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

May 28, 2018

$y = 3 \cos \left(\frac{2 \pi}{4 \pi} x\right) = 3 \cos \left(\frac{x}{2}\right)$

May 28, 2018

The general form for the cosine function is:

$y = A \cos \left(B x + C\right) + D$

The amplitude is: $| A |$

The period is: $P = \frac{2 \pi}{B}$

The phase shift is $\phi = - \frac{C}{B}$

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

$P = 4 \pi$

$4 \pi = \frac{2 \pi}{B}$

$B = \frac{1}{2}$

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:

$y = 3 \cos \left(\frac{1}{2} x\right)$