# How do you write an equation of the cosine function with amplitude of 2, period of 2π/3, phase shift of π/6, and a vertical shift of 1?

Sep 17, 2016

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

#### Explanation:

The general equation of cosine is $y = A \cos \left(B x - C\right) + D$,
where
$A =$ the amplitude

$\frac{2 \pi}{\left\mid B \right\mid} =$ the period

$\frac{C}{B} =$ the phase shift

$D =$ the vertical shift

In this example, the amplitude is given as 2, the period $\frac{2 \pi}{3}$, the phase shift is $\frac{\pi}{6}$, and vertical shift is 1.

$A = 2$

$\frac{2 \pi}{3} = \frac{2 \pi}{\left\mid B \right\mid}$ so $B = 3$

$\frac{\pi}{6} = \frac{C}{B} = \frac{C}{3}$

$\frac{\pi}{6} = \frac{C}{3}$ so $C = \frac{\pi}{2}$

$D = 1$

Giving us the equation

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