Learn9MA0Paper 2 · Pure Mathematics 2Algebra and functionsTransformations of graphs

Algebra and functions · Subtopic 16 of 16

Transformations of Graphs

Translating, stretching and reflecting $y=f(x)$ : what each of $f(x)+a$ , $f(x+a)$ , $af(x)$ and $f(ax)$ does, how to describe it in exam language, and how to combine transformations in the right order.

Cambridge 9709 Paper 1Edexcel YMA01 WMA13Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2≈ 50 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Recognise and describe translations, stretches and reflections of $y=f(x)$ .
Find the image of a point, such as a vertex, under a transformation.
Write the equation of a transformed graph.
Apply combinations of transformations in the correct order, and describe a combination.

Pause and think

Warm-up: the graph of $y=x^2$ has its vertex at $(0,0)$ . Where is the vertex of $y=(x-3)^2$ ? And of $y=x^2-3$ ?

Section 2 of 14

Core idea

Start hereMoving a graph without redrawing it

You know the shape of $y=x^2$ by heart. What does $y=(x-3)^2+2$ look like? You could plot dozens of points, or you could notice it's the same parabola moved $3$ right and $2$ up. Every change to a formula, adding, multiplying, inside or outside the brackets, moves, stretches or flips the graph in a predictable way.

The big idea

Changes outside $f$ act on $y$ and do what you'd expect: $f(x)+a$ up $a$ , $af(x)$ stretch vertically by $a$ , $-f(x)$ flip in the $x$ -axis. Changes inside act on $x$ and do the opposite: $f(x+a)$ left $a$ , $f(ax)$ squash horizontally by $\frac1a$ , $f(-x)$ flip in the $y$ -axis.

1. Translations

Translations of y = x²

f(x)+3

adds

3

to every output, so the graph moves up

3

f(x-2)

moves the graph right

2

, even though there is a minus sign.

Why

f(x-a)

moves the graph right

$\text{Point on }y=f(x):\ (p,\,f(p))$
Any point of the original.
$\text{On }y=f(x-a)\text{, put }x=p+a:\ y=f(p+a-a)=f(p)$
To get the same output we must input $a$ more.
$(p,f(p))\ \longrightarrow\ (p+a,\,f(p))$
Every point moves $a$ to the right.

y=f(x-a)

y=f(x)

translated by

\begin{pmatrix}a\\0\end{pmatrix}

, i.e. right by

a

. Similarly

y=f(x)+b

is a translation by

\begin{pmatrix}0\\b\end{pmatrix}

y = (x − a)², sliding as a changes

a = -3.0

When

a

is positive the curve moves right; when

a

is negative, left. The dashed curve is the original

y=x^2

2. Stretches and reflections

Stretches of y = sin x

2\sin x

reaches

2

instead of

1

(vertical stretch, factor

2

\sin2x

fits two waves into

360^\circ

(horizontal stretch, factor

\tfrac12

\begin{array}{ll}y=af(x)&\text{stretch parallel to the }y\text{-axis, factor }a\\y=f(ax)&\text{stretch parallel to the }x\text{-axis, factor }\tfrac1a\\y=-f(x)&\text{reflection in the }x\text{-axis}\\y=f(-x)&\text{reflection in the }y\text{-axis}\end{array}

Worked exampleWhere does a point go?

$y=f(x)$ has a maximum at $(2,6)$ . Find the maximum point of (a) $y=f(x+1)$ , (b) $y=3f(x)$ , (c) $y=f(2x)$ , (d) $y=-f(x)$ .

Goal: Image of $(2,6)$ under each transformation.
Strategy: Outside changes act on $y$ as written; inside changes act on $x$ inversely.

$f(x+1):\ (2-1,\,6)=(1,6)$
Left $1$ .
$3f(x):\ (2,\,18)$
$y$ trebled.
$f(2x):\ (1,\,6)$
$x$ halved.
$-f(x):\ (2,\,-6)$
Flipped: becomes a minimum.

Answer

(1,6)

;

(2,18)

;

(1,6)

;

(2,-6)

(minimum).

✓ CheckFor (c):

y=f(2x)

x=1

gives

f(2)=6

✓.

3. Combining transformations

Worked exampleDescribe $y=2(x-3)^2+1$ from $y=x^2$

Describe a sequence of transformations from $y=x^2$ to $y=2(x-3)^2+1$ .

Goal: An ordered list.
Strategy: Build the formula from $x^2$ in steps, recording each change.

$x^2\ \to\ (x-3)^2$
Translation $\begin{pmatrix}3\\0\end{pmatrix}$ .
$(x-3)^2\ \to\ 2(x-3)^2$
Stretch parallel to the $y$ -axis, factor $2$ .
$2(x-3)^2\ \to\ 2(x-3)^2+1$
Translation $\begin{pmatrix}0\\1\end{pmatrix}$ .

AnswerTranslate right 3, stretch vertically ×2, translate up 1. Vertex

(0,0)\to(3,1)

✓ CheckVertex of

2(x-3)^2+1

(3,1)

✓.

Common misconceptionOrder can matter Stretching by

2

then adding

1

gives

2f(x)+1

; adding

1

then stretching gives

2(f(x)+1)=2f(x)+2

. Different! Follow the order in which the formula is built.

Pause and think

What transformation maps $y=\sin x$ to $y=\sin3x$ ?

Summary

Outside: acts on $y$ , as expected.
Inside: acts on $x$ , the opposite way.
Track key points; mind the order.

Translation: A slide, described by a column vector.
Stretch: Scaling in one direction by a factor.

HistoryGeometry as the study of transformations. In 1872 the German mathematician Felix Klein proposed that each kind of geometry should be understood through the transformations it allows: moves, reflections, stretches. His "Erlangen programme" changed how mathematicians think about shape, and it is the same idea you are using here, one graph at a time.

In the real worldEvery video game does this. When a character in a game moves, grows or flips to face the other way, the computer is applying translations, stretches and reflections to its coordinates, many times a second. Phone photo apps do the same when you crop, zoom or mirror a picture.

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