# How do you write an equation of the cosine function with amplitude=3, period = 2pi/3, phase shift = 1pi/9, and vertical shift 4?

Nov 14, 2016

In $y = a \cos \left(b \left(x - c\right)\right) + d$:

$| a |$ is the amplitude
$\frac{2 \pi}{b}$ is the period
$c$ is the phase shift
$d$ is the vertical transformation

The amplitude is $3$, so $a = 3$.

The period is $\frac{2 \pi}{3}$, so we solve for $b$.

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

$b = 3$

The phase shift is $+ \frac{\pi}{9}$, so $c = \frac{\pi}{9}$.

The vertical transformation is $+ 4$, so $d = 4$.

$\therefore$The equation is $y = 3 \cos \left(3 \left(x - \frac{\pi}{9}\right)\right) + 4$, which can be written as $y = 3 \cos \left(3 x - \frac{\pi}{3}\right) + 4$

Hopefully this helps!