# What is the phase shift, vertical displacement with respect to y=sinx for the graph y=sin(x+(2pi)/3)+5?

Feb 18, 2018

See below.

#### Explanation:

We can represent a trigonometrical function in the following form:

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

Where:

• $\textcolor{w h i t e}{8} \boldsymbol{a} \textcolor{w h i t e}{88} = \text{amplitude}$

• $\boldsymbol{\frac{2 \pi}{b}} \textcolor{w h i t e}{8} = \text{the period}$ ( note $\boldsymbol{2 \pi}$ is the normal period of the sine function )

• $\boldsymbol{\frac{- c}{b}} \textcolor{w h i t e}{8} = \text{the phase shift}$

• $\textcolor{w h i t e}{8} \boldsymbol{\mathrm{dc}} o l \mathmr{and} \left(w h i t e\right) \left(888\right) = \text{ the vertical shift}$

From example:

$y = \sin \left(x + \frac{2 \pi}{3}\right) + 5$

Amplitude = $\boldsymbol{a} = \textcolor{b l u e}{1}$

Period = $\boldsymbol{\frac{2 \pi}{b}} = \frac{2 \pi}{1} = \textcolor{b l u e}{2 \pi}$

Phase shift = $\boldsymbol{\frac{- c}{b}} = \frac{\frac{- 2 \pi}{3}}{1} = \textcolor{b l u e}{- \frac{2 \pi}{3}}$

Vertical shift = $\boldsymbol{d} = \textcolor{b l u e}{5}$

So $y = \sin \left(x + \frac{2 \pi}{3}\right) + 5 \textcolor{w h i t e}{88}$ is $\textcolor{w h i t e}{888} y = \sin \left(x\right)$:

Translated 5 units in the positive y direction, and shifted $\frac{2 \pi}{3}$ units in the negative x direction.

GRAPH: