0606/23 Oct/Nov 2025, Question 3: Graphs of Modulus Functions

Part (ai) · 2 marks

Rewrite x² − x − 6 as (x + a)² + b, stating the constants a and b.

1. 1x² − x = (x − 1/2)² − 1/4
2. 2x² − x − 6 = (x − 1/2)² − 1/4 − 6
3. 3= (x − 1/2)² − 25/4

Answer(x − 1/2)² − 25/4

Part (aii) · 2 marks

Using your completed-square form, state the coordinates of the stationary point of y = x² − x − 6.

1. 1The minimum of (x − 1/2)² − 25/4 is when x − 1/2 = 0
2. 2Stationary point (1/2, −25/4)

Answer(1/2, −25/4)

Part (b) · 3 marks

On the given axes, sketch y = |x² − x − 6| for x from −4 to 4.

1. 1y = x² − x − 6 = (x − 3)(x + 2) crosses the x-axis at −2 and 3
2. 2Between −2 and 3 the quadratic is negative, so reflect that part in the x-axis
3. 3The reflected arch has maximum (1/2, 25/4) and passes through (0, 6)
4. 4Cusps (sharp points) at (−2, 0) and (3, 0); ends at (−4, 14) and (4, 6)

AnswerCorrect graph: W-shaped with cusps on the x-axis at (−2, 0) and (3, 0), y-intercept (0, 6), the middle part a reflected arch with maximum (1/2, 25/4) in the first quadrant, rising to 14 at x = −4 and 6 at x = 4

Part (c) · 2 marks

Using your sketch, find the range of x satisfying |x² − x − 6| < 4.

1. 1Draw y = 4 on the graph; it meets the graph in four places
2. 2x² − x − 6 = 4 gives x = (1 ± √41)/2 ≈ −2.70, 3.70
3. 3x² − x − 6 = −4 gives x² − x − 2 = 0, x = −1 or 2
4. 4The graph is below y = 4 for −2.7 < x < −1 and 2 < x < 3.7

Answer−2.7 < x < −1, 2 < x < 3.7

More past-paper questions on graphs of modulus functions

• 0606/12 Feb/Mar 2026 Q2 · 6 marks
• 0606/22 May/Jun 2026 Q4 · 7 marks
• 0606/23 May/Jun 2026 Q1 · 5 marks
• 0606/12 Feb/Mar 2025 Q1 · 7 marks
• 0606/23 May/Jun 2025 Q4 · 8 marks