Learn9MA0Paper 2 · Pure Mathematics 2Algebra and functionsCurve sketching and transformations

Algebra and functions · Subtopic 11 of 16

Curve Sketching and Transformations

A factorised cubic crosses the $x$ -axis at each factor's root and touches it at a repeated root. Reciprocal graphs have asymptotes. Changes outside the bracket act vertically and do what you expect; changes inside act horizontally and do the opposite.

Edexcel YMA01 WMA11Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2≈ 40 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Sketch cubics given in factorised form, including repeated roots.
Sketch $y=\dfrac kx$ and $y=\dfrac k{x^2}$ with their asymptotes.
Use the intersections of two graphs to count the solutions of an equation.
Apply $f(x)+a$ , $f(x+a)$ , $af(x)$ and $f(ax)$ to a graph and to a key point.

Pause and think

$y=f(x+3)$ uses the value of $f$ three steps to the right. So does the graph move left or right?

Section 2 of 14

Core idea

Start hereA curve from three facts

Tell me where a cubic crosses the axis and which way it heads, and I can sketch it without plotting a single point. And once you know one graph, moving or stretching it follows simple rules.

The big idea

Sketch from roots, intercepts and end behaviour. Transformations: $f(x)+a$ up $a$ ; $f(x+a)$ left $a$ ; $af(x)$ stretch vertically × $a$ ; $f(ax)$ stretch horizontally × $\frac1a$ .

1. Sketching

y = (x + 2)(x − 1)(x − 3): a cubic from its roots

Roots from the brackets; the

y

-intercept by putting

x=0

(2)(-1)(-3)=6

. A positive

x^3

term means it starts low on the left and ends high on the right.

Worked example $y=(x+2)(x-1)(x-3)$

Sketch.

Goal: Key features.
Strategy: Roots, $y$ -intercept, ends.

$x=-2,1,3$
Each bracket zero.
$y(0)=(2)(-1)(-3)=6$
Put $x=0$ .
$x\to\pm\infty:\ y\approx x^3$
Positive cubic: bottom left to top right.

AnswerUp through

-2

, max, down through

(0,6)

and

1

, min, up through

3

✓ Check

x=2

(4)(1)(-1)=-4<0

, consistent with dipping between 1 and 3 ✓.

y = k/x and y = k/x²: asymptotes

Both approach the axes without touching them.

\tfrac kx

is in opposite quadrants;

\tfrac k{x^2}

is always positive, so both branches are above the axis.

Edge caseRepeated root A squared factor

(x-a)^2

touches the axis at

a

without crossing.

2. Transformations

y = f(x + a): a shift of a to the LEFT

a = 0.0

Adding inside the bracket moves the graph the "wrong" way: the vertex moves to

x=-a

, because now

x=-a

gives what

x=0

used to.

Why

f(x+a)

moves left

$g(x)=f(x+a)$
New graph.
$g(x_0-a)=f(x_0)$
The new graph reaches the old value $f(x_0)$ when $x$ is $a$ smaller.

Every point moves

a

to the left. Inside changes act on

x

"backwards"; outside changes act on

y

directly.

Worked exampleMoving a point

$(4,-6)$ lies on $y=f(x)$ . Find its image on $y=f(2x)$ , $y=3f(x)$ , $y=f(x)-1$ , $y=f(x-3)+2$ .

Goal: Four points.
Strategy: Inside: $x$ inverse; outside: $y$ direct.

$f(2x):\ (2,-6)$
Halve $x$ .
$3f(x):\ (4,-18)$
Triple $y$ .
$f(x)-1:\ (4,-7)$
Down 1.
$f(x-3)+2:\ (7,-4)$
Right 3, up 2.

Answer

(2,-6)

(4,-18)

(4,-7)

(7,-4)

✓ Check

f(2\cdot2)=f(4)=-6

✓.

Common misconception $f(2x)$ is not a stretch by 2 It squashes by factor

\frac12

horizontally.

Pause and think

Asymptotes of $y=\frac3{x-2}$ ?

Summary

Roots, intercept, ends.
Inside changes are reversed; outside are direct.

Asymptote: Line approached but never reached.
Translation: A shift.

HistoryAlgebra meets geometry. In 1637 René Descartes published La Géométrie, showing that equations could be drawn as curves and curves described by equations. Every sketch you draw uses his idea of coordinates.

Fun fact"Not falling together". "Asymptote" comes from the Greek asymptotos, meaning "not falling together". The curve and its asymptote get endlessly close but never meet, like two friends who keep almost bumping into each other.

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