Transformations of Graphs — Pure Mathematics 1 (9709) — Mathematics

Introduction

Transformations of graphs are a core tool in Pure Mathematics 1. Rather than plotting every function from scratch, you can derive the graph of a related function by applying a systematic geometric change — a translation, reflection, or stretch — to one you already know. In the 9709 exam, questions may ask you to describe a transformation in words, sketch a transformed graph (labelling key points), write down the equation of a transformed curve, or determine unknown parameters from a given transformation. Marks are frequently lost through sign errors and confusing horizontal with vertical effects, so precision here is essential.

Core Concept

Starting from a base graph $y = f(x)$ , the syllabus requires you to understand four families of transformation and their combinations.

1. Vertical Translation: $y = f(x) + a$

Every $y$ -value is increased by $a$ . The graph shifts upward by $a$ units (downward if $a < 0$ ). The $x$ -coordinates of all points are unchanged.

In vector notation: translation by $\begin{pmatrix}0\\a\end{pmatrix}$ .

2. Horizontal Translation: $y = f(x + a)$

The graph shifts left by $a$ units (right if $a < 0$ ). Note the counter-intuitive direction: replacing $x$ with $x + a$ moves the graph in the negative $x$ -direction.

In vector notation: translation by $\begin{pmatrix}-a\\0\end{pmatrix}$ .

3. Vertical Stretch / Reflection: $y = af(x)$

Every $y$ -value is multiplied by $a$ . This is a stretch parallel to the $y$ -axis with scale factor $a$ .

If $a > 1$ : the graph is stretched away from the $x$ -axis.
If $0 < a < 1$ : the graph is compressed towards the $x$ -axis.
If $a = -1$ : the graph is reflected in the $x$ -axis.
If $a < 0$ (other values): a stretch combined with a reflection in the $x$ -axis.

Points on the $x$ -axis (where $f(x) = 0$ ) are invariant.

4. Horizontal Stretch / Reflection: $y = f(ax)$

Every $x$ -value is divided by $a$ . This is a stretch parallel to the $x$ -axis with scale factor $\tfrac{1}{a}$ .

If $a > 1$ : the graph is compressed towards the $y$ -axis.
If $0 < a < 1$ : the graph is stretched away from the $y$ -axis.
If $a = -1$ : the graph is reflected in the $y$ -axis.

Points on the $y$ -axis (where $x = 0$ ) are invariant.

Summary Table

Transformation	Effect on graph	Invariant line
$y = f(x) + a$	Translation by $\begin{pmatrix}0\\a\end{pmatrix}$	None
$y = f(x + a)$	Translation by $\begin{pmatrix}-a\\0\end{pmatrix}$	None
$y = af(x)$ , $a > 0$	Stretch $\parallel$ to $y$ -axis, scale factor $a$	$x$ -axis
$y = f(ax)$ , $a > 0$	Stretch $\parallel$ to $x$ -axis, scale factor $\tfrac{1}{a}$	$y$ -axis
$y = -f(x)$	Reflection in the $x$ -axis	$x$ -axis
$y = f(-x)$	Reflection in the $y$ -axis	$y$ -axis

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Introduction

Core Concept

1. Vertical Translation: y=f(x)+ay = f(x) + ay=f(x)+a

2. Horizontal Translation: y=f(x+a)y = f(x + a)y=f(x+a)

3. Vertical Stretch / Reflection: y=af(x)y = af(x)y=af(x)

4. Horizontal Stretch / Reflection: y=f(ax)y = f(ax)y=f(ax)