Chapter 10.5, Problem 14E

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 $y=8\cos \frac{\pi x}{6}-3$ .

Given information: The maximum point, $A\left(0,5\right)$ and minimum point, $B\left(6,-11\right)$ .

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 $M=5$ and the minimum value is $m=-11$ .

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

  $\begin{array}{c}k=\frac{M+m}{2}\\ =\frac{5+\left(-11\right)}{2}\\ =-\frac{6}{2}\\ =-3\end{array}$

At $x=0$ , the graph has its maximum at the $y$ axis. So, it is a cosine function with no horizontal shift.

  $h=0$

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

  $\begin{array}{c}\left|a\right|=\left|\frac{M-m}{2}\right|\\ =\left|\frac{-11-5}{2}\right|\\ =\left|\frac{-16}{2}\right|\\ =8\end{array}$

Calculate the period of the function.

  $\begin{array}{c}\text{Period}=2\left(6-0\right)\\ =12\end{array}$

Substitute $12$ for period in the formula $P=\frac{2\pi }{b}$ to calculate the value of $b$ .

  $\begin{array}{c}12=\frac{2\pi }{b}\\ b=\frac{2\pi }{12}\\ b=\frac{\pi }{6}\end{array}$

Substitute $8$ for $a$ , $-3$ for $k$ , $0$ for $h$ and $\frac{\pi }{6}$ for $b$ in the general equation of cosine.

  $\begin{array}{c}y=8\cos \left(\frac{\pi }{6}\right)\left(x-0\right)+\left(-3\right)\\ y=8\cos \frac{\pi x}{6}-3\end{array}$

Therefore, the sinusoid function for the given maximum and minimum points is $y=8\cos \frac{\pi x}{6}-3$ .