maddyhelps

Math 30-1 · Worksheets

Sinusoidal models · Set A

Ten questions of mixed difficulty, covering Sinusoidal models. Print it, or work through it on screen — the answer key starts on its own page.

All worksheets

Sinusoidal models · Set A

Math 30-1 · maddyhelps.com

Name
Date
Score
/ 10

Circle the best answer for each question. Show your work in the space provided.

  1. A sinusoidal function has a period of 12. What is the value of b, in radians?

    1. a) π/6
    2. b) 12
    3. c) π/12
    4. d) 24π

  2. A rider's height is h(t) = −10 cos(πt/20) + 12, in metres after t seconds. How high is the rider at t = 0?

    1. a) 2 m
    2. b) −10 m
    3. c) 12 m
    4. d) 22 m

  3. In a sinusoidal model of a rider's height on a Ferris wheel, what does the amplitude represent?

    1. a) the time for one rotation
    2. b) the height of the wheel's centre
    3. c) the radius of the wheel
    4. d) the diameter of the wheel

  4. A sinusoidal graph has a maximum of 9 and a minimum of 1. What is the equation of its midline?

    1. a) y = 0
    2. b) y = 4
    3. c) y = 5
    4. d) y = 8

  5. Which equation has an amplitude of 3, a period of π and a midline of y = −2?

    1. a) y = 3 sin 2x − 2
    2. b) y = −2 sin 3x + π
    3. c) y = 2 sin 3x − 3
    4. d) y = 3 sin(½x) − 2

  6. Daylight in a city varies sinusoidally over a year, from a maximum of 17 hours to a minimum of 7 hours. What are the amplitude and period?

    1. a) 5 hours and 365 days
    2. b) 5 hours and 182.5 days
    3. c) 10 hours and 365 days
    4. d) 12 hours and 365 days

  7. A rider's height is h(t) = −10 cos(πt/20) + 12, in metres after t seconds. When does the rider first reach the top?

    1. a) 10 seconds
    2. b) 20 seconds
    3. c) 12 seconds
    4. d) 40 seconds

  8. Which equation produces exactly the same graph as y = sin x?

    1. a) y = −cos x
    2. b) y = cos(x + π/2)
    3. c) y = cos(x − π/2)
    4. d) y = cos 2x

  9. A sinusoidal graph has a maximum at (2, 7) and the next minimum at (8, 1). What is its period?

    1. a) 3
    2. b) 8
    3. c) 12
    4. d) 6

  10. Where does y = cos x reach its maximum, for 0 ≤ x < 2π?

    1. a) x = 0
    2. b) x = π/2
    3. c) x = 3π/2
    4. d) x = π

Answer key · Sinusoidal models · Set A

Math 30-1 · maddyhelps.com

  1. a) π/6 — Period = 2π ÷ b, so b = 2π ÷ period = 2π ÷ 12 = π/6. π/12 would give a period of 24.
  2. a) 2 m — cos 0 = 1, so h(0) = −10 + 12 = 2 m. The negative cosine starts at its minimum — the rider boards at the bottom.
  3. c) the radius of the wheel — The rider goes one radius above and one radius below the centre. The centre's height is the midline, and one rotation is the period.
  4. c) y = 5 — The midline sits halfway between the extremes: (9 + 1) ÷ 2 = 5.
  5. a) y = 3 sin 2x − 2 — Amplitude 3 means a = 3. Period π means b = 2π ÷ π = 2. Midline −2 means d = −2. b = ½ would give a period of 4π.
  6. a) 5 hours and 365 days — Amplitude = (17 − 7) ÷ 2 = 5 hours, and the pattern repeats once a year. 12 hours is the midline, not the amplitude.
  7. b) 20 seconds — The top is where −10 cos(πt/20) is largest, so cos(πt/20) = −1, which first happens when πt/20 = π. That gives t = 20, halfway through a 40-second rotation.
  8. c) y = cos(x − π/2) — Sine is cosine shifted right by a quarter period. Moving cos x's peak from 0 to π/2 lands it on sin x's peak.
  9. c) 12 — From a maximum to the next minimum is half a cycle: 8 − 2 = 6. A full period is twice that, 12.
  10. a) x = 0 — cos 0 = 1, the largest cosine can be. That is why cosine models are handy when something starts at its highest point.