Circular functions: graphs, transformations, modelling practice questions.

State the amplitude of the function f(x) = 3 sin(2x) + 1.

• 1
• 2
• 3
• 6

Answer: 3

The amplitude is |a| where f(x) = a sin(bx) + d, so it is 3. The 2 changes the period and the +1 shifts the graph vertically.

Determine the period of g(x) = cos(x/3).

• 6π
• 2π/3
• 3π
• π/3

Answer: 6π

Period = 2π ÷ b with b = 1/3, giving 2π × 3 = 6π. Dividing x by 3 stretches the graph horizontally.

The graph of y = sin x is transformed to give y = sin(x − π/4). Describe the transformation.

• A translation of π/4 units to the left
• A translation of π/4 units to the right
• A translation of π/4 units vertically downwards
• A horizontal stretch with scale factor π/4

Answer: A translation of π/4 units to the right

Replacing x by (x − c) with c > 0 shifts the graph c units to the right — the sign inside the bracket works opposite to intuition.

Determine the period of y = tan(2x).

• 2π
• π
• π/4
• π/2

Answer: π/2

The basic tangent function has period π (not 2π), so y = tan(2x) has period π ÷ 2 = π/2.

The height in metres of a carriage on a Ferris wheel is modelled by h(t) = 8 sin(πt/15) + 10, where t is in minutes. Calculate the maximum height reached.

• 10 m
• 18 m
• 8 m
• 26 m

Answer: 18 m

The maximum of sin is 1, so h_max = 8 × 1 + 10 = 18 m. The value 10 is the height of the centre, not the top.

A function y = a cos(bx) + d with a > 0 has maximum value 7 and minimum value −1. Determine the value of d.

• 3
• 4
• 7
• −1

Answer: 3

d is the midline: d = (max + min)/2 = (7 + (−1))/2 = 3. The value 4 is the amplitude a, not d.