Rational Functions
A rational function is a ratio of polynomials.
About this topic
A rational function is a ratio of two polynomials, f(x) = p(x)/q(x). Where q(x) = 0 the function is undefined, producing either a vertical asymptote or a hole if the factor cancels. A horizontal asymptote depends on comparing the degrees of the numerator and denominator.
Factor to find intercepts, holes and asymptotes before graphing. Rational functions model situations such as average cost per unit and concentration in mixtures, where something is divided by a changing quantity.
Worked example: f(x) = (x² − 4)/(x² − x − 6). Factor to get (x − 2)(x + 2)/((x − 3)(x + 2)). The factor x + 2 cancels, leaving a hole at x = −2, while x = 3 gives a vertical asymptote. The degrees are equal, so the horizontal asymptote is the ratio of leading coefficients, y = 1. The x-intercept is at x = 2. Plot those features first and the sketch almost draws itself.
Rational functions appear when a company spreads a fixed cost over more units, so average cost = (fixed + variable)/n falls toward the variable cost per unit as n grows. Students often forget that cancelling a factor still leaves a hole, and that the original function is undefined there. Also, a graph can cross a horizontal asymptote in the middle; the asymptote describes the end behaviour as x gets very large or very small.
Key ideas
- Zeros of the denominator give vertical asymptotes
- Compare degrees for horizontal asymptotes
- Holes appear at cancelled factors
Quick check
Use cases: where Rational Functions helps
- Homework: review the key idea behind Rational Functions before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Rational Functions without notes.
- Visual learning: play with the interactive lab to see how Rational Functions changes when you adjust its inputs.
- Teaching a friend: explain Rational Functions in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Rational Functions?
A rational function is a ratio of polynomials.
What will I learn in Rational Functions?
Zeros of the denominator give vertical asymptotes. Compare degrees for horizontal asymptotes. Holes appear at cancelled factors.
Is Rational Functions part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Rational Functions lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Rational Functions?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.