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4024/22 Oct/Nov 2021, Question 3: Plotting Curves from a Table of Values

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 11 marks

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The question, in short

Complete a table for y = x³/2 − 3x − 1, plot its graph for −3 ≤ x ≤ 3, then use it to reason about solutions and a line through (1, 1) and (−2, −1).

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

Part (a) · 1 mark

Fill in the missing y-value in the table for y = x³/2 − 3x − 1 at the given x values.

  1. 1x = −3: (−3)³/2 − 3 × (−3) − 1 = −13.5 + 9 − 1 = −5.5

Answer−5.5 or −5 1/2 or −11/2

Part (b) · 3 marks

Plot the curve y = x³/2 − 3x − 1 on the grid for x from −3 to 3.

  1. 1Plot (−3, −5.5), (−2, 1), (−1, 1.5), (0, −1), (1, −3.5), (2, −3), (3, 3.5)
  2. 2Join with a smooth curve

AnswerCorrect smooth curve

Part (c) · 2 marks

Using the drawn graph, justify why x³ − 6x − 2 = 6 has exactly one solution.

  1. 1x³ − 6x − 2 = 6 → ÷ 2: x³/2 − 3x − 1 = 3
  2. 2The line y = 3 meets the curve only once, so there is only one solution

AnswerLine y = 3 only intersects the graph once

Part (di) · 1 mark

Draw line L through (1, 1) and (−2, −1) on the grid.

  1. 1Plot (1, 1) and (−2, −1) and rule a straight line through them across the grid

AnswerRuled line through (1, 1) and (−2, −1)

Part (dii) · 2 marks

Calculate the gradient of line L.

  1. 1Gradient = (1 − (−1))/(1 − (−2)) = 2/3

Answer2/3

Part (diii) · 2 marks

Determine the x-coordinates where line L meets the curve y = x³/2 − 3x − 1.

  1. 1Read the x-coordinates where line L crosses the curve
  2. 2Line L: y = 2/3 x + 1/3; exactly, 3x³ − 22x − 8 = 0
  3. 3x ≈ −2.5, x ≈ −0.37, x ≈ 2.87

Answer−2.6 to −2.4, −0.45 to −0.3, 2.8 to 2.95

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Also searched as: 4024/22/O/N/21 · 4024_w21_qp_22 · 4024 w21 22 · question 3.
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