Rational functions — asymptotes, holes, and sketches (IB Math AA HL)

Key ideas

• Hole: if (x − a) is a factor of both numerator and denominator, the graph has a missing point at x = a. Find its y-value from the simplified function.
• Vertical asymptote: x = a where the simplified denominator is zero. Check the sign of f(x) just either side to see whether the graph goes up or down.
• Horizontal asymptote: numerator of lower degree gives y = 0; equal degrees give y = (leading coefficient of the top) ÷ (leading coefficient of the bottom).
• Oblique asymptote: when the numerator's degree is one more, divide to write f(x) = mx + c + remainder ÷ denominator. The asymptote is y = mx + c, because the last term tends to 0.
• Intercepts: f(0) for the y-intercept; zeros of the simplified numerator for the x-intercepts.
• Label every asymptote with its equation on a sketch, and show the hole as an open circle.

Worked example

(a) Let f(x) = (x² + x − 2)/(x² − 4). Find the asymptotes, the hole and the intercepts of the graph of f. (b) Let g(x) = (2x² + 3x − 1)/(x + 2). Find the equations of the asymptotes of the graph of g.

1. (a) Factorise: f(x) = (x + 2)(x − 1) ÷ ((x − 2)(x + 2)) = (x − 1)/(x − 2) for x ≠ −2.
2. The factor (x + 2) cancels, so there is a hole at x = −2, with y = (−2 − 1)/(−2 − 2) = 3/4.
3. Vertical asymptote x = 2. Equal degrees with leading coefficients 1 and 1, so the horizontal asymptote is y = 1.
4. Intercepts: f(0) = (−1)/(−2) = 1/2, so (0, 1/2); the numerator is zero at x = 1, so (1, 0).
5. (b) Divide: 2x² + 3x − 1 = (x + 2)(2x − 1) + 1, so g(x) = 2x − 1 + 1/(x + 2).
6. Vertical asymptote x = −2; as x → ±∞, 1/(x + 2) → 0, so the oblique asymptote is y = 2x − 1.

Answer: (a) x = 2 and y = 1; hole at (−2, 3/4); intercepts (0, 1/2) and (1, 0). (b) x = −2 and y = 2x − 1.

Exam-style HL AA questions on this topic

• Polynomials and Rational Functions questions — 79 questions with mark schemes, plus free worked examples
• All HL AA Functions questions by topic

See it on a graph

• Graph of y = 1/x
• Graph of y = (2x + 1)/(x − 1)
• Open the free Grapher

Maths IA ideas that use rational functions

• Is a hanging chain a parabola? Catenary versus quadratic models
• Modelling the Keeling curve: trend plus seasons
• Modelling bacterial growth in the biology lab
• Exponential or logistic? Modelling my country's population
• All HL AA IA ideas

Related in Functions

• Inequalities — rational and modulus
• Polynomial division and the factor theorem
• Logarithm laws and change of base

Revise Rational functions for the exam

• HL AA revision notes — every unit summarised, with the key results to learn
• HL AA past papers — specimen papers and timed practice papers
• Formulas for Functions — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S92 · Problem S27 · Problem S91 · Problem S26. All 22 →

Sibling topic in other IB Maths courses