Modulus Functions

The modulus function (absolute value function) gives the non-negative value of a number or expression. It is denoted |x| and is required knowledge for the Edexcel 9MA0 A-Level specification. You must be able to sketch modulus graphs, solve modulus equations and modulus inequalities.

---

Definition

The modulus (or absolute value) of x is defined as:

|x| = x if x ≥ 0 |x| = −x if x < 0

In words: |x| gives the distance of x from the origin on the number line, so it is always non-negative.

Examples:

• |5| = 5
• |−3| = 3
• |0| = 0

For an expression: |2x − 7| = 2x − 7 when 2x − 7 ≥ 0 (i.e. x ≥ 3.5), and |2x − 7| = −(2x − 7) = 7 − 2x when x < 3.5.

---

The Graph of y = |f(x)|

To sketch y = |f(x)| from the graph of y = f(x):

1. Draw the graph of y = f(x).
2. Any part of the graph that lies below the x-axis (where f(x) < 0) is reflected in the x-axis.
3. Parts already above or on the x-axis remain unchanged.