Chapter 13.4, Problem 19WE

To determine

The equation of curve.

Answer to Problem 19WE

The equation of the curve is $y=3+3\cos \frac{\pi }{4}x$ .

Explanation of Solution

Given:

  

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,

  $a=\frac{M-m}{2}$

From the graph,

The value of curve at origin is maximum.

So, the given curve is a cosine curve.

The curve completes a cycle in interval of 8 . So, the period of the given cycle is 8 .

Therefore,

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

We know that,

The maximum value of the curve is 36 and the minimum value is 0.

Therefore,

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

So, the equation of the graph is

  $\begin{array}{l}y=3+\left(3\right)\cos \left(\frac{\pi }{4}\right)x\\ y=3+3\cos \frac{\pi }{4}x\end{array}$