Chapter 9.4, Problem 47E

To determine

To find: The graph for the given function.

Answer to Problem 47E

The required graph is shown in Figure 1.

Explanation of Solution

Given:

The given function is $g\left(x\right)=-4\cos \left(x+\frac{\pi }{4}\right)-1$ .

Calculation:

Consider the given function is,

  $g\left(x\right)=-4\cos \left(x+\frac{\pi }{4}\right)-1$

Compare the above equation with the general form $g\left(x\right)=a\cos \left(x-h\right)+k$ .

Then,

  $\begin{array}{l}a=\left|-4\right|\\ b=1\\ h=\frac{-\pi }{2}\\ k=-1\end{array}$

From above and from the given functionthe period is obtained as,

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

Then, determine the five key points as,

First,

  $\begin{array}{c}\left(\frac{1}{4}\cdot \frac{2\pi }{b}+h,k\right)=\left(\frac{1}{4}\cdot \frac{2\pi }{1}-\frac{\pi }{4},-1\right)\\ =\left(\frac{\pi }{4},-1\right)\end{array}$

Second,

  $\begin{array}{c}\left(\frac{3}{4}\cdot \frac{2\pi }{b}+h,k\right)=\left(\frac{3}{4}\cdot \frac{2\pi }{1}-\frac{\pi }{4},-1\right)\\ =\left(\frac{5\pi }{4},-1\right)\end{array}$

Third, the maximum is,

  $\begin{array}{c}\left(0+h,a+k\right)=\left(-\frac{\pi }{4},-4,-1\right)\\ =\left(-\frac{\pi }{4},-5\right)\end{array}$

Fourth point the maximum.

  $\begin{array}{c}\left(\frac{2\pi }{b}+h,a+k\right)=\left(\frac{2\pi }{1}-\frac{\pi }{4},-4-1\right)\\ =\left(\frac{7\pi }{4},-5\right)\end{array}$

Fifth point, the minimum.

  $\begin{array}{c}\left(\frac{1}{2}\cdot \frac{2\pi }{b}+h,-a+k\right)=\left(\frac{1}{2}\cdot \frac{2\pi }{1}-\frac{\pi }{4},4-1\right)\\ =\left(\frac{3\pi }{4},3\right)\end{array}$

Reflect the graph of $a<0$ the graph is reflected in the middle as,

  $y=-1$

Then $\left(\frac{\pi }{4},-1\right)$ is $\left(\frac{\pi }{4},1\right)$ , and $\left(\frac{5\pi }{4},-1\right)$ is $\left(\frac{5\pi }{4},1\right)$ .

From the points, the graph for the function is shown in Figure 1.

  

Figure 1