# How do you use transformation to graph the sin function and determine the amplitude and period of y=sin(x+pi/6)?

Dec 3, 2017

See below.

#### Explanation:

If we look at a trig function in the form:

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

Amplitude is $\textcolor{w h i t e}{888} \textcolor{b l u e}{a}$

Period is $\textcolor{w h i t e}{888} \textcolor{b l u e}{\frac{2 \pi}{b}}$

Phase shift is $\textcolor{w h i t e}{888} \textcolor{b l u e}{\frac{- c}{b}}$

Vertical shift is $\textcolor{w h i t e}{888} \textcolor{b l u e}{d}$

From: $y = \sin \left(x + \frac{\pi}{6}\right)$

We can see amplitude is 1. This is the same as for $y = \sin \left(x\right)$

The period is: $\frac{2 \pi}{1} = 2 \pi$. This is the same as $y = \sin x$

Phase shift is: $\frac{- \frac{\pi}{6}}{1} = - \frac{\pi}{6}$ This translates the graph of

$y = \sin x$ $\textcolor{w h i t e}{88} \frac{\pi}{6}$ units to the left.

From the above, we conclude that the graph of $y = \sin \left(x + \frac{\pi}{6}\right)$ is the graph of $y = \sin x$ translated $\frac{\pi}{6}$ units to the left.

Graph of $y = \sin \left(x + \frac{\pi}{6}\right)$ and $y = \sin x$ on the same axes: 