Chapter 3.7, Problem 65E

To determine

To find: the slant asymptote, the vertical asymptotes, and sketch a graph of the function.

Answer to Problem 65E

The quotient $y=x+2$ is the slant asymptote.

Explanation of Solution

Given information:

Given function,

  $r\left(x\right)=\frac{x^2}{x-2}$

Calculation:

Consider the function,

  $r\left(x\right)=\frac{x^2}{x-2}$

Vertical asymptote of $r$ are the lines $x=a$ , where $a$ is a zero of the denominator.

Al slant asymptote occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator. To find the slant asymptote must divide the numerator by the denominator using either long division or synthetic division.

Here, denominator is $x-2$

So,

  $x-2=0$

This implies that,

  $x=2$

Therefore, vertical asymptote is 2.

Using long division, we get

  $\begin{array}{c}x-2\begin{array}{c}\hfill x+2\\ \hfill \overline{)x^2}\end{array}\\ \underset{\_}{x^2-2x}\\ 2x\\ \underset{\_}{2x-4}\\ 4\end{array}$

This implies that,

  $r\left(x\right)=\left(x+2\right)+\frac{4}{x-2}$

Therefore, the quotient $y=x+2$ is the slant asymptote.

Now to draw the graph of the function $r\left(x\right)=\frac{x^2}{x-2}$ , slant asymptote $y=x+2$ and vertical asymptote 2, find the value of the function for different values of $x$ as,

The required table is,

$x$	0	1	$-2$	$-6$	1.5	2.5	3	4	6	10
$r\left(x\right)=\frac{x^2}{x-2}$	0	$-1$	$-1$	$-4.5$	$-4.5$	12.5	9	8	9	12.5

Thus, the required graph is given below,