The modulus function and its graphs

Definition

|x| = x when x ≥ 0 and |x| = -x when x < 0. So |3| = 3 and |-3| = 3. A modulus is never negative.

y = |ax + b| and y = |f(x)|

Sketch y = f(x), then reflect any part below the x-axis in the x-axis. For y = |ax + b| this gives a V shape with its vertex on the x-axis at x = -b/a.

y = f(|x|)

For x ≥ 0, f(|x|) = f(x), so keep that part. Then reflect it in the y-axis to get the part for x < 0. The graph is symmetrical about the y-axis.

Worked example

Sketch y = |2x - 4|. State the vertex and the y-intercept.

1. 2x - 4 = 0 at x = 2, so the vertex is (2, 0).
2. At x = 0, y = |-4| = 4.
3. The part of y = 2x - 4 with x < 2 is reflected upwards.

Answer: A V shape with vertex (2, 0) and y-intercept (0, 4)

Check yourself

1. Find the x-coordinate of the vertex of y = |3x + 6|.
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   -2. 3x + 6 = 0 when x = -2.

2. Find the y-coordinate where y = |2x - 7| meets the y-axis.
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   7. At x = 0, y = |-7| = 7.

3. How is the graph of y = |f(x)| obtained from y = f(x)?
   • Reflect the part for x < 0 in the y-axis
   • Reflect the whole graph in the x-axis
   • Reflect the parts below the x-axis in the x-axis
   • Delete the parts below the x-axis
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   Reflect the parts below the x-axis in the x-axis. Negative outputs become positive; non-negative outputs stay the same.

4. Starting from the graph of y = f(x), describe how to sketch y = f(|x|).
   • Reflect the parts below the x-axis in the x-axis
   • Keep the part for x ≥ 0 and reflect it in the y-axis
   • Reflect the part for x < 0 in the x-axis
   • Translate the graph to the right
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   Keep the part for x ≥ 0 and reflect it in the y-axis. f(|x|) = f(x) when x ≥ 0, and f(|-x|) = f(|x|), so the graph is symmetrical about the y-axis.

5. Find the y-coordinate of the local maximum of y = |x² - 6x + 5| between x = 1 and x = 5.
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   4. y = x² - 6x + 5 has its minimum at (3, -4). The modulus reflects it to (3, 4).

6. f(x) = x² - 4x. Evaluate f(|x|) when x = -3.
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   -3. f(|-3|) = f(3) = 9 - 12 = -3. (Compare |f(-3)| = |21| = 21.)

7. What is the range of y = |x - 2| - 3?
   • y ≥ 0
   • y ≥ 3
   • y ≥ 2
   • y ≥ -3
   Show answer

   y ≥ -3. |x - 2| ≥ 0, so |x - 2| - 3 ≥ -3. The vertex is at (2, -3).

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 3 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.