Chapter 4.4, Problem 27E

Solution

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

Given information:

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

Graph:

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

Compare the function $y=4\sin \frac{x}{4}$ 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 $a=4\text{and}b=\frac{1}{4}$ .

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

  $\text{Period}=\frac{2\pi }{b}$

Substitute the value $b=\frac{1}{4}$ in this formula.

  $\begin{array}{c}\text{Period}=\frac{2\pi }{\frac{1}{4}}\\ =8\pi \end{array}$

Now find the frequency of the function.

  $\text{Frequency}=\frac{b}{2\pi }$

Substitute the value $b=\frac{1}{4}$ in this formula.

  $\begin{array}{c}\text{Frequency}=\frac{\frac{1}{4}}{2\pi }\\ =\frac{1}{8\pi }\end{array}$

Now compare the given function $y=4\sin \frac{x}{4}$ with the standard function $y=\sin x$ .

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

Now the graph of the function $y=4\sin \frac{x}{4}$ is:

  

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