Rational Functions (AA HL)
A rational function is one polynomial divided by another, and its graph is shaped almost entirely by what happens near the values that make the denominator zero and what happens as \(x\) gets very large. This topic covers finding vertical, horizontal and oblique asymptotes, sketching the graph from them, and the special case of the self-inverse reciprocal function.
What the syllabus says
This topic maps onto two points in the official IB Analysis & Approaches syllabus, one at SL and one AHL extension.
| Code | Syllabus content |
|---|---|
| SL2.8 | The reciprocal function \(f(x)=\tfrac1x,\ x\neq0\): its graph and self-inverse nature. Rational functions of the form \(f(x)=\dfrac{ax+b}{cx+d}\) and their graphs, including the equations of the vertical asymptote \(x=-\tfrac dc\) and horizontal asymptote \(y=\tfrac ac\). |
| AHL2.13 | Rational functions of the form \(f(x)=\dfrac{ax+b}{cx^2+dx+e}\) and \(f(x)=\dfrac{ax^2+bx+c}{dx+e}\), and their graphs. Sketches should include all asymptotes (horizontal, vertical and oblique) and any intercepts with the axes. |
The linear-over-linear case is core SL content, examinable at HL; the quadratic/oblique-asymptote case is an AHL-only extension.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a vertical asymptote?
A vertical asymptote is a vertical line \(x=a\) that a graph gets closer and closer to but never crosses, occurring where the denominator of a rational function is zero (and the numerator isn't zero there too).
e.g. \(f(x)=\dfrac{1}{x-3}\) has vertical asymptote \(x=3\), since the denominator is zero there.
What is a horizontal asymptote?
A horizontal asymptote is a horizontal line \(y=k\) that a graph approaches as \(x\to\pm\infty\). For a rational function where numerator and denominator have the same degree, \(k\) is the ratio of the leading coefficients.
e.g. \(f(x)=\dfrac{2x+1}{x-1}\to2\) as \(x\to\pm\infty\), so the horizontal asymptote is \(y=2\).
What is an oblique asymptote?
An oblique (or slant) asymptote is a straight line \(y=mx+c\), with \(m\neq0\), that a graph approaches as \(x\to\pm\infty\). It occurs when the numerator's degree is exactly one higher than the denominator's, and is found by polynomial division.
e.g. \(f(x)=\dfrac{x^2+3x+1}{x+1}=x+2-\dfrac{1}{x+1}\), so the oblique asymptote is \(y=x+2\).
What is the reciprocal function?
The reciprocal function is \(f(x)=\tfrac1x\), the simplest rational function. It's self-inverse, meaning applying it twice returns the original input, and it's undefined at \(x=0\).
e.g. \(f(4)=\tfrac14\), and \(f\!\left(\tfrac14\right)=4\) - applying \(f\) twice returns 4.
What is a removable discontinuity (hole)?
A removable discontinuity, or "hole", happens when a factor cancels from both the numerator and denominator of a rational function. The function is undefined at that single \(x\)-value, but unlike a vertical asymptote it doesn't blow up there - the graph just has a gap.
e.g. \(f(x)=\dfrac{x^2-1}{x-1}=x+1\) for \(x\neq1\), leaving a hole at \(x=1\) (where the simplified line would give \(y=2\)).
Key formulas
Two formulas give the asymptotes of a linear-over-linear rational function directly; oblique asymptotes need polynomial division instead. The two tables below summarise them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The asymptote formulas for \(f(x)=\dfrac{ax+b}{cx+d}\) are given in the formula booklet; there's no closed-form formula for an oblique asymptote, since it depends on carrying out the division.
| Formula | Used for | Booklet? |
|---|---|---|
| \(x=-\dfrac{d}{c}\) | Vertical asymptote of \(\dfrac{ax+b}{cx+d}\) | ✓ Yes |
| \(y=\dfrac{a}{c}\) | Horizontal asymptote of \(\dfrac{ax+b}{cx+d}\) | ✓ Yes |
| Divide, then read off the quotient | Oblique asymptote (numerator one degree higher) | Not in booklet |
Linear-over-linear vs quadratic-involving rational functions
Moving from SL to AHL adds one more type of asymptote and, sometimes, a hole to look out for.
| Feature | \(f(x)=\dfrac{ax+b}{cx+d}\) | \(f(x)=\dfrac{ax^2+bx+c}{dx+e}\) |
|---|---|---|
| Vertical asymptote(s) | One, at \(x=-\tfrac dc\) | One, at \(x=-\tfrac ed\) (unless it cancels to a hole) |
| End behaviour | Horizontal asymptote \(y=\tfrac ac\) | Oblique asymptote, found by division |
| Can cross its own asymptote? | Horizontal one, yes, for some finite \(x\) | Oblique one, yes, for some finite \(x\) |
Types of asymptote
Every rational function's graph is controlled by these three behaviours - which ones apply depends on the degrees of the numerator and denominator.
Vertical
Occurs where the denominator is zero and the numerator isn't - the function is undefined and diverges to \(\pm\infty\) either side.
Horizontal
Occurs when the numerator and denominator have equal degree - the graph flattens towards \(y=\) (ratio of leading coefficients) as \(x\to\pm\infty\).
Oblique
Occurs when the numerator's degree is exactly one more than the denominator's - divide to find the linear part the graph approaches.
Sketching a rational function
A full sketch combines the asymptotes with a handful of easy-to-find extra features.
Intercepts
Set the numerator to zero for \(x\)-intercepts; evaluate \(f(0)\) for the \(y\)-intercept.
Checking for a hole
Before finding the vertical asymptote, check whether numerator and denominator share a common factor - if they do, it's a hole, not an asymptote.
Turning points
For a quadratic-over-linear function, differentiate (usually with the quotient rule) and set \(f'(x)=0\) to find any local maximum or minimum.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Let \(f(x)=\dfrac{2x+1}{x-3}\).
(a) State the equation of the vertical asymptote.
(b) State the equation of the horizontal asymptote.
(c) Find the \(y\)-intercept.
(d) State the domain of \(f.\)
Worked solution
(a) \(x=3.\) A1
(b) As \(x\to\pm\infty,\ f\to2,\) so \(y=2.\) A1
(c) \(f(0)=\dfrac{1}{-3}=-\tfrac13.\) A1
(d) All real \(x\) except \(x=3.\) A1
Let \(f(x)=\dfrac{2x+1}{x-1}\).
(a)(i) State the vertical asymptote.
(a)(ii) State the horizontal asymptote.
(b)(i) Find the \(x\)-intercept.
(b)(ii) Find the \(y\)-intercept.
(c) Sketch the graph.
Worked solution
(a) Asymptotes. VA \(x = 1\) A1 HA \(y = 2\) (equal degrees). A1
(b) Intercepts. \(y\)-int \(f(0) = -1\) M1
\(y\)-int \(=-1\), i.e. \((0,-1).\) A1 \(x\)-int \(2x+1 = 0 \Rightarrow x = -\tfrac12.\) A1
(c) Two branches approaching \(x = 1\) and \(y = 2.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Swapping the vertical and horizontal asymptote formulas. For \(\dfrac{ax+b}{cx+d}\), the vertical asymptote comes from the denominator (\(x=-\tfrac dc\)) and the horizontal one from the leading coefficients (\(y=\tfrac ac\)) - mixing them up is an easy slip under time pressure.
- Claiming a horizontal asymptote when there isn't one. A horizontal asymptote only exists when the numerator's degree is less than or equal to the denominator's - if the numerator's degree is exactly one higher, the function has an oblique asymptote instead.
- Not checking for a common factor before finding the vertical asymptote. If numerator and denominator share a factor, it cancels to leave a hole, not an asymptote, at that \(x\)-value.
- Assuming a graph can never cross its own asymptote. That's only true for vertical asymptotes. A horizontal or oblique asymptote describes end behaviour as \(x\to\pm\infty\), and the graph can cross it for some finite \(x\).
Using your GDC
There isn't a single fixed button sequence for this topic, since asymptotes are usually identified algebraically rather than read off a graph.
On Paper 2, graphing a rational function is a quick way to check your asymptotes and intercepts once you've found them algebraically. Look for a table-of-values or trace feature to see the function's behaviour either side of a suspected vertical asymptote, and use the graph's zoom-out view to check its end behaviour matches the horizontal or oblique asymptote you calculated. Be cautious near a vertical asymptote, though - some calculators join the two branches with a near-vertical line rather than leaving a genuine gap, which can make the graph look misleading if you don't already know where the asymptote is.
See the full GDC guide for calculator-specific graphing and table steps.
Ready to practise properly?
Rational functions questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
How do I find the asymptotes of a rational function?
Set the denominator equal to zero for vertical asymptotes (as long as the numerator isn't also zero there). Compare the degrees of the numerator and denominator for horizontal or oblique asymptotes: same degree gives a horizontal asymptote from the ratio of leading coefficients, numerator one degree higher gives an oblique asymptote found by polynomial division.
What's the difference between a vertical asymptote and a hole?
Both happen where the denominator is zero, but a hole occurs when the same factor also cancels from the numerator. A vertical asymptote is a value the function genuinely blows up towards; a hole is a single missing point where the simplified function would otherwise be defined.
Can a graph cross its own asymptote?
It can cross a horizontal or oblique asymptote for some finite value of \(x\), since those only describe the function's behaviour as \(x\) tends to infinity. A graph can never cross a vertical asymptote, since the function is undefined exactly there.
Can I use my GDC to sketch a rational function?
Yes - graphing the function is the fastest way to check your asymptotes and intercepts on Paper 2. Just be aware that some calculators draw a near-vertical line through an asymptote rather than leaving a genuine gap, so don't rely on the picture alone to identify where the asymptotes are. See the GDC guide for model-specific instructions.
Sub-topics
Rational Functions broken down into its individual skills, each with its own focused page.
Related topics
More Functions topics from the same AA HL syllabus unit, in case you want to keep going.