4MA1/1H Jan 2021, Question 21: Trigonometric Graphs and Equations

Part (ai) · 1 mark

State where the minimum of y = f(x + 3) lies, given the original minimum at (−9, 15).

1. 1The minimum point of y = f(x) is at (−9, 15).
2. 2For y = f(x + 3), the graph shifts left by 3 units.
3. 3−9 − 3 = −12
4. 4The y-coordinate remains unchanged.
5. 5So the minimum point is (−12, 15)

Answer(−12, 15)

Part (aii) · 1 mark

State where the minimum of y = (1/3)f(x) lies, starting from (−9, 15).

1. 1For y = (1/3)f(x), the y-values are scaled by a factor of 1/3.
2. 2The x-coordinate of the minimum remains the same.
3. 3The original y-coordinate is 15.
4. 415 × (1/3) = 5
5. 5So the minimum point is (−9, 5)

Answer(−9, 5)

Part (b) · 2 marks

From the drawn wave of y = a cos(x + b)° on 0 ≤ x ≤ 360, with a > 0 and 0 < b < 360, work out a and b.

1. 1The curve goes up to 2 and down to −2, so a = 2
2. 2At x = 0, y = 0 and the curve goes down, like y = −2 sin x°
3. 32 cos(x + 90)° = −2 sin x°, so b = 90
4. 4a = 2, b = 90

Answera = 2, b = 90

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