Question

How do you write an equation of the cosine function with amplitude 3 and period 4π?

Answer 1

`y = 3 cos({2pi}/{4pi} x) = 3 cos(x/2)`

Answer 2

The general form for the cosine function is:

`y = Acos(Bx+C)+D`

The amplitude is: `|A|`

The period is: `P = (2pi)/B`

The phase shift is `phi=-C/B`

The vertical shift is D

Explanation:

Given:

The amplitude is 3:

`|A| = 3`

The above implies that A could be either positive or negative but we always choose the positive value because the negative value introduces a phase shift:

`A = 3`

Given:

The period is

`P = 4pi`

`4pi = (2pi)/B`

`B = 1/2`

Nothing is said about the phase shift and the vertical shift, therefore, we shall assume that `C` and `D` are 0.

Substitute these values into the general form:

`y = 3cos(1/2x)`