# Slant asymptote

### Slant asymptote

When the polynomial in the numerator is exactly one degree higher than the polynomial in the denominator, there is a slant asymptote in the rational function. To determine the slant asymptote, we need to perform long division.

#### Lessons

For a simplified rational function, when the numerator is exactly one degree higher than the denominator, the rational function has a slant asymptote. To determine the equation of a slant asymptote, we perform long division.

• Introduction
Introduction to slant asymptote

i) What is a slant asymptote?

ii) When does a slant asymptote occur?

iii) Overview: Slant asymptote

• 1.
Algebraically Determining the Existence of Slant Asymptotes

Without sketching the graph of the function, determine whether or not each function has a slant asymptote:

a)
$a(x) = \frac{x^{2} - 3x - 10}{x - 5}$

b)
$b(x) = \frac{x^{2} - x - 6}{x - 5}$

c)
$c(x) = \frac{5x^{3} - 7x^{2} + 10}{x + 1}$

• 2.
Determining the Equation of a Slant Asymptote Using Long Division

Determine the equations of the slant asymptotes for the following functions using long division.

a)
$b(x) = \frac{x^{2} - x - 6}{x - 5}$

b)
$p(x) = \frac{-9x^{2} + x^{4}}{x^{3} - 8}$

• 3.
Determining the Equation of a Slant Asymptote Using Synthetic Division

Determine the equations of the slant asymptotes for the following functions using long division.

a)
$b(x) = \frac{x^{2} - x - 6}{x - 5}$

b)
$f(x) = \frac{x^{2} + 1}{x - 3}$

• 4.
Graphing Rational Functions Incorporating All 3 Kinds of Asymptotes

Sketch the rational function

$f(x) = \frac{2x^{2} - x - 6}{x + 2}$

by determining:

i) points of discontinuity

ii) vertical asymptotes

iii) horizontal asymptotes

iv) slant asymptote