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 Video by Zeeshan Zamurred Functions and Graphs

Edexcel A level Maths: 2.1 The Modulus FunctionWatch the full walkthrough before the notes below.

Open on YouTube

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 $non-negative$ value of $x$ : it strips the sign. So $∣5∣ = 5$ and $∣ - 5∣ = 5$ . Formally it is $x$ if $x \geq 0$ and $- x$ if $x < 0$ .

∣ x ∣ = {x - x x \geq 0 x < 0

Always non-negative — the “size” of

x

3 - 7 = - 4

, and

∣ - 4∣ = 4

Answer

4

The graph of y = |x|

The graph of $y = ∣ x ∣$ is a $V$ with its vertex at the origin: it is $y = x$ for $x \geq 0$ 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 $y = ∣ a x + b ∣$ , draw the line $y = a x + b$ , then $reflect$ the part below the $x$ -axis up above it. The graph touches the $x$ -axis where $a x + b = 0$ .

2 x - 6 = 0 \Rightarrow x = 3

2Minimum of a modulus is 0.

Answertouches x-axis at

(3, 0)

; minimum value

0

Formula recap

∣ x ∣ \geq 0

Modulus is never negative.

∣ x ∣ = x (x \geq 0), - x (x < 0)

Piecewise definition.

y = ∣ a x + b ∣ : 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

Ready to lock in The Modulus Function? Pick a mode and earn XP & Dobloons.