# How do you find period, amplitude, phase shift and midline of y = 3 + 2cos(x + (pi/4))?

Jul 15, 2017

$\text{see explanation}$

#### Explanation:

$\text{the standard form of the "color(blue)"cosine function}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a \cos \left(b x + c\right) + d} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where amplitude "=|a|," period } = \frac{2 \pi}{b}$

$\text{phase shift "=-c/b," vertical shift } = d$

$\text{here "a=2,b=1,c=pi/4" and } d = 3$

$\Rightarrow \text{amplitude "=|2|=2," period } = \frac{2 \pi}{1} = 2 \pi$

$\text{phase shift } = - \frac{\frac{\pi}{4}}{1} = - \frac{\pi}{4}$

$\text{maximum value } = 3 + 2 = 5$

$\text{minimum value } = 3 - 2 = 1$

$\Rightarrow \text{midline } y = \frac{1}{2} \left(5 + 1\right) = 3$