Modulus Functions and Transformations

This lesson covers the modulus function and its graphs, solving modulus equations and inequalities, and a comprehensive review of graph transformations.

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The Modulus Function

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

|x| = x if x ≥ 0
|x| = −x if x < 0

In other words, |x| makes any negative value positive. It represents the distance of x from zero on the number line.

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Graph of y = |f(x)|

To sketch y = |f(x)|:

1. Sketch y = f(x).
2. Reflect any parts below the x-axis upwards (in the x-axis).

Worked Example 1

Sketch y = |2x − 3|.

First sketch y = 2x − 3 (a straight line crossing the x-axis at x = 3/2).

The part where y < 0 (x < 3/2) is reflected upward. The result is a V-shape with vertex at (3/2, 0).

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Graph of y = f(|x|)

To sketch y = f(|x|):

1. Sketch y = f(x) for x ≥ 0.
2. Reflect this in the y-axis to get the part for x < 0.