Chapter 13.4, Problem 25WE

To determine

The equation of the curve.

Answer to Problem 25WE

The equation of the curve is $y=4+3\sin 2x$ .

Explanation of Solution

Given:

  $\begin{array}{l}M=7\\ m=1\\ \text{Period}\text{=}\pi \end{array}$

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,

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

We know that,

The maximum value of the curve is 7 and the minimum value is 1.

Therefore,

  $\begin{array}{l}c=\frac{M+m}{2}=\frac{7+\left(1\right)}{2}=4\\ a=\frac{M-m}{2}=\frac{7-\left(1\right)}{2}=3\end{array}$

So, the equation of the graph is

  $\begin{array}{l}y=4+\left(3\right)\sin \left(2\right)x\\ y=4+3\sin 2x\end{array}$