Question

What is the general equation of a cosine function with an amplitude of 3, a period of 4 pi, and a horizontal shift of -pi?

Answer 1

The general form of the function is
$f(x) = Acos( \frac{2 \pi }{T}x-\phi)$
where
A =  amplitude
T = period
φ = phase shift.

Given:
A = 3
T = 4π
φ = -π

Answer:
$f(x) = 3 cos( \frac{x}{2}+ \pi )$

Answer 2

The parent function is y = cos x, which has amplitude of one and period of 2π (Refer to the first diagram attached below)

To increase the amplitude to three, we will have to stretch the graph vertically.
The function f(x) then become f(x) = 3 cos (x) 
(Refer to the second diagram)

To increase the period from 2π to 4π, we stretch the graph horizontally.
The function f(x) then become f(x) = 3 cos (0.5x)
(Refer to the third diagram)

Lastly, to shift the graph by π to the left, we translate the graph and the function is f(x) = 3 cos (0.5x + π)