# What is the amplitude of y=-2/3sinx and how does the graph relate to y=sinx?

Feb 11, 2018

See below.

#### Explanation:

We can express this in the form:

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

Where:

• $\textcolor{w h i t e}{88} \boldsymbol{a}$ is the amplitude.
• $\textcolor{w h i t e}{88} \boldsymbol{\frac{2 \pi}{b}}$ is the period.
• $\textcolor{w h i t e}{8} \boldsymbol{- \frac{c}{b}}$ is the phase shift.
• $\textcolor{w h i t e}{888} \boldsymbol{d}$ is the vertical shift.

From our example:

$y = - \frac{2}{3} \sin \left(x\right)$

We can see the amplitude is $\boldsymbol{\frac{2}{3}}$, amplitude is always expressed as an absolute value. i.e.

$| - \frac{2}{3} | = \frac{2}{3}$

$\boldsymbol{y = \frac{2}{3} \sin x}$ is $\boldsymbol{y = \sin x}$ compressed by a factor of $\frac{2}{3}$ in the y direction.

$\boldsymbol{y = - \sin x}$ is $\boldsymbol{y = \sin x}$ reflected in the x axis.

So:

$\boldsymbol{y = - \frac{2}{3} \sin x}$ is $\boldsymbol{y = \sin x}$ compressed by a factor $\frac{2}{3}$in the direction of the y axis and reflected in the x axis.

Graphs of the different stages:   