Chapter 10.5, Problem 15E

Solution

To write: The function for the sinusoid with the given maximum point $A$ and minimum point $B$ .

The sinusoid function for the given maximum and minimum points is $y=2\sin \frac{1}{4}\left(x+2\pi \right)-2$ .

Given information: The maximum point, $A\left(0,0\right)$ and minimum point $B\left(4\pi ,-4\right)$ .

Formula used: The general equation of the sinusoid function is given as:

  $y=a\sin b\left(x-h\right)+k$ or $y=a\cos b\left(x-h\right)+k$

Here, $a$ is amplitude, $h$ is the horizontal shift, $k$ is the vertical shift and $\frac{2\pi }{b}$ is the period.

Calculation:

From the given points the maximum value is $M=0$ and the minimum value is $m=-4$ .

Calculated the vertical shift of the graph by the mean of the maximum and minimum values.

  $\begin{array}{c}k=\frac{M+m}{2}\\ =\frac{0+\left(-4\right)}{2}\\ =-\frac{4}{2}\\ =-2\end{array}$

The maximum point lies at the origin. So, it is a sine function with horizontal shift $-2\pi$ .

  $h=-2\pi$

The amplitude can be calculated by the half of the difference between maximum and minimum.

  $\begin{array}{c}\left|a\right|=\left|\frac{M-m}{2}\right|\\ =\left|\frac{0-\left(-4\right)}{2}\right|\\ =\left|\frac{4}{2}\right|\\ =2\end{array}$

Calculate the period of the function.

  $\begin{array}{c}\text{Period}=2\left(4\pi -0\right)\\ =8\pi \end{array}$

Substitute $8\pi$ for period in the formula $P=\frac{2\pi }{b}$ to calculate the value of $b$ .

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

Substitute $2$ for $a$ , $-2$ for $k$ , $-2\pi$ for $h$ and $\frac{1}{4}$ for $b$ in the general equation of sine.

  $\begin{array}{c}y=2\sin \left(\frac{1}{4}\right)\left(x-\left(-2\pi \right)\right)+\left(-2\right)\\ y=2\sin \frac{1}{4}\left(x+2\pi \right)-2\end{array}$

Therefore, the sinusoid function for the given maximum and minimum points is $y=2\sin \frac{1}{4}\left(x+2\pi \right)-2$ .