Chapter 4.2, Problem 92E

To determine

Find the asymptotes of the graph of the rational function.

Answer to Problem 92E

Vertical Asymptote: $x=\frac{5+\sqrt{89}}{4}$; $x=\frac{5-\sqrt{89}}{4}$; No Horizontal Asymptote The Slant Asymptote $y=(\frac{1}{2}x-\frac{7}{4})$

Explanation of Solution

Given: the rational function $f(x)=\frac{x^3-6x^2+x-1}{2x^2-5x-8}$

Concept Used:

There are three kinds of asymptotes: horizontal, vertical and oblique.

An asymptote is a line that the graph of a function approaches but never actually touches. There are vertical asymptotes, horizontal asymptotes, and oblique asymptotes. Oblique asymptotes are also called slant asymptotes. Vertical and horizontal asymptotes are vertical and horizontal lines, respectively. An oblique, or slant, asymptote is an asymptote that is neither vertical nor horizontal

  

Calculation:

To find the vertical asymptotes, simply make denominator equal to zero and solve for x.

For Horizontal Asymptotes,when the degree of the numerator is greater than the degree of the denominator there is no Horizontal Asymptote. .

A slant (oblique) asymptote occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator.

To find the slant asymptote you must divide the numerator by the denominator using either long division or synthetic division.

By Long division we get: $f(x)=\frac{x^3-6x^2+x-1}{2x^2-5x-8}=(\frac{1}{2}x-\frac{7}{4})+\frac{\frac{55x}{4}+13}{2x^2-5x-8}$

The slant or Oblique asymptote: $y=(\frac{1}{2}x-\frac{7}{4})$

Vertical Asymptote:   $f(x)=\frac{x^3-6x^2+x-1}{2x^2-5x-8}$ Denominator = 0   $2x^2-5x-8=0$ By using quadratic formula: Vertical Asymptotes are:   $x=\frac{5+\sqrt{89}}{4}$ and $x=\frac{5-\sqrt{89}}{4}$ Horizontal Asymptotes: For:   $f(x)=\frac{x^3-6x^2+x-1}{2x^2-5x-8}$; When the degree of the numerator is greater than the degree of the denominator there is no Horizontal Asymptote. Vertical Asymptotes are:   $x=\frac{5+\sqrt{89}}{4}$ and $x=\frac{5-\sqrt{89}}{4}$ No Horizontal Asymptote. The slant or Oblique asymptote:   $y=(\frac{1}{2}x-\frac{7}{4})$

Thus, Vertical Asymptote: $x=\frac{5+\sqrt{89}}{4}$; $x=\frac{5-\sqrt{89}}{4}$; No Horizontal Asymptote The Slant Asymptote $y=(\frac{1}{2}x-\frac{7}{4})$