Lemmara logoLemmaraWorked solution

0580/32 May/Jun 2024, Question 6: Quadratic Graphs and the Parabola

Cambridge IGCSE Mathematics 0580 · Paper 32 (Core) · Calculator · 14 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Complete a table for y = −x² + 8x + 1, plot its graph for 0 ≤ x ≤ 8, find its symmetry line, then draw and analyse a line with gradient 1/2 through (2, 7).

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

Part (ai) · 3 marks

Fill in the missing y-values in the table for y = −x² + 8x + 1 at the given x-values.

  1. 1x = 0: y = 1
  2. 2x = 1: y = −1 + 8 + 1 = 8
  3. 3x = 3: y = −9 + 24 + 1 = 16
  4. 4x = 5: y = −25 + 40 + 1 = 16
  5. 5x = 7: y = −49 + 56 + 1 = 8

Answer| x | 0 | 1 | 3 | 5 | 7 | | y | 1 | 8 | 16 | 16 | 8 |

Part (aii) · 4 marks

Plot the curve y = −x² + 8x + 1 on the grid for x from 0 to 8.

  1. 1Plot (0, 1), (1, 8), (2, 13), (3, 16), (4, 17), (5, 16), (6, 13), (7, 8), (8, 1)
  2. 2Join with a smooth ∩-shaped curve with its maximum at (4, 17)

Answercorrect curve

Part (aiii) · 1 mark

State the equation of the vertical line of symmetry of the drawn curve.

  1. 1The curve is symmetrical about its highest point (4, 17): x = 4

Answerx = 4

Part (bi) · 2 marks

Draw the straight line with gradient 1/2 passing through (2, 7) for 0 ≤ x ≤ 8.

  1. 1From (2, 7), gradient 1/2 means up 1 for every 2 across
  2. 2Rule the line through (0, 6), (2, 7) and (8, 10)

AnswerCorrect line

Part (bii) · 2 marks

Find the equation of this straight line in the form y = mx + c.

  1. 17 = 1/2 × 2 + c → c = 6
  2. 2y = 1/2 x + 6

Answery = 1/2 x + 6

Part (biii) · 2 marks

Give the coordinates where this straight line meets the curve y = −x² + 8x + 1.

  1. 1Read off where the line crosses the curve
  2. 2(Check: −x² + 8x + 1 = 1/2 x + 6 gives x² − 7.5x + 5 = 0, x ≈ 0.74 or 6.76)
  3. 3(0.7, 6.4) and (6.8, 9.4)

Answer(0.6 to 0.9, 6.2 to 6.5), (6.6 to 6.9, 9.2 to 9.5)

Stuck on a step? Read the free lesson: Quadratic Graphs and the Parabola

Practise more quadratic graphs and the parabola questions

Real 0580 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on quadratic graphs and the parabola

Also searched as: 0580/32/M/J/24 · 0580_s24_qp_32 · 0580 s24 32 · question 6.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.