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AP Precalculus Quiz

AP Precalculus Quiz: Periodic Phenomena

Practice Periodic Phenomena in AP Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 16

0 of 16 answered

The average monthly temperature in a city is modeled by a periodic function, T(m)T(m)T(m), where mmm is the month number. The function has a period of 12 months. The function is increasing from m=1m=1m=1 (January) to m=7m=7m=7 (July).

Given that T(m)T(m)T(m) is periodic with a period of 12, which of the following statements must be true?

Select an answer to continue

What this quiz covers

This quiz focuses on Periodic Phenomena, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The average monthly temperature in a city is modeled by a periodic function, T(m)T(m)T(m), where mmm is the month number. The function has a period of 12 months. The function is increasing from m=1m=1m=1 (January) to m=7m=7m=7 (July).

Given that T(m)T(m)T(m) is periodic with a period of 12, which of the following statements must be true?

  1. The function is decreasing from m=7m=7m=7 to m=13m=13m=13.
  2. The function is increasing from m=13m=13m=13 to m=19m=19m=19. (correct answer)
  3. The function is decreasing from m=19m=19m=19 to m=25m=25m=25.
  4. The function is increasing from m=12m=12m=12 to m=18m=18m=18.

Explanation: The behavior of a periodic function repeats every period. The function is increasing on the interval from m=1m=1m=1 to m=7m=7m=7. This behavior will repeat 12 months later. The interval starting 12 months after m=1m=1m=1 is m=1+12=13m=1+12=13m=1+12=13, and the interval ending 12 months after m=7m=7m=7 is m=7+12=19m=7+12=19m=7+12=19. Therefore, the function must be increasing from m=13m=13m=13 to m=19m=19m=19.

Question 2

Let g(t)g(t)g(t) be a periodic function with a period of 6. A minimum value of the function occurs at t=2t=2t=2.

Which of the following must be true?

  1. A maximum value must occur at t=5t=5t=5.
  2. A minimum value must occur at t=14t=14t=14. (correct answer)
  3. The function must be increasing on the interval (2,5)(2, 5)(2,5).
  4. The function must be decreasing on the interval (8,11)(8, 11)(8,11).

Explanation: Since the function is periodic with period 6, if a minimum occurs at t=2t=2t=2, then other minima must occur at t=2+6kt = 2 + 6kt=2+6k for integer values of kkk. For k=2k=2k=2, a minimum occurs at t=2+6(2)=14t = 2 + 6(2) = 14t=2+6(2)=14. The locations of maxima and intervals of increase/decrease depend on the specific shape of the function within a period and cannot be determined from the given information alone.

Question 3

The function f(x)=xsin⁡(x)f(x) = x \sin(x)f(x)=xsin(x) models a phenomenon.

Why is the function f(x)f(x)f(x) not a periodic function?

  1. The function does not have a constant period because the zeros are not equally spaced.
  2. The function is not periodic because its output values do not repeat exactly over any interval.
  3. The function is not periodic because the amplitude of the oscillations is not constant. (correct answer)
  4. The function is not periodic because it is a product of a linear and a trigonometric function.

Explanation: For a function to be periodic, the output values must repeat exactly over successive intervals of a fixed length (the period). While the function f(x)=xsin⁡(x)f(x) = x \sin(x)f(x)=xsin(x) oscillates, the factor of xxx causes the amplitude of the oscillations to increase as ∣x∣|x|∣x∣ increases. Since the maximum and minimum values change in each oscillation, the function's values do not repeat exactly, and therefore it is not periodic. Distractor D is not a sufficient reason, as some products can be periodic.

Question 4

Consider a periodic function fff with period k>0k>0k>0.

Which of the following equations is the definition of a periodic function fff with period kkk?

  1. f(x+k)=f(x)f(x+k) = f(x)f(x+k)=f(x) for all xxx in the domain. (correct answer)
  2. f(kx)=f(x)f(kx) = f(x)f(kx)=f(x) for all xxx in the domain.
  3. f(x)=f(−x)f(x) = f(-x)f(x)=f(−x) for all xxx in the domain.
  4. f(x+k)=f(x)+kf(x+k) = f(x) + kf(x+k)=f(x)+k for all xxx in the domain.

Explanation: The definition of a periodic function fff with period kkk is that the function's values repeat every kkk units. This is formally expressed as f(x+k)=f(x)f(x+k) = f(x)f(x+k)=f(x) for all xxx in the function's domain, where kkk is the smallest positive constant for which this is true. Distractor C is the definition of an even function. Distractors B and D describe other types of function transformations, not periodicity.

Question 5

A machine part moves up and down in a periodic fashion. Its height, h(t)h(t)h(t), in centimeters at time ttt seconds is modeled by a periodic function. The part moves from its minimum height of 5 cm to its maximum height of 25 cm and back to its minimum height in a total of 8 seconds.

What is the period of the function h(t)h(t)h(t)?

  1. 4 seconds
  2. 8 seconds (correct answer)
  3. 16 seconds
  4. 20 seconds

Explanation: The period of a periodic function is the length of the smallest interval over which the function completes one full cycle. The description states that the part moves from its minimum, to its maximum, and back to its minimum in 8 seconds. This is the definition of one full cycle. Therefore, the period is 8 seconds. Distractor A is the half-period (time from min to max).

Question 6

The function P(t)P(t)P(t) models the population of a certain species of insect, where ttt is the number of months since the beginning of a study. The function is periodic, repeating its pattern every 12 months. Over the interval 2<t<52 < t < 52<t<5, the population is decreasing.

Based on the periodic nature of the insect population, on which of the following intervals must the population also be decreasing?

  1. 5<t<85 < t < 85<t<8
  2. 8<t<118 < t < 118<t<11
  3. 14<t<1714 < t < 1714<t<17 (correct answer)
  4. 20<t<2320 < t < 2320<t<23

Explanation: Since the function is periodic with a period of 12 months, the behavior of the function on any interval will repeat 12 months later. The interval 14<t<1714 < t < 1714<t<17 corresponds to the interval (2+12)<t<(5+12)(2+12) < t < (5+12)(2+12)<t<(5+12). Therefore, the population must also be decreasing on this interval. The other intervals do not correspond to a simple period shift of the given interval.

Question 7

A function fff models a periodic phenomenon. The function has a period of 12. It is known that f(5)=10f(5) = 10f(5)=10 and f(8)=3f(8) = 3f(8)=3.

Based on the properties of the periodic function fff, which of the following statements must be true?

  1. f(17)=10f(17) = 10f(17)=10 (correct answer)
  2. f(12)=10f(12) = 10f(12)=10
  3. f(20)=10f(20) = 10f(20)=10
  4. f(5)=−10f(5) = -10f(5)=−10

Explanation: A periodic function with period kkk satisfies the property f(x+k)=f(x)f(x + k) = f(x)f(x+k)=f(x). Since the period is 12, f(5+12)=f(5)f(5 + 12) = f(5)f(5+12)=f(5). Therefore, f(17)=f(5)=10f(17) = f(5) = 10f(17)=f(5)=10. Distractor B incorrectly assumes the function value at the period is equal to a known value. Distractor C incorrectly uses f(8+12)=f(20)=3f(8+12) = f(20) = 3f(8+12)=f(20)=3, not 10. Distractor D confuses periodicity with properties of odd functions.

Question 8

The height of a tide at a particular harbor is modeled by a periodic function. The time between a high tide of 14 feet and the next low tide of 2 feet is 6.2 hours.

What is the period of the function that models the tide's height?

  1. 3.1 hours
  2. 6.2 hours
  3. 12.4 hours (correct answer)
  4. 24.8 hours

Explanation: The time between a consecutive maximum (high tide) and minimum (low tide) represents half of one full cycle. Therefore, the period is twice this duration. The period is 2×6.2=12.42 \times 6.2 = 12.42×6.2=12.4 hours. Distractor B is the half-period. Distractor A is a quarter-period. Distractor D is two full periods.

Question 9

The function f(x)f(x)f(x) is periodic. The function completes 4 full cycles over the interval 0≤x≤200 \le x \le 200≤x≤20.

What is the period of the function f(x)f(x)f(x)?

  1. 4
  2. 5 (correct answer)
  3. 10
  4. 20

Explanation: The period is the length of one cycle. If 4 cycles are completed over an interval of length 20, the length of one cycle is the total length divided by the number of cycles. Period = $20/4=5\$20 / 4 = 5$20/4=5. Distractor A is the number of cycles. Distractor D is the length of the entire interval shown. Distractor C is the length of two cycles.

Question 10

The voltage, V(t)V(t)V(t), in an alternating current (AC) circuit is a periodic function of time ttt in milliseconds. The function repeats its pattern every 16 milliseconds. A maximum voltage of 120 volts is reached at t=4t=4t=4 milliseconds.

At which of the following times will the voltage also be at a maximum of 120 volts?

  1. t=12t = 12t=12 milliseconds
  2. t=16t = 16t=16 milliseconds
  3. t=20t = 20t=20 milliseconds (correct answer)
  4. t=28t = 28t=28 milliseconds

Explanation: The function is periodic with a period of 16 milliseconds. If a maximum occurs at t=4t=4t=4, subsequent maxima will occur at t=4+16kt = 4 + 16kt=4+16k for any integer kkk. For k=1k=1k=1, the next maximum is at t=4+16=20t = 4 + 16 = 20t=4+16=20 milliseconds. Other options do not follow this pattern.

Question 11

A Ferris wheel completes one full rotation every 8 minutes. The function h(t)h(t)h(t) models a passenger's height above the ground, in meters, ttt minutes after the ride begins. The ride starts at the lowest point. The rate of change of the passenger's height is positive and increasing for the first minute of the ride.

Which of the following statements must also be true for the function h(t)h(t)h(t)?

  1. The rate of change of height is negative and increasing for the first minute of the second rotation.
  2. The rate of change of height is positive and increasing for the first minute of the second rotation. (correct answer)
  3. The rate of change of height is positive and decreasing for the first minute of the second rotation.
  4. The rate of change of height is negative and decreasing for the first minute of the second rotation.

Explanation: The period of the function is 8 minutes. This means all characteristics of the function, including the behavior of its rate of change, repeat every 8 minutes. The second rotation begins at t=8t=8t=8. The first minute of the second rotation is the interval 8<t<98 < t < 98<t<9. The behavior on this interval must be identical to the behavior on the first minute of the ride, 0<t<10 < t < 10<t<1. Therefore, the rate of change is positive and increasing.

Question 12

The number of visitors to a ski resort, V(m)V(m)V(m), is a periodic function of the month, mmm, where m=1m=1m=1 corresponds to January. The period is 12 months. The maximum number of visitors occurs in February (m=2m=2m=2), and the minimum occurs in August (m=8m=8m=8).

Which statement best describes the behavior of V(m)V(m)V(m) from April to June (from m=4m=4m=4 to m=6m=6m=6)?

  1. The number of visitors is increasing, and the rate of increase is slowing down.
  2. The number of visitors is decreasing, and the rate of decrease is speeding up.
  3. The number of visitors is increasing towards the maximum value.
  4. The number of visitors is decreasing towards the minimum value. (correct answer)

Explanation: The maximum is in February (m=2m=2m=2) and the minimum is in August (m=8m=8m=8). The interval from April (m=4m=4m=4) to June (m=6m=6m=6) falls between the maximum and the minimum. Therefore, the number of visitors must be decreasing during this time as it moves toward the minimum value in August.

Question 13

The function g(x)g(x)g(x) models a periodic phenomenon. A complete cycle of the function is observed on the interval [−3,5][-3, 5][−3,5].

What is the period of the function g(x)g(x)g(x)?

  1. 2
  2. 4
  3. 5
  4. 8 (correct answer)

Explanation: The period is the length of one complete cycle. The length of the interval [−3,5][-3, 5][−3,5] is the difference between the endpoints: 5−(−3)=85 - (-3) = 85−(−3)=8. Since one complete cycle occurs over this interval, the period of the function is 8. Distractor A is the average of the endpoints. Distractor C is one of the endpoints.

Question 14

A sound wave's pressure variation is modeled by a periodic function. The time it takes for the wave to complete one full oscillation from a pressure peak, through a trough, and back to a peak is called the period. The frequency of a sound wave is the number of cycles it completes per second, measured in Hertz (Hz).

If a sound wave has a frequency of 500 Hz, what is its period in seconds?

  1. 0.002 seconds (correct answer)
  2. 0.02 seconds
  3. 50 seconds
  4. 500 seconds

Explanation: Frequency and period are reciprocals of each other. If the frequency is 500 cycles per second, the period (time per cycle) is 1/5001 / 5001/500 seconds per cycle. 1/500=0.0021/500 = 0.0021/500=0.002. Therefore, the period is 0.002 seconds. The other answers represent miscalculations or confusion between period and frequency.

Question 15

A traffic light at an intersection is red for 60 seconds, green for 55 seconds, and yellow for 5 seconds. This sequence repeats continuously.

What is the period of the function that models the state of the traffic light?

  1. 55 seconds
  2. 60 seconds
  3. 115 seconds
  4. 120 seconds (correct answer)

Explanation: The period is the total time for one complete, repeating cycle. The cycle consists of the red, green, and yellow light durations. The total time for one cycle is the sum of these durations: 60+55+5=12060 + 55 + 5 = 12060+55+5=120 seconds. Therefore, the period of the traffic light's cycle is 120 seconds.

Question 16

A function h(t)h(t)h(t) models a periodic relationship. The function values repeat every 10 units. The function reaches a maximum value of 8 at t=3t=3t=3 and a minimum value of -2 at t=8t=8t=8.

Which of the following must also be a time when the function reaches its maximum value?

  1. t=8t=8t=8
  2. t=13t=13t=13 (correct answer)
  3. t=18t=18t=18
  4. t=23t=23t=23

Explanation: The function is periodic with a period of 10. A maximum value occurs at t=3t=3t=3. Subsequent maxima will occur at intervals of 10 units from this time. Therefore, another maximum must occur at t=3+10=13t = 3 + 10 = 13t=3+10=13. A minimum occurs at t=8t=8t=8, so another minimum would occur at t=8+10=18t=8+10=18t=8+10=18.