Rational functions and asymptotes (IB Math AA SL)

Key ideas

• The reciprocal function f(x) = 1/x has the asymptotes x = 0 and y = 0, and it is its own inverse.
• Vertical asymptote: set the denominator to zero. For f(x) = (ax + b)/(cx + d) that is x = −d/c.
• Horizontal asymptote: for large x the constants stop mattering, so f(x) gets close to ax/cx = a/c, giving y = a/c.
• Intercepts: the y-intercept is f(0) = b/d, and the x-intercept is where the numerator is zero, x = −b/a.
• Domain: every real x except x = −d/c. Range: every real y except y = a/c.
• Write asymptotes as equations (x = 2, y = 3), not as single numbers: examiners take marks off for answers in the wrong form.

Worked example

f(x) = (3x − 1)/(x − 2). Write down the equations of the asymptotes and the coordinates of the intercepts.

1. Denominator zero: x − 2 = 0, so the vertical asymptote is x = 2.
2. Ratio of the x-coefficients: 3/1, so the horizontal asymptote is y = 3.
3. y-intercept: f(0) = (−1)/(−2) = 1/2, the point (0, 1/2).
4. x-intercept: 3x − 1 = 0, so x = 1/3, the point (1/3, 0).

Answer: Asymptotes x = 2 and y = 3; intercepts (0, 1/2) and (1/3, 0). The domain is x ≠ 2 and the range is y ≠ 3.

Exam-style SL AA questions on this topic

• Rational and Reciprocal Functions questions — 45 questions with mark schemes, plus free worked examples
• All SL AA Functions questions by topic

Do it on your GDC

A calculator is allowed on AA Paper 2, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Graphs: intersections, zeros, max and min on your calculator

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 and asymptotes

• Is a hanging chain a parabola? Catenary versus quadratic models
• Modelling the Keeling curve: trend plus seasons
• Which function best describes the arch of a famous bridge?
• Why do stopping distances grow so fast? Modelling the Highway Code table
• All SL AA IA ideas

Related in Functions

• Domain, range, and function composition
• Function transformations — translations, stretches, reflections
• Inverse functions and one-to-one mappings

Revise Rational functions and asymptotes for the exam

• SL AA revision notes — every unit summarised, with the key results to learn
• SL AA past papers — specimen papers and timed practice papers
• Formulas for Functions — on the one-page SL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• SL 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 S88 · Problem S23 · Problem S87 · Problem S22. All 22 →

Sibling topic in other IB Maths courses