AP PreCalculus Practice Quiz: Periodic Phenomena

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 14 questions to check your progress.

Question 1 of 14

Which of the following best describes a periodic relationship between an input and an output?

A)As input values increase, output values consistently increase.B)As input values increase, output values demonstrate a repeating pattern over successive equal-length intervals.C)As input values increase, output values approach a single, constant value.D)As input values increase, the output values change unpredictably.

All Questions (14)

Which of the following best describes a periodic relationship between an input and an output?

A) As input values increase, output values consistently increase.

B) As input values increase, output values demonstrate a repeating pattern over successive equal-length intervals.

C) As input values increase, output values approach a single, constant value.

D) As input values increase, the output values change unpredictably.

Correct Answer: B

Based on the provided content, "A periodic relationship can be identified... if, as the input values increase, the output values demonstrate a repeating pattern over successive equal-length intervals."

In the formal definition of a periodic function, f(x+k) = f(x), what does the smallest positive value k represent?

A) The maximum value of the function.

B) The starting point of the function's pattern.

C) The period of the function.

D) The average value of the function.

Correct Answer: C

The provided content states, "The period of the function is the smallest positive value k such that f(x+k) = f(x) for all x in the domain."

A Ferris wheel starts with a passenger at the bottom, reaches the top in 45 seconds, and returns to the bottom 45 seconds later, completing one full rotation. This process repeats. What is the period of the function describing the passenger's height over time?

A) 45 seconds

B) 90 seconds

C) 22.5 seconds

D) 135 seconds

Correct Answer: B

The period is the length of one full, repeating cycle. The passenger goes from the bottom to the top (45s) and back to the bottom (45s), completing the cycle. The total length of this repeating pattern is 45 + 45 = 90 seconds.

The output values of a function at successive integer inputs are recorded as follows: 5, 8, 10, 8, 5, 2, 0, 2, 5, 8, 10... Based on this data, what is the estimated period of the function?

A) 4

B) 5

C) 8

D) 10

Correct Answer: C

The pattern of output values is 5, 8, 10, 8, 5, 2, 0, 2. The next value is 5, which is where the pattern begins to repeat. The length of this repeating interval is 8 values. Therefore, the period is estimated to be 8.

Which of the following scenarios describes a periodic relationship?

A) The height of a tree as it grows from a seedling to maturity over 20 years.

B) The distance a car travels from its starting point while maintaining a constant speed on a straight highway.

C) The height of a point on the rim of a bicycle wheel as the bicycle moves at a constant speed.

D) The balance of a savings account that earns compound interest annually.

Correct Answer: C

The height of a point on a rotating wheel demonstrates a repeating pattern over successive equal-length intervals of time. The other scenarios describe relationships that are consistently increasing, not repeating.

The water level in a harbor is a periodic function of time. At 3:00 AM, the water is at its highest level. It then recedes, reaching its lowest level at 9:15 AM. It returns to its highest level at 3:30 PM. What is the period of the water level function?

A) 6 hours and 15 minutes

B) 12 hours and 30 minutes

C) 12 hours

D) 24 hours

Correct Answer: B

The period is the length of one full cycle. The cycle described is from one high point to the next high point. The time from the first high point (3:00 AM) to the next high point (3:30 PM, which is 15:30 in 24-hour time) is 12 hours and 30 minutes.

A child is on a swing. A function h(t) models the child's height above the ground over time t, in seconds. The child starts at the lowest point, 2 feet above the ground, at t=0. They swing to their highest point of 8 feet, then back to the lowest point, completing one full swing every 4 seconds. Based on this verbal representation, which feature would a graph of this periodic relationship have?

A) A minimum value at t=4 seconds.

B) A maximum value at t=4 seconds.

C) A minimum value at t=0 seconds.

D) A maximum value at t=0 seconds.

Correct Answer: C

The verbal representation states that the child starts at the lowest point at t=0. Therefore, a graph of the function would show a minimum value at the beginning of the interval, t=0.

A relationship is considered periodic if the output values...

A) ...increase or decrease without bound.

B) ...remain constant over the entire domain.

C) ...form a pattern that repeats over equal-length input intervals.

D) ...can be modeled by a straight line.

Correct Answer: C

The provided content defines a periodic relationship as one where "output values demonstrate a repeating pattern over successive equal-length intervals" as input values increase.

A machine part's temperature is measured every minute, producing the following sequence of output values (°C): 50, 55, 58, 55, 50, 45, 42, 45, 50, 55, 58... What is the estimated period of this function?

A) 6 minutes

B) 7 minutes

C) 8 minutes

D) 9 minutes

Correct Answer: C

To estimate the period, we must find where the pattern of output values begins to repeat. The sequence is 50, 55, 58, 55, 50, 45, 42, 45. The next value is 50, which is the same as the first value. The subsequent values (55, 58...) also match the beginning of the sequence. Therefore, the repeating pattern has a length of 8 values, and the period is 8 minutes.

The depth of water at a pier is a periodic function of time. The function f(t) has a period of 12 hours. If f(2) represents the maximum depth at 2:00 AM, what must also be true according to the definition f(x+k) = f(x)?

A) The depth will be at its minimum at 8:00 AM.

B) The depth will be at its maximum at 2:00 PM.

C) The depth will be at its maximum at 8:00 PM.

D) The depth will be at its minimum at 2:00 PM.

Correct Answer: B

The period k is 12 hours. The function has a maximum at x=2 (2:00 AM). According to the definition f(x+k) = f(x), the function will have the same value at x+k. Here, f(2 + 12) = f(14). A time of 14:00 corresponds to 2:00 PM. Therefore, the depth will also be at its maximum at 2:00 PM.

When identifying a periodic relationship, what is the crucial characteristic of the "successive intervals" over which the output pattern repeats?

A) They must be of increasing length.

B) They must be of equal length.

C) They must be of decreasing length.

D) They must correspond to positive output values only.

Correct Answer: B

The provided content explicitly states that a periodic relationship demonstrates a repeating pattern "over successive equal-length intervals."

An automated light in a factory turns on at 7:00 AM. It stays on for 3 hours, then turns off for 5 hours. This cycle repeats continuously. What is the period of the function that models whether the light is on or off?

A) 3 hours

B) 5 hours

C) 8 hours

D) 12 hours

Correct Answer: C

The period is the length of one complete, repeating cycle. The cycle consists of the light being on (3 hours) and then off (5 hours). The total length of this repeating pattern is 3 + 5 = 8 hours. The start time of 7:00 AM does not affect the length of the period.

A sound wave's pressure P(t) repeats its pattern every 0.04 seconds. It is also observed that P(t) = P(t + 0.08) and P(t) = P(t + 0.12). Based on the formal definition, what is the period of the function P(t)?

A) 0.12 seconds

B) 0.08 seconds

C) 0.04 seconds

D) 0.02 seconds

Correct Answer: C

The period is defined as the *smallest positive value* k for which f(x+k) = f(x). While the pattern also repeats over intervals of 0.08s and 0.12s (which are multiples of the period), the smallest interval over which it repeats is given as 0.04 seconds. Therefore, the period is 0.04 seconds.

To construct a graph of the periodic relationship describing the amount of moonlight on the ground over the course of several months, what would be the most appropriate labels for the input and output axes?

A) Input: Moon's phase (e.g., full, new); Output: Time (days)

B) Input: Time (days); Output: Amount of light (e.g., brightness level)

C) Input: Amount of light; Output: Time (days)

D) Input: Moon's position in the sky; Output: Time of night

Correct Answer: B

A graph of a relationship plots the output value as a function of the input value. In this scenario, the amount of light changes as time increases. Therefore, time is the input, and the amount of light is the output. The output values demonstrate a repeating pattern as the moon goes through its phases.