Modeling Periodic Phenomena (with Tangent & Sinusoids) Quiz

1) A swimmer's vertical bobbing is modeled by y(t) = 1.2 sin(πt) + 0.8. What is the maximum height?

2.4

1.2

0.8

2

Given: max = D + |A|. Goal: compute.

Step 1: 0.8 + 1.2 = 2.0.

So, the final answer is 2.0.

Explanation

Given: max = D + |A|. Goal: compute.

Step 1: 0.8 + 1.2 = 2.0.

So, the final answer is 2.0.

2) The function y = 6 cos((π/3)(t - 1)) + 2. Which is true?

Amplitude 6, midline y = 2, phase shift right 1, period 6

Amplitude 6, midline y = 2, phase shift left 1, period 6

Amplitude 6, midline y = 2, phase shift right 1, period 6/π

Amplitude 6, midline y = 2, phase shift right 1, period 6π

Given: A = 6, D = 2, B = π/3. Goal: features.

Step 1: Period = 2π/(π/3) = 6; (t − 1) ⇒ shift right 1.

So, the final answer is amplitude 6, midline y = 2, shift right 1, period 6.

Explanation

Given: A = 6, D = 2, B = π/3. Goal: features.

Step 1: Period = 2π/(π/3) = 6; (t − 1) ⇒ shift right 1.

So, the final answer is amplitude 6, midline y = 2, shift right 1, period 6.

3) A sound wave is modeled by s(t) = A cos(2πft) with amplitude A = 0.1 and frequency f = 440. What is the period?

1/440

2π/440

440

2π

Given: period T = 1/f. Goal: T.

Step 1: T = 1/440.

So, the final answer is 1/440.

Explanation

Given: period T = 1/f. Goal: T.

Step 1: T = 1/440.

So, the final answer is 1/440.

4) A sinusoidal graph oscillates between -7 and 1 with midline at -3. If one cycle takes length 5 and the graph reaches its minimum at t = 0, which model fits?

Y = −3 + 4 cos((2π/5)t)

Y = −3 − 4 cos((2π/5)t)

Y = −3 + 4 sin((2π/5)t)

Y = −3 − 4 sin((2π/5)t)

Given: amplitude 4, midline −3, min at t = 0. Goal: pick phase.

Step 1: Cosine with negative amplitude starts at minimum.

Step 2: ω = 2π/5.

So, the final answer is y = −3 − 4 cos((2π/5)t).

Explanation

Given: amplitude 4, midline −3, min at t = 0. Goal: pick phase.

Step 1: Cosine with negative amplitude starts at minimum.

Step 2: ω = 2π/5.

So, the final answer is y = −3 − 4 cos((2π/5)t).

5) P(t) = 3 + 2 sin(4πt). Frequency (cycles per unit t)?

2π

4π

2

1/(2π)

Given: 4πt = 2π(2t). Goal: frequency.

Step 1: f = 2.

So, the final answer is 2.

Explanation

Given: 4πt = 2π(2t). Goal: frequency.

Step 1: f = 2.

So, the final answer is 2.

6) A sinusoid has zeros at t = 0, t = 4, t = 8, ... and increases immediately after each zero. The midline is y = 2 and amplitude 5. Which model fits?

Y = 2 + 5 cos((π/4)t)

Y = 2 + 5 sin((π/4)t)

Y = 2 − 5 sin((π/4)t)

Y = 2 − 5 cos((π/4)t)

Given: zeros every 4 ⇒ half-period = 4 ⇒ period 8 ⇒ ω = π/4. Goal: choose sign.

Step 1: Increasing after zero ⇒ +sin form.

Step 2: Add midline 2 and amplitude 5.

So, the final answer is y = 2 + 5 sin((π/4)t).

Explanation

Given: zeros every 4 ⇒ half-period = 4 ⇒ period 8 ⇒ ω = π/4. Goal: choose sign.

Step 1: Increasing after zero ⇒ +sin form.

Step 2: Add midline 2 and amplitude 5.

So, the final answer is y = 2 + 5 sin((π/4)t).

7) The sinusoid y = 8 sin((π/6)t) - 2 is shifted how from y = 8 sin((π/6)t)?

Horizontal shift right 2

Horizontal shift left 2

Vertical shift down 2

Vertical shift up 2

Given: subtract 2. Goal: describe shift.

Step 1: “− 2” moves graph down by 2.

So, the final answer is vertical shift down 2.

Explanation

Given: subtract 2. Goal: describe shift.

Step 1: “− 2” moves graph down by 2.

So, the final answer is vertical shift down 2.

8) A seasonal water level is modeled by W(t) = 12 - 3 cos((2π/365)t). What is the range?

[9, 15]

[12, 15]

[0, 12]

[9, 12]

Given: amplitude 3, midline 12. Goal: range.

Step 1: Range = [12 − 3, 12 + 3] = [9, 15].

So, the final answer is [9, 15].

Explanation

Given: amplitude 3, midline 12. Goal: range.

Step 1: Range = [12 − 3, 12 + 3] = [9, 15].

So, the final answer is [9, 15].

9) You observe a sinusoid with period 3 seconds and amplitude 10. Which B gives the correct period in y = 10 sin(Bt)?

2π/3

3

2π

π/3

Given: period T = 3. Goal: B.

Step 1: B = 2π/T = 2π/3.

So, the final answer is 2π/3.

Explanation

Given: period T = 3. Goal: B.

Step 1: B = 2π/T = 2π/3.

So, the final answer is 2π/3.

10) A sinusoid has midline y = -1, amplitude 4, and passes upward through the midline at t = 0. Which model fits?

Y = −1 + 4 cos(ωt)

Y = −1 + 4 sin(ωt)

Y = −1 − 4 sin(ωt)

Y = −1 − 4 cos(ωt)

Given: upward through midline at 0 ⇒ sine with +A. Goal: choose form.

Step 1: Use y = −1 + 4 sin(ωt).

So, the final answer is y = −1 + 4 sin(ωt).

Explanation

Given: upward through midline at 0 ⇒ sine with +A. Goal: choose form.

Step 1: Use y = −1 + 4 sin(ωt).

So, the final answer is y = −1 + 4 sin(ωt).

11) A simple harmonic motion is modeled by y = 3 sin(2x) + 1. What is the amplitude and midline?

Amplitude 3, midline y = 1

Amplitude 2, midline y = 3

Amplitude 1, midline y = 3

Amplitude 3, midline y = 0

Given: y = A sin(Bx) + D. Goal: read A and D.

Step 1: A = 3 ⇒ amplitude 3; D = 1 ⇒ midline y = 1.

So, the final answer is amplitude 3, midline y = 1.

Explanation

Given: y = A sin(Bx) + D. Goal: read A and D.

Step 1: A = 3 ⇒ amplitude 3; D = 1 ⇒ midline y = 1.

So, the final answer is amplitude 3, midline y = 1.

12) A pendulum's displacement is modeled by θ(t) = 0.2 cos(5t). What is the frequency in cycles per second?

5/(2π)

5

1/5

2π/5

Given: angular frequency ω = 5 rad/s. Goal: f.

Step 1: f = ω/(2π) = 5/(2π).

So, the final answer is 5/(2π).

Explanation

Given: angular frequency ω = 5 rad/s. Goal: f.

Step 1: f = ω/(2π) = 5/(2π).

So, the final answer is 5/(2π).

13) A sinusoid y = -2 sin(4x) + 5 is horizontally stretched/compressed compared to y = sin(x). What is its period?

π/2

π

2π

4π

Given: B = 4. Goal: period.

Step 1: Period = 2π/|B| = 2π/4 = π/2.

So, the final answer is π/2.

Explanation

Given: B = 4. Goal: period.

Step 1: Period = 2π/|B| = 2π/4 = π/2.

So, the final answer is π/2.

14) The function T(t) = 18 + 7 cos((2π/24)(t - 6)) models daily temperature in °C. What is the time of maximum temperature?

T = 6

T = 0

T = 12

T = 18

Given: maximum when cos(⋯) = 1. Goal: t.

Step 1: (2π/24)(t − 6) = 0 ⇒ t = 6.

So, the final answer is t = 6.

Explanation

Given: maximum when cos(⋯) = 1. Goal: t.

Step 1: (2π/24)(t − 6) = 0 ⇒ t = 6.

So, the final answer is t = 6.

15) A sinusoidal model fits data with max 15, min 3, and period 10. If using a cosine that starts at maximum at t = 0, which is correct?

Y = 9 cos((π/5)t) + 9

Y = 6 cos((π/5)t) + 9

Y = 9 cos((2π/10)t) + 9

Y = 6 cos((2π/10)t) + 9

Given: amplitude 6, midline 9, ω = 2π/10 = π/5. Goal: model.

Step 1: Cosine at max at t = 0 ⇒ +cos.

So, the final answer is y = 6 cos((π/5)t) + 9.

Explanation

Given: amplitude 6, midline 9, ω = 2π/10 = π/5. Goal: model.

Step 1: Cosine at max at t = 0 ⇒ +cos.

So, the final answer is y = 6 cos((π/5)t) + 9.

16) The function y = A sin(Bx) + D has amplitude 6, period 4, and midline y = -3. Which equation matches?

Y = 6 sin(πx/2) − 3

Y = 6 sin(2πx/4) − 3

Y = 6 sin((π/2)x) − 3

Y = 6 sin(πx) − 3

Given: amplitude 6 ⇒ A = 6; period 4 ⇒ B = 2π/4 = π/2; midline −3 ⇒ D = −3.

Step 1: Compose y = 6 sin((π/2)x) − 3.

So, the final answer is y = 6 sin((π/2)x) − 3.

Explanation

Given: amplitude 6 ⇒ A = 6; period 4 ⇒ B = 2π/4 = π/2; midline −3 ⇒ D = −3.

Step 1: Compose y = 6 sin((π/2)x) − 3.

So, the final answer is y = 6 sin((π/2)x) − 3.

17) A Ferris wheel has a center height of 60 ft and radius 50 ft. One full revolution takes 80 seconds, starting at the bottom at t = 0. Which model for height h(t) is appropriate?

H(t) = 60 − 50 cos((π/40)t)

60 + 50 sin((π/40)t − π/2)

60 + 50 cos((π/40)t + π)

60 + 50 cos((π/40)t)

Given: center 60, radius 50, period 80. Goal: model starting at bottom.

Step 1: ω = 2π/80 = π/40; bottom at t=0 ⇒ cos(0) = 1 but need height 60 − 50.

Step 2: Use minus sign to start at minimum: 60 − 50 cos((π/40)t).

So, the final answer is h(t) = 60 − 50 cos((π/40)t).

Explanation

Given: center 60, radius 50, period 80. Goal: model starting at bottom.

Step 1: ω = 2π/80 = π/40; bottom at t=0 ⇒ cos(0) = 1 but need height 60 − 50.

Step 2: Use minus sign to start at minimum: 60 − 50 cos((π/40)t).

So, the final answer is h(t) = 60 − 50 cos((π/40)t).

18) The graph of a sinusoid has maximum 12, minimum 4, and period 8. Which model fits if it starts at its maximum at t = 0?

Y = 8 cos((π/4)t) + 8

Y = 4 cos((π/4)t) + 8

Y = 4 cos((π/2)t) + 8

Y = 8 cos((π/2)t) + 8

Given: amplitude = (12 − 4)/2 = 4, midline = (12 + 4)/2 = 8. Goal: ω.

Step 1: Period 8 ⇒ ω = 2π/8 = π/4.

Step 2: Cosine starting at max: y = 4 cos((π/4)t) + 8.

So, the final answer is y = 4 cos((π/4)t) + 8.

Explanation

Given: amplitude = (12 − 4)/2 = 4, midline = (12 + 4)/2 = 8. Goal: ω.

Step 1: Period 8 ⇒ ω = 2π/8 = π/4.

Step 2: Cosine starting at max: y = 4 cos((π/4)t) + 8.

So, the final answer is y = 4 cos((π/4)t) + 8.

19) A tide height (in feet) is modeled by H(t) = 5 + 2.5 sin((π/6)t - π/2), where t is hours. What is the midline and amplitude?

Midline 0, amplitude 5

Midline 2.5, amplitude 5

Midline 7.5, amplitude 5

Midline 5, amplitude 2.5

Midline D = 5; amplitude |A| = 2.5.

Explanation

Midline D = 5; amplitude |A| = 2.5.

20) Y = −4 cos(πx) + 2. What is the period?

1

2

4

2π

Period = 2π/|B| with B = π ⇒ 2π/π = 2.

Explanation

Period = 2π/|B| with B = π ⇒ 2π/π = 2.

Modeling Periodic Phenomena (with Tangent & Sinusoids) Quiz

1) A swimmer's vertical bobbing is modeled by y(t) = 1.2 sin(πt) + 0.8. What is the maximum height?

2)

What first name or nickname would you like us to use?

2) The function y = 6 cos((π/3)(t - 1)) + 2. Which is true?

3) A sound wave is modeled by s(t) = A cos(2πft) with amplitude A = 0.1 and frequency f = 440. What is the period?

4) A sinusoidal graph oscillates between -7 and 1 with midline at -3. If one cycle takes length 5 and the graph reaches its minimum at t = 0, which model fits?

5) P(t) = 3 + 2 sin(4πt). Frequency (cycles per unit t)?

6) A sinusoid has zeros at t = 0, t = 4, t = 8, ... and increases immediately after each zero. The midline is y = 2 and amplitude 5. Which model fits?

7) The sinusoid y = 8 sin((π/6)t) - 2 is shifted how from y = 8 sin((π/6)t)?

8) A seasonal water level is modeled by W(t) = 12 - 3 cos((2π/365)t). What is the range?

9) You observe a sinusoid with period 3 seconds and amplitude 10. Which B gives the correct period in y = 10 sin(Bt)?

10) A sinusoid has midline y = -1, amplitude 4, and passes upward through the midline at t = 0. Which model fits?

11) A simple harmonic motion is modeled by y = 3 sin(2x) + 1. What is the amplitude and midline?

12) A pendulum's displacement is modeled by θ(t) = 0.2 cos(5t). What is the frequency in cycles per second?

13) A sinusoid y = -2 sin(4x) + 5 is horizontally stretched/compressed compared to y = sin(x). What is its period?

14) The function T(t) = 18 + 7 cos((2π/24)(t - 6)) models daily temperature in °C. What is the time of maximum temperature?

15) A sinusoidal model fits data with max 15, min 3, and period 10. If using a cosine that starts at maximum at t = 0, which is correct?

16) The function y = A sin(Bx) + D has amplitude 6, period 4, and midline y = -3. Which equation matches?

17) A Ferris wheel has a center height of 60 ft and radius 50 ft. One full revolution takes 80 seconds, starting at the bottom at t = 0. Which model for height h(t) is appropriate?

18) The graph of a sinusoid has maximum 12, minimum 4, and period 8. Which model fits if it starts at its maximum at t = 0?

19) A tide height (in feet) is modeled by H(t) = 5 + 2.5 sin((π/6)t - π/2), where t is hours. What is the midline and amplitude?

20) Y = −4 cos(πx) + 2. What is the period?