as fast. You’re multiplying it by k. So your period is

going to be short. It’s going to take you less

distance for the whole argument to get to the same point

on the unit circle. So let’s think

about it this way– so if we wanted to say 2

pi over k is equal to 8, well, what is our k? Well, we could take the

reciprocal of both sides. We get k over 2 pi

is equal to 1/8. Multiply both sides by 2pi. And we get k is

equal to– let’s see. This is 1. This is 4. k is equal to pi/4. And we are done. And you can verify that by

trying out some of these points right over here. This function is equal to 3

sine of pi over 4x minus 2.