Question

Write an equation of the cosine function with amplitude 2 and period 6π.

Answer 1

The general equation of cosine function is:

$y=a\cos (bx+c)+d$

where:

a = amplitude

$\frac{2\pi}{b}=period$$\frac{-c}{b}=horizontal\text{ shift}$

d = vertical shift

In this case, we have no vertical nor horizontal shifts, then:

c = 0

d = 0

The amplitude is 2, then:

a = 2

The period is 6π, then:

$\begin{gathered} \frac{2\pi}{b}=6\pi \\ 2\pi=6\pi\cdot b \\ \frac{2\pi}{6\pi}=b \\ \frac{1}{3}=b \end{gathered}$

Substituting this information into the general equation, we get:

$\begin{gathered} y=2\cos (\frac{1}{3}x+0)+0 \\ y=2\cos (\frac{1}{3}x) \end{gathered}$