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Sequences, functions and graphs · Subtopic 8 of 11

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 2Edexcel 4MA1 Papers 1H, 2H≈ 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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