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4PM1/01 Jan 2023, Question 3: Rational Functions and Asymptotes

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 9 marks

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The question, in short

For the curve y = (ax + 3)/(1 − 2x), x ≠ 1/2, with horizontal asymptote y = 4, find a, the vertical asymptote, axis crossings, and sketch the curve.

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

Part (a) · 2 marks

Find the constant a given that the horizontal asymptote of C is y = 4.

  1. 1As x → ∞, y → a/(−2)
  2. 2a/(−2) = 4
  3. 3a = −8

Answera = −8

Part (b) · 1 mark

State the equation of the vertical asymptote of C.

  1. 1The denominator 1 − 2x = 0 when x = 1/2

Answerx = 1/2

Part (c) · 2 marks

Find the coordinates where C meets (i) the x-axis and (ii) the y-axis.

  1. 1y = (3 − 8x)/(1 − 2x)
  2. 2y = 0: 3 − 8x = 0, x = 3/8
  3. 3x = 0: y = 3

Answer(i) (3/8, 0) (ii) (0, 3)

Part (d) · 4 marks

Sketch C on the given axes, clearly showing both asymptotes and the axis-crossing coordinates.

  1. 1Draw the asymptotes x = 1/2 and y = 4
  2. 2Left branch: approaches y = 4 from below as x → −∞, passes through (0, 3) and (3/8, 0), goes to −∞ as x → 1/2
  3. 3Right branch: comes down from +∞ near x = 1/2 and approaches y = 4 from above

AnswerTwo-branch curve with asymptotes x = 1/2 and y = 4: left branch below y = 4, passing through (0, 3) and (3/8, 0) and going down towards x = 1/2; right branch above y = 4 to the right of x = 1/2

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