# Slant asymptote

- Lesson: 18:13
- Lesson: 2a4:13
- Lesson: 2b2:10
- Lesson: 2c0:54
- Lesson: 3a3:38
- Lesson: 3b8:49
- Lesson: 4a3:56
- Lesson: 4b2:26
- Lesson: 515:00

### Slant asymptote

#### 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.

- 1.Introduction to slant asymptote
i) What is a slant asymptote?

ii) When does a slant asymptote occur?

iii) Overview: Slant asymptote

- 2.
**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)$c(x) = \frac{5x^{3} - 7x^{2} + 10}{x + 1}$ - 3.
**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)$p(x) = \frac{-9x^{2} + x^{4}}{x^{3} - 8}$ - 4.
**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)$f(x) = \frac{x^{2} + 1}{x - 3}$ - 5.
**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