# How do you use transformation to graph the sin function and determine the amplitude and period of y=sin(3x)?

Dec 9, 2017

See below.

#### Explanation:

We can find the transformation of $\sin \left(x\right)$ to $\sin \left(3 x\right)$ using the following equation:

$y = a \sin \left(b x + c\right) + d$

Where:

Amplitude is $\textcolor{w h i t e}{88} a$

Period is $\textcolor{w h i t e}{88} \frac{2 \pi}{b}$

Phase shift is $\textcolor{w h i t e}{88} \frac{- c}{b}$

Vertical shift is $\textcolor{w h i t e}{88} d$

$\therefore$

For $\textcolor{w h i t e}{88} y = \sin \left(3 x\right)$

Amplitude is 1, (This is the same as $\sin \left(x\right)$)

Period is $\textcolor{w h i t e}{88} \frac{2 \pi}{b} = \frac{2 \pi}{3}$ ( period of $\sin \left(x\right)$ is $2 \pi$)

( This is a compression in the horizontal direction by a factor of 3 )

Phase shift is $\textcolor{w h i t e}{88} \frac{- c}{b} = \frac{0}{3} = 0$ ( No phase shift )

Vertical shift is $\textcolor{w h i t e}{88} d = 0$ ( No vertical shift )

Graph of $y = \sin \left(x\right) \mathmr{and} y = \sin \left(3 x\right)$ 