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0606/22 May/Jun 2026, Question 4: Modulus Equations and Inequalities

Cambridge IGCSE Additional Mathematics 0606 · Paper 22 (Calculator) · Calculator · 7 marks

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The question, in short

Sketch y = |x² − 5| on the given axes, giving axis intercepts, then find possible expressions for f(x) = ax + b given the solutions of |f(x)| = 4.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 3 marks

Sketch the graph of y = |x² − 5| on the axes and state where it crosses the x- and y-axes.

  1. 1y = x² − 5 is a U-shaped parabola, vertex (0, −5), crossing the x-axis at ±√5
  2. 2The modulus reflects the part below the x-axis upwards
  3. 3Between −√5 and √5 the curve becomes an arch with maximum (0, 5)
  4. 4Cusps at (±√5, 0); outer branches unchanged

AnswerW-shaped curve: cusps on the x-axis at x = −√5 and x = √5, local maximum (0, 5) on the y-axis, outer branches curving upwards; intercepts (0, 5), (−√5, 0), (√5, 0)

Part (b) · 4 marks

Given f(x) = ax + b with non-zero integer a and b, and |f(x)| = 4 at x = 2.5 and x = −1.5, find possible expressions for f(x).

  1. 1|f(2.5)| = 4 and |f(−1.5)| = 4; a line cannot take the same value twice, so f(2.5) = 4, f(−1.5) = −4 or f(2.5) = −4, f(−1.5) = 4
  2. 22.5a + b = 4 and −1.5a + b = −4: 4a = 8, a = 2, b = −1
  3. 32.5a + b = −4 and −1.5a + b = 4: 4a = −8, a = −2, b = 1
  4. 4f(x) = 2x − 1 or f(x) = −2x + 1

Answerf(x) = 2x − 1 or f(x) = −2x + 1

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Also searched as: 0606/22/M/J/26 · 0606_s26_qp_22 · 0606 s26 22 · question 4.
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