Question

Vertical Translations? Help!

Answer 1

In order to find out what the value is when x=π, we have to model the function. It says it is a cosine wave, and a amplitude of 1/3 so we start as
1/3(cos(x))

It says a period of π/3. So the period of a cosine function is
2π/n

Where n is cos(nx)

So to find n, we set the period and that equal to each other

$\frac{\pi}{3}=\frac{2\pi}{n}$

Finding n we get 6. So now we have

1/3(cos(6x))

The shift is -2π. When we have a shift of something, we put the opposite sign with the x. So now it's
1/3(cos(6x+2π))

Factor out a 6 and we get

1/3(cos(6(x+π/3))

If we add the period to x, nothing happens so it remains
1/3(cos(6x))

The vertical shift is -3 so we just add that to everything so it becomes

1/3(cos(6x))-3

Plugging in x=π we get
1/3(cos(6π))-3

Cos(6π) is 1 so it is 1/3 - 3, or -8/3