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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9231Paper 1 · Further Pure Mathematics 1Rational functions and graphsRational functions and asymptotes

Rational functions and graphs · Subtopic 1 of 1

Rational Functions and their Graphs

A rational function $y=\frac{P(x)}{Q(x)}$ has vertical asymptotes where $Q(x)=0$ , and its behaviour for large $|x|$ (horizontal or oblique asymptote) comes from dividing. Writing $y=f(x)$ as a quadratic in $x$ and using the discriminant gives the range and turning points.

Cambridge 9231 Paper 1AQA 7367 Paper 1AQA 7367 Paper 2≈ 40 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find vertical, horizontal and oblique asymptotes.
Sketch rational functions with all key features.
Use the discriminant to find the range and turning points.

Pause and think

What happens to $y=\dfrac{2x+1}{x-3}$ when $x$ is a million?

Section 2 of 14

Core idea

Start hereA curve that runs from a wall

Graph $y=\frac{2x+1}{x-3}$ and it shoots up near $x=3$ like a wall it can't touch, while far away it flattens out at height 2. Those invisible guides, asymptotes, are what you find first when sketching any rational function.

The big idea

Vertical asymptotes where the denominator is zero; long-run behaviour from division (horizontal or oblique asymptote); the range from the discriminant of $y=f(x)$ treated as an equation in $x$ .

1. Asymptotes

Horizontal asymptote of

\frac{ax+b}{cx+d}

$y=\frac{a+\frac bx}{c+\frac dx}$
Divide top and bottom by $x$ .
$\to\frac ac\text{ as }|x|\to\infty$
$\frac bx,\frac dx\to0$ .

Worked exampleOblique asymptote

Find the asymptotes of $y=\frac{x^2+3x+4}{x+1}$ .

Goal: All asymptotes.
Strategy: Denominator zero; polynomial division.

$x=-1$
Vertical.
$x^2+3x+4=(x+1)(x+2)+2$
Divide.
$y=x+2+\frac2{x+1}$
Remainder term $\to0$ .

Answer

x=-1

and

y=x+2

✓ Check

x=100

y=\frac{10304}{101}=102.02\approx102

✓.

2. Range and turning points

Discriminant method

$y=\frac{x^2-5}{x-3}\Rightarrow yx-3y=x^2-5$
Multiply out.
$x^2-yx+(3y-5)=0$
Quadratic in $x$ .
$y^2-4(3y-5)\ge0$
Real $x$ exists only if the discriminant is $\ge0$ .
$(y-2)(y-10)\ge0\Rightarrow y\le2\text{ or }y\ge10$
Solve in $y$ .

The boundary values

2

and

10

are the

y

-coordinates of the turning points.

Worked exampleGreatest value of $\frac x{x^2+1}$

Find the range.

Goal: Range.
Strategy: Discriminant.

$yx^2-x+y=0$
Rearrange.
$1-4y^2\ge0$
Discriminant.
$-\tfrac12\le y\le\tfrac12$
Solve.

AnswerRange

[-\frac12,\frac12]

; greatest value

\frac12

x=1

✓ Check

\frac1{1+1}=\frac12

✓.

Sketching checklist

Drawing diagram…

Each feature usually earns a mark.

Edge caseCrossing a horizontal asymptote Allowed:

y=\frac x{x^2+1}

crosses

y=0

at the origin.

Summary

Vertical: denominator zero.
Divide for long-run behaviour.
Discriminant gives range.

Oblique asymptote: A slanted line approached for large |x|.

HistoryAsymptote. The word comes from Greek "not falling together". Apollonius used it around 200 BC for the lines a hyperbola approaches but never meets.

Fun factCrossing an asymptote. Graphs can cross horizontal or oblique asymptotes (often in the middle). They just can't cross vertical ones, because the function isn't defined there.

Section 3 of 14

Key formulae

Linear over linear

y=\frac{ax+b}{cx+d}:\quad x=-\frac dc,\quad y=\frac ac

Oblique asymptote

\frac{x^2+px+q}{x+r}=x+m+\frac{k}{x+r}\ \Rightarrow\ y=x+m

Range via discriminant

y=f(x)\ \to\ Ax^2+Bx+C=0\ (\text{in }x),\quad B^2-4AC\ge0

TrickTest just beside each asymptote. Plug in

x=3.01

and

x=2.99

(for an asymptote at

x=3

). The signs tell you which way the graph shoots off on each side.

Section 4 of 14

Worked examples

Example 1 · Oblique asymptote0 / 3 steps

Find the asymptotes of

y=\dfrac{x^2+3x+4}{x+1}

Example 2 · Range0 / 3 steps

Find the set of values taken by

y=\dfrac{x^2-5}{x-3}

Section 5 of 14

Explore the graph

Linear over linear

Loading the interactive graph…

Try this

Move $d$ : the vertical asymptote is at $x=-d$ .
Move $a$ : the horizontal asymptote is $y=a$ .
For which $b$ does the curve collapse to a straight line? (When $b=ad$ .)

$y=\frac{ax+b}{x+d}$ has asymptotes $x=-d$ and $y=a$ .

In the real worldLens formula.

\frac1v=\frac1f-\frac1u

gives

v=\frac{fu}{u-f}

: a rational function. Its vertical asymptote at

u=f

is why an object at the focal point forms no image.

Section 6 of 14

Common mistakes

IncorrectHorizontal asymptote

y=\frac bd

Correct

y=\frac ac

For large

x

only the leading terms matter.

IncorrectGraph crossing a vertical asymptote

CorrectIt can't

The function is undefined there.

IncorrectSolving

B^2-4AC\ge0

for

x

CorrectIt's an inequality in

y

You need the

y

-values for which some real

x

exists.

The curve: getting closer and closer

The asymptote: you will never actually reach me.

Section 7 of 14

Exam tip

Exam tipLabel everything. Sketch marks need asymptote equations written on the lines and intercept coordinates on the axes.

Section 8 of 14

Guided practice

SA-FP-RAT-G011 markL1 FoundationNon-calculatorAsymptotes

State the equation of the vertical asymptote of

y=\dfrac{2x+1}{x-3}

, giving the value

x=k

. State

k

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FP-RAT-0011 markL1 FoundationNon-calculatorAsymptotes

State the value

c

for the horizontal asymptote

y=c

y=\dfrac{2x+1}{x-3}

Salam Academy Original Practice · Proposed mark scheme

SA-FP-RAT-0023 marksL2 StandardNon-calculatorAsymptotes

Find the oblique asymptote of

y=\dfrac{x^2+3x+4}{x+1}

, giving

y

in terms of

x

Salam Academy Original Practice · Proposed mark scheme

SA-FP-RAT-0033 marksL3 ExamNon-calculatorRange

Find the greatest value of

y=\dfrac{x}{x^2+1}

Salam Academy Original Practice · Proposed mark scheme

SA-FP-RAT-0042 marksL2 StandardNon-calculatorAsymptotes

Find the

x

-values of the vertical asymptotes of

y=\dfrac{x^2-4}{x^2-1}

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-FP-RAT-0054 marksL4 ChallengingNon-calculatorRange

Find the

y

-coordinates of the turning points of

y=\dfrac{x^2-5}{x-3}

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Vertical: denominator zero.
Horizontal/oblique: divide.
Range: discriminant in $y$ .

Section 12 of 14

Quick quiz

Q1 · SA-FP-RAT-Q011 markL1 FoundationNon-calculatorAsymptotes

The horizontal asymptote of

y=\dfrac{3x-1}{x+2}

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 9231 AS & A Level Further Mathematics
Where: Paper 1 · 1.2 Rational functions and graphs
Expected: Sketch rational functions with linear or quadratic numerator and denominator; asymptotes; use the discriminant to find the range or turning points.
Calculator: Permitted on all papers; exact answers are expected where the question asks.
Typical style: A sketch question expects asymptotes drawn and labelled with their equations, and all intercepts.
Open in: 9231 · Paper 1

Section 14 of 14

Past paper questions

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 9231, Current syllabus (official specification).