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WMA11/01 Jan 2022, Question 2: Completed Square Form and Turning Points

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 6 marks

The question, in short

For f(x) = 11 − 4x − 2x², rewrite in completed-square form, draw the curve C showing its y-intercept, and state its axis of symmetry.

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

Part (a) · 3 marks

Find integers a, b and c so that 11 − 4x − 2x² equals a + b(x + c)².

  1. 1f(x) = 11 − 2(x² + 2x)
  2. 2x² + 2x = (x + 1)² − 1
  3. 3f(x) = 11 − 2[(x + 1)² − 1] = 11 − 2(x + 1)² + 2
  4. 4f(x) = 13 − 2(x + 1)²

Answerf(x) = 13 − 2(x + 1)², so a = 13, b = −2, c = 1

Part (b) · 2 marks

Draw y = f(x), marking the coordinates where the curve meets the y-axis.

  1. 1The coefficient of x² is negative, so the curve is ∩-shaped
  2. 2x = 0 gives y = 11, so it crosses the y-axis at (0, 11)
  3. 3From (a) the maximum point is (−1, 13), to the left of the y-axis
  4. 4Since the maximum is above the x-axis, the curve crosses the x-axis twice: x = −1 ± √6.5

AnswerA ∩-shaped parabola crossing the y-axis at (0, 11), with its maximum point (−1, 13) to the left of the y-axis, crossing the x-axis once on each side of the origin (at x = −1 ± √6.5)

Part (c) · 1 mark

Give the equation of the mirror line of C.

  1. 1The line of symmetry passes through the maximum point (−1, 13)
  2. 2x = −1

Answerx = −1

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