Chapter 4.6, Problem 51E

(a)

To determine

To compare: the graphs when $n$ is even and $n$ is odd.

Explanation of Solution

Given information:

Use a graphing calculator to graph the function $f\left(x\right)={\left(x+3\right)}^n$ for $n=2,3,4,5,6$ and 7.

Calculation:

Input equation $Y_1={\left(X+3\right)}^2$ using key $\left(X+3\right)\wedge 2$ . The press GRAPH key.

  

Input equation $Y_1={\left(X+3\right)}^3$ using key $\left(X+3\right)\wedge 3$ . The press GRAPH key.

  

Input equation $Y_1={\left(X+3\right)}^4$ using key $\left(X+3\right)\wedge 4$ . The press GRAPH key.

  

Input equation $Y_1={\left(X+3\right)}^5$ using key $\left(X+3\right)\wedge 5$ . The press GRAPH key.

  

Input equation $Y_1={\left(X+3\right)}^6$ using key $\left(X+3\right)\wedge 6$ . The press GRAPH key.

  

Input equation $Y_1={\left(X+3\right)}^7$ using key $\left(X+3\right)\wedge 7$ . The press GRAPH key.

  

For all function $f\left(x\right)=\infty$ as $x\to \infty$ . When $n$ is even $f\left(x\right)=\infty$ as $x\to -\infty$ but when $n$ is odd, $f\left(x\right)=-\infty$ as $x\to -\infty$ .

(b)

To determine

To describe: the behavior of the graph near the zero $x=-3$ as $n$ increases.

Explanation of Solution

Given information:

Use a graphing calculator to graph the function $f\left(x\right)={\left(x+3\right)}^n$ for $n=2,3,4,5,6$ and 7.

Calculation:

As $n$ increases, the graph becomes more flat near the zero $x=-3$ .

(c)

To determine

To describe: the behavior of the graph of $g\left(x\right)={\left(x-4\right)}^{20}$ near $x=4$ using results from parts (a) and (b).

Explanation of Solution

Given information:

Use a graphing calculator to graph the function $f\left(x\right)={\left(x+3\right)}^n$ for $n=2,3,4,5,6$ and 7.

Calculation:

The graph of $g$ becomes more vertical and straight near $x=4$ .