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WMA11/01 Jun 2023, Question 9: Transformations of Graphs

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

The question, in short

From two given trig graphs, write equations for f(x) and g(x), then sketch transformed curves y = f(x + π/4) and y = g(x) − 2, marking axis intercepts.

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

Part (ia) · 2 marks

Use the cosine-type graph with max 3 and min −3 to write an expression for f(x).

  1. 1The graph has a maximum at x = 0, so it is a cosine curve
  2. 2The amplitude is 3 and the period is 2π
  3. 3f(x) = 3 cos x

Answerf(x) = 3 cos x

Part (ib) · 3 marks

Sketch y = f(x + π/4) for −2π < x < 2π, showing coordinates where the curve meets the axes.

  1. 1y = f(x + π/4) is y = 3 cos(x + π/4): translate Figure 3 by π/4 to the left
  2. 2x-intercepts: x + π/4 = ±π/2, ±3π/2, so x = −7π/4, −3π/4, π/4, 5π/4
  3. 3y-intercept: y = 3 cos(π/4) = 3√2/2
  4. 4Sketch the wave between −2π and 2π with these intercepts, max 3 and min −3

AnswerThe curve y = 3 cos(x + π/4): Figure 3 translated π/4 to the left, max 3 and min −3, crossing the x-axis at (−7π/4, 0), (−3π/4, 0), (π/4, 0), (5π/4, 0) and the y-axis at (0, 3√2/2).

Part (iia) · 2 marks

Use the sine-type graph with max 1, min −1 and period π to write an expression for g(x).

  1. 1The graph passes through the origin going up, so it is a sine curve
  2. 2The amplitude is 1 and the period is π (two cycles in 2π)
  3. 3g(x) = sin 2x

Answerg(x) = sin 2x

Part (iib) · 2 marks

Sketch y = g(x) − 2 for −2π < x < 2π, showing the y-intercept coordinates.

  1. 1y = g(x) − 2 is Figure 4 translated 2 units down
  2. 2The curve now oscillates between −3 and −1, entirely below the x-axis
  3. 3y-intercept: sin 0 − 2 = −2, so (0, −2)

AnswerThe curve y = sin 2x − 2: Figure 4 moved down 2 units, oscillating between −1 and −3 (entirely below the x-axis), with y intercept (0, −2).

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