Chapter 9, Problem 34SGR

To determine

To graph: The rational function $f\left(x\right)=\frac{x+2}{{\left(x+5\right)}^2}$ .

Answer to Problem 34SGR

The graph is drawn below.

Explanation of Solution

Given information:

The given rational function is $f\left(x\right)=\frac{x+2}{{\left(x+5\right)}^2}$ .

Calculation:

If $f\left(x\right)=\frac{a\left(x\right)}{b\left(x\right)}$ , $a\left(x\right)$ and $b\left(x\right)$ are polynomial function with no common factor other than 1, and $b\left(x\right)\ne 0$ , then $f\left(x\right)$ has vertical asymptotes whenever $b\left(x\right)=0$ .

Therefore, for the function $f\left(x\right)=\frac{x+2}{{\left(x+5\right)}^2}$ .

  $\begin{array}{l}f\left(x\right)=\frac{x+2}{{\left(x+5\right)}^2}\\ f\left(x\right)=\frac{x+2}{\left(x+5\right)\left(x+5\right)}\end{array}$

So the function is undefined for $x=-5$ .

Therefore, $x=-5$ is the vertical asymptote. Also,

Since, the degree of $a\left(x\right)$ is less than the degree of $b\left(x\right)$ .

Therefore, horizontal asymptotes is the line $y=0$ .

Also,

  $\begin{array}{l}x+2=0\\ x=-2\end{array}$

There is a zero at $x=-2$ .

The graph of the function with vertical asymptote is drawn below: