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WMA11/01 Oct 2021, Question 9: Transformations of Graphs

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

Given f(x) = √x with P(9, 3) on the curve, sketch and label two transformed graphs, then find where they intersect and give the exact coordinates of Q.

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

Part (a) · 3 marks

On a copy of the graph, draw and label y = f(2x) and y = f(x) + 3, marking the image of P on each.

  1. 1y = f(2x) is a stretch parallel to the x-axis, scale factor ½: P(9, 3) → (4.5, 3)
  2. 2y = f(x) + 3 is a translation 3 units up: P(9, 3) → (9, 6)
  3. 3Sketch y = f(2x) starting at O, above y = √x; sketch y = f(x) + 3 starting at (0, 3)

Answery = f(2x): curve from O, above y = √x, a stretch towards the y-axis, with P moved to (4.5, 3). y = f(x) + 3: same shape as y = √x moved up 3, starting at (0, 3), with P moved to (9, 6).

Part (b) · 3 marks

Using the intersection Q of the two transformed graphs, show its x coordinate satisfies √x = 3(√2 + 1).

  1. 1√(2x) = √x + 3
  2. 2√2·√x − √x = 3, so (√2 − 1)√x = 3
  3. 3√x = 3/(√2 − 1) × (√2 + 1)/(√2 + 1)
  4. 4√x = 3(√2 + 1)/(2 − 1) = 3(√2 + 1)

AnswerShown: √(2x) = √x + 3 gives √x(√2 − 1) = 3, so √x = 3/(√2 − 1) = 3(√2 + 1)

Part (c) · 3 marks

Using the result from part b, determine the exact coordinates of Q in simplest form.

  1. 1x = [3(√2 + 1)]² = 9(√2 + 1)²
  2. 2= 9(2 + 2√2 + 1) = 27 + 18√2
  3. 3y = √x + 3 = 3√2 + 3 + 3 = 6 + 3√2
  4. 4Q = (27 + 18√2, 6 + 3√2)

AnswerQ(27 + 18√2, 6 + 3√2)

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