Chapter 9.4, Problem 41E

To determine

To find: The graph for the given function.

Answer to Problem 41E

The required graph is shown in Figure 1.

Explanation of Solution

Given:

The given function is $g\left(x\right)=-\cos x+3$ .

Calculation:

Consider the given function is,

  $g\left(x\right)=-\cos x+3$

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

Then,

  $\begin{array}{l}a=\left|-1\right|\\ b=1\\ h=0\\ k=3\end{array}$

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

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

Thus, the vertical shift is $k=3$ and the horizontal shift is $h=0$ .

Then, determine the five key points as,

First,

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

Second,

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

Third, the maximums

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

Fourth point the maximum.

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

Fifth point, the minimum.

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

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

  $y=3$

Then $\left(\frac{\pi }{2},3\right)$ is $\left(\frac{\pi }{2},-3\right)$ and $\left(\frac{3\pi }{2},3\right)$ is $\left(\frac{3\pi }{2},-3\right)$ .

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

  

Figure 1