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0580/32 May/Jun 2023, Question 7: Quadratic Graphs and the Parabola

Cambridge IGCSE Mathematics 0580 · Paper 32 (Core) · Calculator · 13 marks

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The question, in short

From a given straight line on a grid, determine its equation, add the line y = 1, and locate the crossing point; then complete a table, plot y = x² + x − 8, state its symmetry line and solve x² + x − 8 = 0 graphically.

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

Part (ai) · 2 marks

Work out the equation of the drawn line L, giving it as y = mx + c.

  1. 1L crosses the y-axis at (0, −2), so c = −2
  2. 2L passes through (0, −2) and (2, 1): gradient = (1 − (−2)) ÷ (2 − 0) = 3/2 = 1.5
  3. 3y = 1.5x − 2

Answer1.5x − 2

Part (aii) · 1 mark

Add the horizontal line y = 1 to the grid.

  1. 1Draw a horizontal ruled line through y = 1 across the grid

AnswerCorrect ruled line

Part (aiii) · 1 mark

State the coordinates where line L and y = 1 meet.

  1. 11.5x − 2 = 1 gives 1.5x = 3, x = 2
  2. 2(2, 1)

Answer(2, 1)

Part (bi) · 2 marks

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

  1. 1x = −2: y = 4 − 2 − 8 = −6
  2. 2x = 1: y = 1 + 1 − 8 = −6
  3. 3x = 4: y = 16 + 4 − 8 = 12

Answer| x | −2 | 1 | 4 | | y | −6 | −6 | 12 |

Part (bii) · 4 marks

Plot the curve y = x² + x − 8 on the grid for x from −4 to 4.

  1. 1Plot (−4, 4), (−3, −2), (−2, −6), (−1, −8), (0, −8), (1, −6), (2, −2), (3, 4), (4, 12)
  2. 2Join with a smooth U-shaped curve, lowest point at (−0.5, −8.25)

AnswerCorrect and accurate curve

Part (biii) · 1 mark

Give the equation of the mirror line of the plotted curve.

  1. 1y = −8 at x = −1 and x = 0, so the line of symmetry is halfway
  2. 2x = (−1 + 0) ÷ 2 = −1/2

Answerx = −1/2

Part (biv) · 2 marks

Using your graph, find the x-values where x² + x − 8 equals zero.

  1. 1Read off where the curve crosses y = 0 (the x-axis)
  2. 2Check: x = (−1 ± √33)/2 = −3.37… or 2.37…
  3. 3x ≈ −3.4 and x ≈ 2.4

Answer−3.5 to −3.3, 2.3 to 2.5

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Also searched as: 0580/32/M/J/23 · 0580_s23_qp_32 · 0580 s23 32 · question 7.
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