2.1PureCore

The Modulus Function

The modulus (or absolute value) of a number is its size, ignoring sign — written $|x|$ . It is never negative, and its graph is the distinctive V-shape that underpins the rest of this chapter.

25 min Functions and Graphs

Edexcel A level Maths: 2.1 The Modulus FunctionCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Understand modulus as “distance from zero”
Evaluate the modulus of expressions
Sketch y = |x| and y = |ax + b|
Read values off a modulus graph

What modulus means

The modulus $|x|$ is the $\textbf{non-negative}$ value of $x$ : it strips the sign. So $|5| = 5$ and $|-5| = 5$ . Formally it is $x$ if $x \ge 0$ and $-x$ if $x < 0$ .

|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}

Always non-negative — the “size” of

x

Example 1Evaluate

|3 - 7|

3 - 7 = -4

, and

|-4| = 4

Answer

4

The graph of y = |x|

The graph of $y = |x|$ is a $\textbf{V}$ with its vertex at the origin: it is $y = x$ for $x \ge 0$ and the reflection $y = -x$ for $x < 0$ . It never goes below the $x$ -axis.

Tip — Any modulus graph sits entirely on or above the x-axis — the output is never negative.

y = |ax + b|

To sketch $y = |ax + b|$ , draw the line $y = ax + b$ , then $\textbf{reflect}$ the part below the $x$ -axis up above it. The graph touches the $x$ -axis where $ax + b = 0$ .

Example 1Where does

y = |2x - 6|

meet the x-axis, and what is its minimum value?

2x - 6 = 0 \Rightarrow x = 3

2Minimum of a modulus is 0.

Answer

touches x-axis at

(3, 0)

; minimum value

0

Formula recap

|x| \ge 0

Modulus is never negative.

|x| = x \ (x \ge 0), \ -x \ (x < 0)

Piecewise definition.

y = |ax+b|: \text{reflect below-axis part up}

Sketching rule.

Common mistakes to avoid

Thinking |−5| = −5.

Modulus removes the sign: |−5| = 5.

Drawing a modulus graph dipping below the x-axis.

It is reflected up; the graph is always ≥ 0.

Key takeaways

|x| is the size of x, always ≥ 0.
y = |x| is a V-shape with vertex at the origin.
For y = |ax + b|, reflect the part of the line below the x-axis upwards.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

$|-8|$ equals:
- $8$
- $-8$
- $0$
- $16$
Show answer and explanation
Answer: $8$
Modulus removes the sign.
$|3 - 10|$ equals:
- $7$
- $-7$
- $13$
- $3$
Show answer and explanation
Answer: $7$
|−7| = 7.
The graph of $y = |x|$ is:
- A V-shape
- A straight line
- A parabola
- A circle
Show answer and explanation
Answer: A V-shape
Vertex at the origin.
A modulus graph is always:
- On or above the x-axis
- Below the x-axis
- Symmetric about the y-axis only
- Negative
Show answer and explanation
Answer: On or above the x-axis
Output ≥ 0.
$y = |2x - 6|$ touches the x-axis at:
- $x = 3$
- $x = 6$
- $x = -3$
- $x = 0$
Show answer and explanation
Answer: $x = 3$
2x − 6 = 0.
The minimum value of $y = |ax + b|$ is:
- $0$
- $b$
- $-b$
- $a$
Show answer and explanation
Answer: $0$
Modulus bottoms out at 0.

Next step

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