4MA1/2H Nov 2021, Question 21: Trigonometric Graphs and Equations

Part (a) · 1 mark

Write down the coordinates of the maximum point A on the curve y = 9 − 3(x + 2)².

1. 19 − 3(x + 2)² is largest when (x + 2)² = 0, so x = −2
2. 2f(−2) = 9, so A = (−2, 9)

Answer(−2, 9)

Part (b) · 1 mark

Find the equation of the curve after translating C by the vector (4, 0).

1. 1A translation 4 to the right replaces x with x − 4
2. 2y = 9 − 3((x − 4) + 2)²
3. 3y = 9 − 3(x − 2)²

Answery = 9 − 3(x − 2)²

Part (c) · 1 mark

Describe fully the single transformation mapping y = 9 − 3(x + 2)² onto y = 3(x + 2)² − 9.

1. 13(x + 2)² − 9 = −(9 − 3(x + 2)²), so T is y = −f(x)
2. 2y = −f(x) is a reflection in the x-axis (y = 0)

AnswerReflection in the line y = 0 or x-axis

Part (d) · 3 marks

From the given cosine graph for −180 ≤ x ≤ 360, find the values of a, b and c.

1. 1From the graph: maximum 5, minimum −1
2. 2c = (5 + (−1)) ÷ 2 = 2 and a = 5 − 2 = 3
3. 3cos is 1 when x − b = 0; the maximum is at x = 270, so b = 270 or −90
4. 4a = 3, b = −90, c = 2

Answera = 3, b = −90, c = 2 or a = −3, b = 90, c = 2

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