Combining graph transformationsEdexcel International A Level Maths: Revision notes

Section 1

Single transformations of y = f(x)

Four building blocks, each with a direction:

$y=f(x)+a$ : translation $\begin{pmatrix} 0 \\ a \end{pmatrix}$ (up by $a$ ).
$y=f(x+a)$ : translation $\begin{pmatrix} -a \\ 0 \end{pmatrix}$ (left by $a$ ).
$y=af(x)$ : stretch scale factor $a$ parallel to the $y$ -axis.
$y=f(ax)$ : stretch scale factor $\frac1a$ parallel to the $x$ -axis. Reflections are special cases: $y=-f(x)$ reflects in the $x$ -axis and $y=f(-x)$ reflects in the $y$ -axis.

Key termstranslationstretchreflection

Common mistake

Thinking $f(x+a)$ moves the graph right. It moves it left by $a$ .

Exam tip

Changes inside the brackets affect $x$ and behave 'backwards'; changes outside affect $y$ and behave as expected.

Section 2

Combining transformations

Treat the transformations on the $x$ - and $y$ -coordinates separately. For a point $(p,q)$ on $y=f(x)$ , the image on $y=2f(3x)$ is $\left(\frac p3,2q\right)$ . On $y=f(-x)+1$ it is $(-p,q+1)$ . On $y=3-f\left(\frac x2\right)$ it is $(2p,3-q)$ . Order matters only when two transformations act on the same coordinate. $2f(x)+1$ means stretch first, then translate; $2(f(x)+1)$ means translate first, then stretch.

Key termsimageorder of transformations

Common mistake

Applying $-f(x)+5$ as translate first. The reflection comes first, then the translation $\begin{pmatrix} 0 \\ 5 \end{pmatrix}$ .

Section 3

Worked example with a quadratic

Let $f(x)=x^2-2x-3=(x-1)^2-4$ , with minimum $(1,-4)$ and roots $-1$ , $3$ . $y=2f(x+1)$ : move left $1$ , stretch $y$ by $2$ : minimum $(0,-8)$ , roots $\pm2$ ; equation $y=2x^2-8$ . $y=5-f(x)$ : reflect in the $x$ -axis, then up $5$ : maximum $(1,9)$ ; equation $y=-x^2+2x+8$ with roots $-2$ , $4$ . Always verify with one point: $5-f(1)=5+4=9$ .

Key termsturning point

Exam tip

Transform the key points (turning point, intercepts) and then draw or state the curve; do not re-derive the whole curve.

Section 4

Trigonometric graphs

The graph of $y=3+\sin2x$ comes from $y=\sin x$ by a stretch of scale factor $\frac12$ parallel to the $x$ -axis (period $\pi$ ) and a translation up by $3$ . It oscillates between $2$ and $4$ . For $y=-\cos\left(x+\frac{\pi}{4}\right)$ : translate $y=\cos x$ left by $\frac{\pi}{4}$ , then reflect in the $x$ -axis. It starts at $-\cos\frac{\pi}{4}=-\frac{\sqrt2}{2}$ when $x=0$ . In general the amplitude is unchanged by $f(ax)$ but the period becomes $\frac{2\pi}{a}$ .

Key termsperiodamplitude

Common mistake

Giving the period of $\sin2x$ as $4\pi$ . It is $\pi$ .

Section 5

Describing transformations in the exam

A full description needs the type, the direction or axis, and the size (scale factor or vector). 'Stretch scale factor $\frac12$ parallel to the $x$ -axis' is complete; 'squash it' is not. When asked for the order, give the sequence in which transformations are applied to $y=f(x)$ . The graph of $y=f(ax+b)$ is not required.

Key termsscale factor

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Combining graph transformations

The curve $y=f(x)$ has a maximum point at $(2,5)$ and crosses the $y$ -axis at $(0,1)$ .
Find the coordinates of the turning point of the curve $y=3-f\left(\frac{x}{2}\right)$ and state whether it is a maximum or minimum.2 marks
The point $A(4,-6)$ lies on the curve $y=f(x)$ .
Write down the coordinates of the image of $A$ on the curve $y=-f(-x)$ .2 marks
The function $g$ is defined by $g(x)=3+\sin2x$ for $0\leqslant x\leqslant2\pi$ , where $x$ is in radians. It is formed by transforming $y=\sin x$ .
Describe the sequence of transformations that maps $y=\sin x$ to $y=g(x)$ , and state the period of $g$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).