# How do you find the amplitude, period, vertical and phase shift and graph y=3-1/2costheta?

Jun 1, 2017

Amplitude: $\frac{1}{2}$
Period: $2 \pi$
Vertical shift: $3$
Phase shift: $\pi$

#### Explanation:

Given: $3 - \frac{1}{2} \cos \left(\theta\right)$

Begin by observing that multiplication of a sinusoid by negative is a phase shift of $\pi$

Using the general from,

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

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

In the given equation $B = 1$, therefore, we can rewrite the equation without a negative sign if we insert $\pi$ phase shift:

$\pi = \frac{C}{1}$

$C = \pi$

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

Period $T = \frac{2 \pi}{B}$

$T = 2 \pi$

The amplitude is $A = \frac{1}{2}$

Vertical shift $D = 3$