Chapter 13.4, Problem 22WE

To determine

The equation of the curve.

Answer to Problem 22WE

The equation of the curve is $y=\sin \frac{x}{4}$ .

Explanation of Solution

Given:

  $\begin{array}{l}M=1\\ m=-1\\ \text{Period}\text{=}8\pi \end{array}$

Calculations:

Suppose general form of a function,

  $y=c+a\sin bx$ ,

Here, a is the amplitude of the function.

And period is $\frac{2\pi }{b}$ .

Also,

The average of sum of maximum and minimum gives,

  $c=\frac{M+m}{2}$

The average of difference of maximum and minimum value gives amplitude,

  $a=\frac{M-m}{2}$

Therefore,

  $\begin{array}{l}\frac{2\pi }{b}=8\pi \\ b=\frac{2\pi }{8\pi }\\ b=\frac{1}{4}\end{array}$

We know that,

The maximum value of the curve is 0 and the minimum value is -2.

Therefore,

  $\begin{array}{l}c=\frac{M+m}{2}=\frac{1+\left(-1\right)}{2}=0\\ a=\frac{M-m}{2}=\frac{1-\left(-1\right)}{2}=1\end{array}$

So, the equation of the graph is

  $\begin{array}{l}y=0+\left(1\right)\sin \left(\frac{1}{4}\right)x\\ y=\sin \frac{x}{4}\end{array}$