# How do you find the amplitude and period of y=4sin3x?

Aug 5, 2015

Amplitude of $4 \sin \left(3 x\right)$ is $4$
Period of $4 \sin \left(3 x\right)$ is $6 \pi$

#### Explanation:

$y = \sin \left(\theta\right)$ has an amplitude of $1$
$\textcolor{w h i t e}{\text{XXXX}}$$y$ varies between $+ 1$ and $- 1$
If $y = 4 \sin \left(\theta\right)$
$\textcolor{w h i t e}{\text{XXXX}}$then $y$ will vary between $+ 4$ and $- 4$
$\textcolor{w h i t e}{\text{XXXX}}$$\rightarrow y = 4 \sin \left(\theta\right)$ has an amplitude of $4$ (for any $\theta$)

$y = k \cdot \sin \left(\theta\right)$ has a period of $2 \pi$
$\textcolor{w h i t e}{\text{XXXX}}$$y$ completes one cycle while $\theta$ increases by $2 \pi$
if $3 x = \theta$
$\textcolor{w h i t e}{\text{XXXX}}$$y$ completes one cycle while $3 x$ increases by $2 \pi$
$\textcolor{w h i t e}{\text{XXXX}}$or (expressed in another way)
$\textcolor{w h i t e}{\text{XXXX}}$$y$ completes $\frac{1}{3}$ of a cycle while $x$ increase by $2 \pi$

$\textcolor{w h i t e}{\text{XXXX}}$$\Rightarrow y$ completes 1 cycle while $x$ increases by $3 \times 2 \pi = 6 \pi$