# How do you find the amplitude, period, vertical and phase shift and graph y=3+1/2cos(theta+pi)?

Jan 18, 2017
• amplitude $= \frac{1}{2}$

• period $= 2 \pi$

• vertical shift $= + 3$

• phase shift $= - \pi$

#### Explanation:

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

maybe start by re-writing it as

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

So:

y is shifted up by 3, the vertical shift, and $\theta$ is shifted left by $\pi$

$\implies Y = \frac{1}{2} \cos \left(\Theta\right)$, where $Y = y - 3 , \Theta = \theta + \pi$, and the function $\cos \left(\Theta\right)$ has period $2 \pi$ and $\frac{1}{2} \cos \left(\Theta\right)$ has amplitude $\frac{1}{2}$.

We conclude that:

• amplitude $= \frac{1}{2}$

• period $= 2 \pi$

• vertical shift $= + 3$

• phase shift $= - \pi$