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0580/41 May/Jun 2021, Question 10: Quadratic Graphs and the Parabola

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 12 marks

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The question, in short

Using the given quadratic y = 3 + 4x − x² over −1 ≤ x ≤ 5, fill missing table values, plot its curve, find a k giving no solutions, and solve a related equation graphically.

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

Part (a) · 3 marks

Work out the missing y-values in the table for the stated x-values using y = 3 + 4x − x².

  1. 1x = −0.5: 3 − 2 − 0.25 = 0.75
  2. 2x = 0: 3; x = 2: 3 + 8 − 4 = 7
  3. 3x = 4: 3 + 16 − 16 = 3; x = 4.5: 3 + 18 − 20.25 = 0.75

Answer| x | −0.5 | 0 | 2 | 4 | 4.5 | | y | 0.75 | 3 | 7 | 3 | 0.75 |

Part (b) · 4 marks

Plot the curve of y = 3 + 4x − x² on the grid for x from −1 to 5.

  1. 1Plot (−1, −2), (−0.5, 0.75), (0, 3), (1, 6), (2, 7), (3, 6), (4, 3), (4.5, 0.75), (5, −2)
  2. 2Join with a smooth ∩-shaped curve, maximum at (2, 7)

Answercorrect curve

Part (c) · 1 mark

State one whole-number k so that 3 + 4x − x² = k has no solutions.

  1. 1The maximum value of y is 7
  2. 2A line y = k above 7 does not meet the curve, e.g. k = 8

AnswerAny integer ≥ 8

Part (d) · 4 marks

Sketch the curve, then add a straight line on the grid to solve −1 + 9/2x − x² = 0.

  1. 1−1 + 9/2 x − x² = 0, add 4 − 1/2 x to both sides: 3 + 4x − x² = 4 − 1/2 x
  2. 2Draw the line y = 4 − 1/2 x (through (0, 4) and (4, 2))
  3. 3Read the x-values where it crosses the curve
  4. 4x ≈ 0.23 or x ≈ 4.27

Answerx = 0.2 to 0.3 or x = 4.2 to 4.3

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Also searched as: 0580/41/M/J/21 · 0580_s21_qp_41 · 0580 s21 41 · question 10.
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