Chapter 4.4, Problem 14E

Solution

To calculate: The amplitude, period, and frequency of the function $y=2\cos \frac{x}{3}$ 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 $6\pi ,\frac{1}{6\pi },\text{and}2$ respectively.

Given information:

The given function is $y=2\cos \frac{x}{3}$ .

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: $y=2\cos \frac{x}{3}$

Compare the function $y=2\cos \frac{x}{3}$ with the standard function $y=a\cos bx$ .

Then the values are $a=2\text{and}b=\frac{1}{3}$ .

Here $a$ is the amplitude of the function and its value is 2.

Now find the period of the function.

  $\text{Period}=\frac{2\pi }{b}$

Substitute the value $b=\frac{1}{3}$ in this formula.

  $\begin{array}{c}\text{Period}=\frac{2\pi }{\frac{1}{3}}\\ =6\pi \end{array}$

Now find the frequency of the function.

  $\text{Frequency}=\frac{b}{2\pi }$

Substitute the value $b=\frac{1}{3}$ in this formula.

  $\begin{array}{c}\text{Frequency}=\frac{\frac{1}{3}}{2\pi }\\ =\frac{1}{6\pi }\end{array}$

Now use the above information about the amplitude, period, and frequency to draw the graph: