# How do you find the period for y=4sin(2x)+1?

Aug 22, 2015

$y = 4 \sin \left(2 x\right) + 1$ has a period of $\pi$

#### Explanation:

Starting from a base equation:
$\textcolor{w h i t e}{\text{XXXX}} {y}_{\theta} = \sin \left(\theta\right) = \textcolor{red}{1} \cdot \sin \left(\textcolor{b l u e}{\theta}\right) + \textcolor{g r e e n}{0}$
$\textcolor{w h i t e}{\text{XXXXXXXX}}$has an amplitude of $\textcolor{red}{1}$
$\textcolor{w h i t e}{\text{XXXXXXXX}}$is centered on $y = \textcolor{g r e e n}{0}$
$\textcolor{w h i t e}{\text{XXXXXXXX}}$and has a period of $2 \pi$
To say that ${y}_{\theta}$ has a period of $2 \pi$
$\textcolor{w h i t e}{\text{XXXX}}$means that ${y}_{\theta}$ has a repeating pattern for $\left(0 + k\right) \le \theta \le \left(2 \pi + k\right)$ for any constant $k$

$\textcolor{w h i t e}{\text{XXXX}} y = \textcolor{red}{4} \sin \left(\textcolor{b l u e}{2 x}\right) + \textcolor{g r e e n}{1}$
$\textcolor{w h i t e}{\text{XXXXXXXX}}$has an amplitude of $\textcolor{red}{4}$
$\textcolor{w h i t e}{\text{XXXXXXXX}}$is centered on $y = \textcolor{g r e e n}{1}$
$\textcolor{w h i t e}{\text{XXXXXXXX}}$and has a repeating pattern for
$\textcolor{w h i t e}{\text{XXXXXXXXXXX}} \left(0 + k\right) \le 2 x \le \left(2 \pi + k\right)$
$\textcolor{w h i t e}{\text{XXXXXXXXXXX}}$or
$\textcolor{w h i t e}{\text{XXXXXXXXXXX}} \left(0 + \hat{k}\right) \le x \le \left(\pi + \hat{k}\right)$ for
constant $\hat{k}$
$\textcolor{w h i t e}{\text{XXXXXXX}}$ That is, it has a period of $\pi$