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0580/31 May/Jun 2023, Question 6: Quadratic Graphs and the Parabola

Cambridge IGCSE Mathematics 0580 · Paper 31 (Core) · Calculator · 14 marks

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The question, in short

Using y = 5 + 3x − x², fill in missing table values, plot its curve from x = −2 to 5, state its symmetry line, then handle y = 2x + 1 and find where the two graphs meet.

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 y = 5 + 3x − x² at the given x values.

  1. 1x = −2: y = 5 + 3(−2) − (−2)² = 5 − 6 − 4 = −5
  2. 2x = 0: y = 5
  3. 3x = 1: y = 5 + 3 − 1 = 7
  4. 4x = 3: y = 5 + 9 − 9 = 5
  5. 5x = 4: y = 5 + 12 − 16 = 1

Answer| x | −2 | 0 | 1 | 3 | 4 | | y | −5 | 5 | 7 | 5 | 1 |

Part (b) · 4 marks

Plot the curve y = 5 + 3x − x² on the grid across the interval −2 ≤ x ≤ 5.

  1. 1Plot (−2, −5), (−1, 1), (0, 5), (1, 7), (2, 7), (3, 5), (4, 1), (5, −5)
  2. 2Join with a smooth curve, highest point between x = 1 and x = 2 (at x = 1.5, y = 7.25)

AnswerCorrect and accurate curve

Part (c) · 1 mark

Give the equation of the mirror line of the curve y = 5 + 3x − x².

  1. 1Symmetric pairs: y = 7 at x = 1 and x = 2, y = 5 at x = 0 and x = 3
  2. 2Line of symmetry halfway: x = (1 + 2) ÷ 2 = 1.5
  3. 3x = 1.5

Answerx = 1.5

Part (di) · 2 marks

Fill in the empty y entries in the table for y = 2x + 1 at x = −1, 0 and 2.

  1. 1x = −1: y = 2(−1) + 1 = −1
  2. 2x = 0: y = 1
  3. 3x = 2: y = 2(2) + 1 = 5

Answer| x | −1 | 0 | 2 | | y | −1 | 1 | 5 |

Part (dii) · 1 mark

Draw the straight line y = 2x + 1 on the grid from x = −2 to x = 5.

  1. 1Plot (−1, −1), (0, 1), (2, 5)
  2. 2Draw a ruled straight line through them from x = −2 (y = −3) to x = 5 (y = 11)

AnswerCorrect ruled line

Part (e) · 3 marks

Sketch both graphs over −2 ≤ x ≤ 5 and state the coordinates of their two crossing points.

  1. 1Read off where the line crosses the curve
  2. 2Check: 2x + 1 = 5 + 3x − x² gives x² − x − 4 = 0, x = (1 ± √17)/2
  3. 3x ≈ −1.56, y ≈ −2.1 and x ≈ 2.56, y ≈ 6.1
  4. 4Points ≈ (−1.6, −2.1) and (2.6, 6.1)

Answer(−1.7 to −1.4, −2.4 to −1.8), (2.4 to 2.7, 5.8 to 6.4)

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Also searched as: 0580/31/M/J/23 · 0580_s23_qp_31 · 0580 s23 31 · question 6.
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