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0580/42 May/Jun 2021, Question 2: Exponential Functions

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 11 marks

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The question, in short

Using a table of values for y = 2 × 0.5ˣ − 1, complete the table, plot the curve, then use a straight-line method to solve a related equation and reason about integer k and the asymptote.

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

Part (ai) · 2 marks

Fill in the two missing y-values in the table for y = 2 × 0.5ˣ − 1 at x = 0 and x = 2.

  1. 1x = 0: 2 × 0.5⁰ − 1 = 2 − 1 = 1
  2. 2x = 2: 2 × 0.25 − 1 = −0.5

Answer| x | 0 | 2 | | y | 1 | −0.5 |

Part (aii) · 4 marks

Plot the graph of y = 2 × 0.5ˣ − 1 on the grid for −1 ≤ x ≤ 2 using the completed table.

  1. 1Plot (−1, 3), (−0.5, 1.83), (0, 1), (0.5, 0.41), (1, 0), (1.5, −0.29), (2, −0.5)
  2. 2Join with a smooth decreasing curve

AnswerCorrect curve

Part (b) · 3 marks

Sketch the curve, draw a suitable straight line, and use it to solve 2 × 0.5ˣ + 2x − 3.5 = 0 for −1 ≤ x ≤ 2.

  1. 12 × 0.5ˣ − 1 = 2.5 − 2x (rearranging the equation)
  2. 2Draw the line y = 2.5 − 2x (through (0, 2.5) and (1, 0.5))
  3. 3Read the x-value where it crosses the curve: x ≈ 1.36

Answer1.3 to 1.4

Part (c) · 2 marks

State the largest integer k for which 2 × 0.5ˣ − 1 = k has no solutions, and give the asymptote equation.

  1. 12 × 0.5ˣ is always positive, so y = 2 × 0.5ˣ − 1 is always greater than −1
  2. 2The highest possible value of k is −1
  3. 3The asymptote is y = −1

AnswerThe highest possible value of k is −1 The equation of the asymptote is y = −1

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Also searched as: 0580/42/M/J/21 · 0580_s21_qp_42 · 0580 s21 42 · question 2.
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