Chapter 3, Problem 78RE

To determine

To graph the rational function and show the x-andy-intercepts and asymptotes.

Explanation of Solution

Given information:

The rational function is $r\left(x\right)=\frac{1}{{(x+2)}^2}$ .

Graph:

Factor: $r\left(x\right)=\frac{1}{{(x+2)}^2}$

x-intercepts: the x-intercept are the zeros of the numerator,

here x-intercept does not exist.

y-intercepts: to find y-intercept, substitute $x=0$ in the original form of the function,

  $\begin{array}{l}r\left(0\right)=\frac{1}{{(0+2)}^2}\\ r\left(0\right)=\frac{1}{4}\end{array}$

So, y-intercept is $\frac{1}{4}$ .

Horizontal asymptote: here the degree of numerator is less than degree of denominator,

So horizontal asymptote is $y=0$ .

Vertical asymptote: the vertical asymptote is occurs where denominator is zero,

  $\begin{array}{l}{\left(x+2\right)}^2=0\\ x+2=0\\ x=-2\end{array}$

So, the vertical asymptote is $x=-2$ .

Slant asymptote: : the slant asymptote is occurs when the numerator degree is one more than denominator, here the degree of denominator is greater than degree od numerator, so slant asymptote does not exist.

Use the above information together with some additional values which is show in table below

To sketch the graph,

x	y
-3	1
-1	1
1	.11
2	0.625

The graph is obtained as:

Interpretation:

From the above graph it can be observed that the x-intercept does not exist, y-intercept is $\frac{1}{4}$ , horizontal asymptote $y=0$ and vertical asymptote is $x=-2$ and slant asymptote does not exist.