Chapter 13.4, Problem 26WE

To determine

The equation of the curve.

Answer to Problem 26WE

The equation of the curve is $y=3\sin \left(\frac{\pi }{2}\right)x-3$ .

Explanation of Solution

Given:

  $\begin{array}{l}M=0\\ m=-6\\ \text{Period}\text{=}4\end{array}$

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,

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

We know that,

The maximum value of the curve is 0 and the minimum value is -6.

Therefore,

  $\begin{array}{l}c=\frac{M+m}{2}=\frac{0+\left(-6\right)}{2}=-3\\ a=\frac{M-m}{2}=\frac{0-\left(-6\right)}{2}=3\end{array}$

So, the equation of the graph is

  $\begin{array}{l}y=-3+\left(3\right)\sin \left(\frac{\pi }{2}\right)x\\ y=3\sin \left(\frac{\pi }{2}\right)x-3\end{array}$