Denominator = 0
For the SL form, solve \(cx+d=0\) and check the function is genuinely undefined there.
IB MATHEMATICS AA · TOPIC 2 FUNCTIONS · PREMIUM
Asymptotes & Rational Functions
This chapter keeps asymptotes visual and practical. At SL, students focus on linear-over-linear rational functions, vertical and horizontal asymptotes, intercepts, inverse functions and links with logarithmic and exponential graphs. Higher-degree rational functions, cancelled factors and oblique asymptotes are clearly marked as HL material.
THE SL CORE
Find the structure first, then sketch.
For a linear-over-linear function, the key information is compact: the denominator gives the vertical asymptote, the ratio of leading coefficients gives the horizontal asymptote, and the intercepts anchor the branches. Students then use this information to construct the graph instead of copying a calculator picture.
Leading-coefficient ratio
For equal linear degrees, the graph approaches \(y=\frac{a}{c}\) for large positive or negative \(x\).
Anchor the sketch
Use the numerator for \(x\)-intercepts and substitute \(x=0\) for the \(y\)-intercept whenever it is defined.
Logs ↔ exponentials
A vertical asymptote of a logarithm becomes a horizontal asymptote of its inverse exponential graph.
HL MATERIAL
Oblique asymptotes come from polynomial division.
When the numerator degree is exactly one larger than the denominator degree, the quotient from polynomial long division gives a straight-line asymptote. The notes show the division line by line: divide the leading terms, multiply back, subtract, repeat, then separate the remainder.
\[f(x)=q(x)+\frac{r(x)}{d(x)}\]
The quotient is the line the graph approaches for large \(|x|\). This entire higher-degree rational-function section is marked clearly so SL students can skip it.
INSIDE THE PACK
Shorter notes, harder practice.
19 pages
Simple asymptote rules, clear graphs, SL/HL separation, long division, crossing asymptotes and logarithm/exponential connections.
25 questions · 10 pages
Many distinct cases, 3 SL + 2 HL past-paper structure adaptations, four linked synthesis tasks and one very difficult final HL challenge.
17 pages
Detailed algebra, long division, exact answers, GDC values and clear solution graphs for the sketching questions.
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Need Exponents & Logarithms first?
The Exponents & Logarithms pack develops the inverse-function ideas used again here when comparing logarithmic and exponential asymptotes.