Pre-Calculus 12 · Mathematics · Lesson 2.3
Rational functions
- Level
- Grade 12
- Reading time
- 4 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
Asymptotes.
What is an asymptote of a rational function?
Short answer
An asymptote is a line that the graph of a rational function approaches but never reaches. A vertical asymptote occurs where the denominator is zero and the numerator is not, marking values the function cannot take. A horizontal asymptote describes the function’s end behaviour, determined by comparing the degrees of the numerator and denominator.
Concept preview
What you will be able to do
- 01Find the domain, vertical asymptotes, and holes of a rational function
- 02Determine horizontal or slant asymptotes by comparing degrees
- 03Solve rational equations and inequalities and sketch the graph
A rational function is a ratio of polynomials, with . Its graph can break apart at the zeros of the denominator, race off toward invisible guide lines called asymptotes, and even have single missing points called holes. Everything about its shape comes from comparing the numerator and denominator, so we analyse them factor by factor.
School Plan
Rational functions
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Related topics
Frequently asked questions
- Where is a vertical asymptote?
- At x-values that make the denominator zero while the numerator is non-zero, since the function is undefined there.
- How do you find a horizontal asymptote?
- Compare the degrees of numerator and denominator; the ratio of leading coefficients gives it when the degrees are equal.