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4024/22 May/Jun 2020, Question 7: Exponential Functions

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 14 marks

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The question, in short

Complete a table for y = 4ˣ, plot its curve, estimate a gradient by tangent, and relate a linear equation to intersections; also find the maximum point of a sketched quadratic.

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

Part (ai) · 1 mark

Fill in the missing y-values in the table for y = 4ˣ at x = 0 and x = 0.5.

  1. 14⁰ = 1
  2. 24⁰.5 = √4 = 2

Answer1, 2

Part (aii) · 3 marks

Plot the graph of y = 4ˣ for x from 0 to 3 on the given grid.

  1. 1Plot (0, 1), (0.5, 2), (1, 4), (1.5, 8), (2, 16), (2.5, 32), (3, 64)
  2. 2Join with a smooth curve

AnswerCorrect curve

Part (aiii) · 2 marks

Draw a tangent at x = 2 and use it to estimate the curve's gradient there.

  1. 1Draw the tangent at (2, 16)
  2. 2Gradient = rise ÷ run from two points on the tangent
  3. 3≈ 22 (exact value 16 ln 4 = 22.18…)

Answer18 to 27

Part (aiva) · 2 marks

Compare 3(4ˣ) + ax + b = 0 with intersections of y = 4ˣ and y = 20x − 12 to find a and b.

  1. 14ˣ = 20x − 12
  2. 23(4ˣ) = 60x − 36
  3. 33(4ˣ) − 60x + 36 = 0
  4. 4a = −60, b = 36

Answera = −60, b = 36

Part (aivb) · 3 marks

Draw y = 20x − 12 on the grid and use intersections to find all solutions of the equation.

  1. 1Draw y = 20x − 12 through (0.6, 0) and (3, 48)
  2. 2Read the x-coordinates where it crosses y = 4ˣ
  3. 3x ≈ 0.74 and x ≈ 2.69

Answer0.7 to 0.8, 2.65 to 2.75

Part (b) · 3 marks

For the sketched quadratic with x-intercepts −2 and 4 and y-intercept 8, find the maximum point coordinates p and q.

  1. 1Roots −2 and 4: y = (4 − x)(x + 2)
  2. 2Maximum is halfway between the roots: p = (−2 + 4)/2 = 1
  3. 3q = (4 − 1)(1 + 2) = 9
  4. 4p = 1, q = 9

Answerp = 1, q = 9

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