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4MA1/1H Jun 2019, Question 22: Trigonometric Graphs and Equations

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1H (Higher) · Calculator · 6 marks

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The question, in short

Draw a transformed sine curve on the given grid, then rewrite a quadratic in completed-square form and describe the single transformation mapping y = x² to the resulting curve.

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

Part (a) · 2 marks

On the grid, sketch y = 2sin(x + 30)° for x from 0 to 360, using the printed y = sin x° curve as reference.

  1. 1shift the graph of y = sinx° left by 30° to get y = sin(x + 30)°
  2. 2then stretch vertically by scale factor 2 to get y = 2sin(x + 30)°
  3. 3plot key points: (60, 2), (150, 0), (240, −2), (330, 0)
  4. 4draw smooth curve through these points
  5. 5ensure curve does not pass through (0, 0)

AnswerA sine curve through (60, 2), (150, 0), (240, −2) and (330, 0), not through (0, 0)

Part (bi) · 2 marks

Express x² − 6x + 10 as (x − a)² + b, giving integer values of a and b.

  1. 1start with x² − 6x + 10
  2. 2complete the square: x² − 6x becomes (x − 3)² − 9
  3. 3so x² − 6x + 10 = (x − 3)² − 9 + 10
  4. 4(x − 3)² + 1

Answer(x − 3)² + 1

Part (bii) · 2 marks

Using your completed-square form, state fully the single transformation taking y = x² to y = x² − 6x + 10.

  1. 1from (b)(i), y = x² − 6x + 10 is y = (x − 3)² + 1
  2. 2this is a transformation of y = x²
  3. 3replace x with x − 3: horizontal shift right by 3
  4. 4add 1: vertical shift up by 1
  5. 5combined as translation vector (3, 1)

Answertranslation of the vector 3 right and 1 up

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