Chapter 4.4, Problem 16E

Solution

To calculate: The amplitude, period, and frequency of the function $y=-4\sin \frac{2x}{3}$ and use the information to sketch a graph of the function in the window $\left[-3\pi ,3\pi \right]\text{by}\left[-4,4\right]$ .

The resultant values of period, frequency and amplitude are $3\pi ,\frac{1}{3\pi },\text{and}-4$ respectively.

Given information:

The given function is $y=-4\sin \frac{2x}{3}$ .

Formula used: The period formula is $\text{Period}=\frac{2\pi }{b}$ and the formula of frequency is $\text{Frequency}=\frac{b}{2\pi }$ .

Calculation:

Consider the given function: $y=-4\sin \frac{2x}{3}$

Compare the function $y=-4\sin \frac{2x}{3}$ with the standard function $y=asinbx$ .

Then the values are $a=-4\text{and}b=\frac{2}{3}$ .

Here $a$ is the amplitude of the function and its value is $-4$ .

Now find the period of the function.

  $\text{Period}=\frac{2\pi }{b}$

Substitute the value $b=\frac{2}{3}$ in this formula.

  $\begin{array}{c}\text{Period}=\frac{2\pi }{\frac{2}{3}}\\ =3\pi \end{array}$

Now find the frequency of the function.

  $\text{Frequency}=\frac{b}{2\pi }$

Substitute the value $b=\frac{2}{3}$ in this formula.

  $\begin{array}{c}\text{Frequency}=\frac{\frac{2}{3}}{2\pi }\\ =\frac{1}{3\pi }\end{array}$

Now use the above information about the amplitude, period, and frequency to draw the graph in the window $\left[-3\pi ,3\pi \right]\text{by}\left[-4,4\right]$: