# Slant asymptote

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##### Intros
###### Lessons
1. Introduction to slant asymptote

i) What is a slant asymptote?

ii) When does a slant asymptote occur?

iii) Overview: Slant asymptote

##### Examples
###### Lessons
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:

1. $a(x) = \frac{x^{2} - 3x - 10}{x - 5}$
2. $b(x) = \frac{x^{2} - x - 6}{x - 5}$
3. $c(x) = \frac{5x^{3} - 7x^{2} + 10}{x + 1}$
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.

1. $b(x) = \frac{x^{2} - x - 6}{x - 5}$
2. $p(x) = \frac{-9x^{2} + x^{4}}{x^{3} - 8}$
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.

1. $b(x) = \frac{x^{2} - x - 6}{x - 5}$
2. $f(x) = \frac{x^{2} + 1}{x - 3}$
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

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###### Topic Notes
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.

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.