# Slant asymptote #### All You Need in One Place

Everything you need for better marks in primary, GCSE, and A-level classes. #### Learn with Confidence

We’ve mastered the UK’s national curriculum so you can study with confidence. #### Instant and Unlimited Help

24/7 access to the best tips, walkthroughs, and practice questions.

0/1
##### Intros
###### Lessons
1. Introduction to slant asymptote

i) What is a slant asymptote?

ii) When does a slant asymptote occur?

iii) Overview: Slant asymptote

0/8
##### 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

###### Free to Join!
StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun - with achievements, customizable avatars, and awards to keep you motivated.
• #### Easily See Your Progress We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.
• #### Make Use of Our Learning Aids   ###### Practice Accuracy

Get quick access to the topic you're currently learning.

See how well your practice sessions are going over time.

Stay on track with our daily recommendations.

• #### Earn Achievements as You Learn   Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.
• #### Create and Customize Your Avatar   Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
###### 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.