Rational Functions and Asymptotes
Graphs with vertical and horizontal asymptotes.
Key idea
Graphs with vertical and horizontal asymptotes.
In depth
Where the denominator is zero but the numerator is not, the graph has a vertical asymptote. As x grows large, the graph approaches a horizontal asymptote determined by comparing degrees. A common factor creates a hole instead.
Worked example
f(x) = (2x + 1)/(x − 3): vertical asymptote x = 3, horizontal asymptote y = 2.
Formula
Real-world uses
- Modelling drug concentration over time and average cost per item as production grows
Common mistake
Treating a hole as an asymptote.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Rational Functions and Asymptotes?
Graphs with vertical and horizontal asymptotes. It is a topic at the advanced level in Algebra.
What is the formula for Rational Functions and Asymptotes?
The key formula is y = a/x.
Where is Rational Functions and Asymptotes used in real life?
Real-world examples: Modelling drug concentration over time and average cost per item as production grows.
What should I learn before Rational Functions and Asymptotes?
Make sure you are comfortable with Rational Equations, the previous topic in Algebra.
What is a common mistake with Rational Functions and Asymptotes?
Treating a hole as an asymptote.
How can I practise Rational Functions and Asymptotes?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Rational Functions and Asymptotes?
Move on to Radical Expressions next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.