Chapter 9.4, Problem 26E

To determine

To find: The graph for the given function.

Answer to Problem 26E

The graph is shown in Figure 1.

Explanation of Solution

Given:

The given function is $g\left(x\right)=\cos x-4$ .

Calculation:

Consider the given function is,

  $g\left(x\right)=\cos x-4$

Consider the general form is,

  $y=a\sin \left(x-h\right)+k$

From above and from the given function. The amplitude is 1, the period is obtained as,

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

Thus, the vertical shift is $-4$ and the horizontal shift is 0.

Then, determine the five key points as,

First,

  $\begin{array}{c}\left(\frac{1}{4}\times \frac{2\pi }{b},0+k\right)=\left(\frac{1}{4}\times \frac{2\pi }{1},0-4\right)\\ =\left(\frac{\pi }{2},-4\right)\end{array}$

Second,

  $\begin{array}{c}\left(\frac{3}{4}\times \frac{2\pi }{b},0+k\right)=\left(\frac{3}{4}\times \frac{2\pi }{1},0-4\right)\\ =\left(\frac{3\pi }{2},-4\right)\end{array}$

Third,

  $\left(h,a+k\right)=\left(0,-3\right)$

Fourth point the maximum.

  $\begin{array}{c}\left(\frac{2\pi }{b},a+k\right)=\left(0,1-4\right)\\ =\left(2\pi ,-3\right)\end{array}$

Fifth point, the minimum.

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

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

  

Figure 1