Chapter 4.4, Problem 26E

Solution

To graph: Three periods of the function $y=20\sin 4x$ .

Given information:

The given function is $y=20\sin 4x$ .

Graph:

Consider the given function: $y=20\sin 4x$

Compare the function $y=20\sin 4x$ with the standard function $y=a\sin bx$ and find the values of amplitude, period, and frequency to draw the graph.

Then the values are $a=20\text{and}b=4$ .

Here $a$ is the amplitude of the function and its value is 20.

  $\text{Period}=\frac{2\pi }{b}$

Substitute the value $b=4$ in this formula.

  $\begin{array}{c}\text{Period}=\frac{2\pi }{4}\\ =\frac{\pi }{2}\end{array}$

Now find the frequency of the function.

  $\text{Frequency}=\frac{b}{2\pi }$

Substitute the value $b=4$ in this formula.

  $\begin{array}{c}\text{Frequency}=\frac{4}{2\pi }\\ =\frac{2}{\pi }\end{array}$

Now compare the given function $y=20\sin 4x$ with the standard function $y=\sin x$ .

It can be said that the graph of the function $y=20\sin 4x$ is vertical stretch of the graph $y=\sin x$ by the factor 20 and horizontally shrink of the graph of $y=\sin x$ by a factor of $\frac{1}{4}$ .

Now the graph of the function $y=20\sin 4x$ is:

  

Interpretation: The graph of function $y=20\sin 4x$ is vertical stretch of the function $y=\sin x$ the factor of 20 and horizontal shrink by the factor of $\frac{1}{4}$ .