Slant asymptote - Rational Functions

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.

Related concepts:
Curve sketching

Lessons

Notes:

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.
Algebraically Determining the Existence of Slant Asymptotes
Without sketching the graph of the function, determine whether or not each function has a slant asymptote:
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.
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.

Slant asymptote

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Get Started Now

Try reviewing these fundamentals first

AlgebraPolynomial long division

AlgebraPolynomial synthetic division

Nope, got it.

Play next lesson

Nope, got it.

Play next lesson

That's it, no more lessons

Goto next topic

Still don't get it?

Nope, I got it.

Still don't get it?

Review these basic concepts…

Polynomial long division

Polynomial synthetic division

Nope, I got it.

Play next lesson or Practice this topic

Start now and get better math marks!

Keep exploring StudyPug

Start now and get better math marks!

Keep exploring StudyPug

Intro Lesson:

Selected

Lesson: 1a

Lesson: 1b

Lesson: 1c

Lesson: 2a

Lesson: 2b

Lesson: 3a

Lesson: 3b

Lesson: 4

Intro

Learn

Practice

Do better in math today

Don't just watch, practice makes perfect.

Do better in math today

Don't just watch, practice makes perfect.

Lessons

Notes:

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.

Intro Lesson

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)=x2−3x−10x−5a(x) = \frac{x^{2} - 3x - 10}{x - 5}a(x)=x−5x2−3x−10​

b)

b(x)=x2−x−6x−5b(x) = \frac{x^{2} - x - 6}{x - 5} b(x)=x−5x2−x−6​

c)

c(x)=5x3−7x2+10x+1c(x) = \frac{5x^{3} - 7x^{2} + 10}{x + 1}c(x)=x+15x3−7x2+10​

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)=x2−x−6x−5b(x) = \frac{x^{2} - x - 6}{x - 5}b(x)=x−5x2−x−6​

b)

p(x)=−9x2+x4x3−8p(x) = \frac{-9x^{2} + x^{4}}{x^{3} - 8}p(x)=x3−8−9x2+x4​

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)=x2−x−6x−5b(x) = \frac{x^{2} - x - 6}{x - 5}b(x)=x−5x2−x−6​

b)

f(x)=x2+1x−3f(x) = \frac{x^{2} + 1}{x - 3}f(x)=x−3x2+1​

4.

Graphing Rational Functions Incorporating All 3 Kinds of Asymptotes Sketch the rational function f(x)=2x2−x−6x+2f(x) = \frac{2x^{2} - x - 6}{x + 2}f(x)=x+22x2−x−6​ by determining: i) points of discontinuity ii) vertical asymptotes iii) horizontal asymptotes iv) slant asymptote

Slant asymptote

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Algebra
Polynomial long division

Algebra
Polynomial synthetic division

or

Introduction to slant asymptote
i) What is a slant asymptote?

ii) When does a slant asymptote occur?

iii) Overview: Slant asymptote

Algebraically Determining the Existence of Slant Asymptotes
Without sketching the graph of the function, determine whether or not each function has a slant asymptote:

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

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

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

Determining the Equation of a Slant Asymptote Using Long Division
Determine the equations of the slant asymptotes for the following functions using long division.

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

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

Determining the Equation of a Slant Asymptote Using Synthetic Division
Determine the equations of the slant asymptotes for the following functions using long division.

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

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

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