The Modulus Function — Pure Mathematics 3 (9709) — Mathematics

Introduction

The modulus function (also called the absolute value function) is one of the first topics in Pure Mathematics 3 and underpins a great deal of work with inequalities throughout the course. In 9709 examinations, questions routinely ask you to sketch a modulus graph, solve a modulus equation algebraically, or solve a modulus inequality — often within a single part. Mastering the three core skills (graph, equation, inequality) from the outset saves marks across the paper.

Core Concept

What is $|x|$ ?

The modulus of a real number $x$ is its distance from zero on the number line. It is always non-negative.

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

For example: $|5| = 5$ , $|-3| = 3$ , $|0| = 0$ .

Graph of $y = |ax + b|$

Start from the straight line $y = ax + b$ . The modulus operation reflects any portion below the $x$ -axis upwards, leaving the portion above unchanged.

Steps to sketch $y = |ax + b|$ :

Find the zero of $ax + b$ : set $ax + b = 0$ , giving $x = -\tfrac{b}{a}$ . This is the vertex (corner point) of the graph, located at $\left(-\tfrac{b}{a},\ 0\right)$ .
Draw the original line $y = ax + b$ lightly.
Keep all parts where $y \geq 0$ unchanged.
Reflect all parts where $y < 0$ in the $x$ -axis (negate those $y$ -values).

The result is a V-shape (or inverted-V if $a < 0$ before reflection, but after reflection it always opens upward with a vertex on the $x$ -axis).

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Introduction

Core Concept

What is ∣x∣|x|∣x∣?

Graph of y=∣ax+b∣y = |ax + b|y=∣ax+b∣

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What is $|x|$ ?

Graph of $y = |ax + b|$