📝 Chapter 8: Sinusoidal Functions

For \( y = 4\sin(3x) - 2 \), state amplitude, period, vertical shift, and range.

The period of \( y = \sin(4x) \) is:

Sketch one period of \( y = 2\cos(x) + 3 \). Mark the maximum, minimum, and midline.

For \( y = -3\sin(2(x - 45°)) + 1 \), state all transformations applied to the parent \( y = \sin x \).

For \( y = 5\sin(x) - 3 \), the maximum value is:

If \( y = \sin(x) \) is shifted 60° to the left, the new equation is:

A sinusoidal function has maximum 9 at \( x = 30° \) and minimum 1 at \( x = 90° \). Determine an equation in the form \( y = a\cos(k(x - d)) + c \). Show all reasoning.

Show that \( y = \sin(x + 90°) = \cos(x) \) using either a transformation argument or substitution into special angles.

A graph passes through \( (0, 5) \), \( (90°, 1) \), \( (180°, -3) \), \( (270°, 1) \), \( (360°, 5) \). Determine an equation of the sinusoidal function modelling this graph.

Define amplitude, period, phase shift, and midline. Use \( y = 4\sin(2(x - 30°)) + 5 \) as your worked example.

Compare and contrast \( y = \sin x \) and \( y = \cos x \). State two similarities and two differences.

Describe step by step how to transform \( y = \cos x \) into \( y = -2\cos(\tfrac{1}{3}(x - 60°)) + 4 \). State the order of transformations.

Why must we factor out \( k \) inside the argument before reading off the phase shift? Use \( y = \sin(2x - 60°) \) as your example.

A Ferris wheel has a diameter of 40 m, with the lowest point 2 m above the ground. It completes one revolution every 80 seconds. A rider boards at the lowest point at \( t = 0 \). (a) Write a sinusoidal function for the rider's height \( h(t) \). (b) Find the height at \( t = 30 \) s. (c) When is the rider first at maximum height?

The water depth in a harbour varies sinusoidally between 1.5 m at low tide and 7.5 m at high tide. High tide occurs at \( t = 6 \) hours, and the period is 12 hours. (a) Write \( h(t) \). (b) Find the depth at \( t = 0 \). (c) Find all times in \( [0, 24] \) hours when the depth is exactly 6 m.

In a Canadian city, the daylight (in hours) varies sinusoidally between 8 (winter solstice, day 355) and 16 (summer solstice, day 172). (a) Write \( D(t) \) where \( t \) is the day of year. (b) Find \( D(81) \) (approximate the spring equinox).