Rational functions and graphs

CIE A-Level Further Mathematics· 30 min read

1. Core Definitions and Intercepts★★☆☆☆⏱ 10 min

📘 Definition

Rational Function

f(x) = \frac{P(x)}{Q(x)}

A function written as the ratio of two polynomials $P (x)$ (numerator) and $Q (x)$ (denominator), where $Q (x)$ is not the zero polynomial. The domain excludes all $x$ that make $Q (x) = 0$ .

Example:

$f (x) = \frac{x ^{2} - 1}{x - 2}$ is a rational function; $f (x) = \frac{x}{x + 1}$ is not.

To start analyzing any rational function, first calculate the intercepts, the key points where the graph crosses the axes.

📐 Worked Example

Find all intercepts of $f (x) = \frac{x ^{2} - 2 x - 3}{2 x - 4}$

1
Find the y-intercept by substituting $x = 0$ :
2
$f(0) = \frac{0 - 0 - 3}{0 - 4} = \frac{3}{4}$
3
The y-intercept is at $(0, \frac{3}{4})$ .
4
Find x-intercepts by setting the numerator equal to zero (check denominator is non-zero):
5
$x^2 - 2x - 3 = 0 \implies (x-3)(x+1) = 0 \implies x = 3, x = -1$
6
Denominator at $x = 3$ is $2 \neq = 0$ , at $x = - 1$ is $- 6 \neq = 0$ , so x-intercepts are at $(3, 0)$ and $(- 1, 0)$ .

Exam tip:

Always check for common factors between numerator and denominator: a shared root creates a hole, not an x-intercept.

2. Vertical and Horizontal Asymptotes★★☆☆☆⏱ 15 min

📘 Definition

Asymptote

A line that the graph of a rational function approaches arbitrarily closely as $x$ or $f (x)$ approaches $\pm \infty$ .

Vertical asymptotes occur at $x$ -values that make the denominator zero (after removing removable discontinuities). For horizontal asymptotes, we compare the degrees of the numerator ( $de g P$ ) and denominator ( $de g Q$ ):

If $de g P < de g Q$ : horizontal asymptote at $y = 0$
If $de g P = de g Q$ : horizontal asymptote at $y = \frac{a _{n}}{b _{m}}$ , where $a_{n}$ = leading coefficient of $P$ , $b_{m}$ = leading coefficient of $Q$
If $de g P > de g Q$ : no horizontal asymptote

📐 Worked Example

Find all vertical and horizontal asymptotes of $f (x) = \frac{3 x ^{2} - 3}{2 x ^{2} - 8}$

1
Factor numerator and denominator to check for common factors:
2
$f(x) = \frac{3(x-1)(x+1)}{2(x-2)(x+2)}$
3
No common factors, so denominator is zero at $x = 2$ and $x = - 2$ , so vertical asymptotes are $x = 2$ and $x = - 2$ .
4
Compare degrees: both numerator and denominator are degree 2, so calculate horizontal asymptote:
5
$y = \frac{\text{Leading coefficient of } P}{\text{Leading coefficient of } Q} = \frac{3}{2}$
6
Horizontal asymptote is $y = \frac{3}{2}$ .

3. Oblique (Slant) Asymptotes★★★☆☆⏱ 15 min

Oblique asymptotes only occur when the degree of the numerator is exactly one greater than the degree of the denominator. We find the equation using polynomial division.

📘 Definition

Oblique Asymptote

A non-vertical, non-horizontal straight line that the graph approaches as $x \to \pm \infty$ . The equation is given by the quotient of the polynomial division (the remainder can be ignored as it tends to zero).

📐 Worked Example

Find the oblique asymptote of $f (x) = \frac{2 x ^{2} - 3 x + 1}{x + 2}$

1
Confirm degrees: numerator degree 2, denominator degree 1, difference is 1, so oblique asymptote exists.
2
Divide numerator by denominator:
3
$2x^2 - 3x + 1 = (x+2)(2x - 7) + 15$
4
Rewrite $f (x)$ :
5
$f(x) = 2x - 7 + \frac{15}{x+2}$
6
As $x \to \pm \infty$ , $\frac{15}{x + 2} \to 0$ , so $f (x)$ approaches $y = 2 x - 7$ , which is the oblique asymptote.

4. Full Sketching Process★★★★☆⏱ 20 min

Follow this consistent step-by-step process to produce an exam-quality sketch that includes all required key features:

Factor numerator and denominator, identify any holes (removable discontinuities)
Find all x-intercepts and the y-intercept
Find all vertical, horizontal or oblique asymptotes
Check for any intersection points between the curve and non-vertical asymptotes
Test the sign of $f (x)$ in regions separated by vertical asymptotes, then sketch

📐 Worked Example

List all key features for a sketch of $f (x) = \frac{x ^{2} + x}{x - 2}$

1
Factor: $f (x) = \frac{x ( x + 1 )}{x - 2}$ , no common factors, no holes.
2
Intercepts: roots at $x = 0$ and $x = - 1$ , so intercepts at $(0, 0)$ and $(- 1, 0)$ .
3
Asymptotes: vertical asymptote at $x = 2$ . Degree of numerator is 1 greater than denominator, so find oblique asymptote by division:
4
$x^2 + x = (x-2)(x+3) + 6 \implies \text{Oblique asymptote } y = x+3$
5
Check for intersection between $f (x)$ and $y = x + 3$ :
6
$\frac{x^2+x}{x-2} = x+3 \implies x^2 + x = x^2 +x -6 \implies 0 = -6$
7
No solution, so the curve never crosses the oblique asymptote.

Exam tip:

When asked to find the number of solutions to an equation, rearrange to get the rational function equal to a straight line, then count intersections on your sketch.

5. Common Pitfalls

Wrong move:

Marking a hole (removable discontinuity) as a vertical asymptote

Why:

Common factors create holes, not asymptotes, and you will lose marks for incorrect asymptote labels

Correct move:

Always factor both numerator and denominator first, cancel common factors, and only mark asymptotes at remaining roots of the denominator

Wrong move:

Claiming a horizontal asymptote exists when the degree of the numerator is greater than the denominator

Why:

Horizontal asymptotes only exist when the degree of the numerator is less than or equal to the degree of the denominator

Correct move:

Compare degrees first: if the difference is 1, you have an oblique asymptote instead of a horizontal one

Wrong move:

Using the remainder from polynomial division as the oblique asymptote

Why:

The remainder tends to zero as $x$ approaches infinity, so it does not form the asymptote equation

Correct move:

The oblique asymptote is always the linear quotient term from the division, ignore the remainder

Wrong move:

Assuming curves never cross horizontal asymptotes

Why:

Unlike vertical asymptotes, curves can cross horizontal asymptotes at finite values of $x$ , only approaching them as $x \to \pm \infty$

Correct move:

Always check for intersections by setting $f (x)$ equal to the asymptote equation to find crossing points if they exist

Wrong move:

Forgetting to mark intercepts on your sketch

Why:

Exam markers require all key features (intercepts, asymptotes, holes) to be clearly labelled to get full marks

Correct move:

Always calculate and label both x-intercepts and the y-intercept before drawing the curve

6. Quick Reference Cheatsheet

Degree Comparison	Type of Non-Vertical Asymptote	How to Find Equation
$de g (P) < de g (Q)$	Horizontal	$y = 0$
$de g (P) = de g (Q)$	Horizontal	$y = \frac{leading P}{leading Q}$
$de g (P) = de g (Q) + 1$	Oblique	Quotient from $P \div Q$ , ignore remainder
$de g (P) > de g (Q) + 1$	None (for 9231)	Not assessed in CIE 9231

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Sketch rational function, find asymptotes
2023 · 1
Find oblique asymptote, intersection points
2021 · 1
Solve inequality using graph sketching

Going deeper

syllabusOfficial CIE 9231 SyllabusFull syllabus reference

What's Next

Mastery of rational function graphing is a foundational skill for many topics in CIE A-Level Further Mathematics, including solving rational inequalities, sketching polar curves, and finding areas under curves. The polynomial division and graphical analysis skills you learned here will also support you when you work with parametric curves and reciprocal graphs. A strong understanding of this topic also helps you quickly answer common exam questions asking for the number of solutions to an equation, relying on graphical interpretation rather than lengthy algebraic calculation.