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AP PreCalculus Practice Quiz: Change in Tandem

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

Test your understanding with short quizzes. This quiz has 15 questions to check your progress.

Question 1 of 15

According to the provided definition, what is a key characteristic of a function?

All Questions (15)

According to the provided definition, what is a key characteristic of a function?

A) Each input value is mapped to multiple output values.

B) Each input value is mapped to exactly one output value.

C) The output values must always be greater than the input values.

D) The graph of the relation must be a straight line.

Correct Answer: B

The definition explicitly states that a function is a mathematical relation where 'each input value is mapped to exactly one output value'.

A function is defined as increasing over an interval of its domain. What does this imply about its input and output values?

A) As the input values increase, the output values always decrease.

B) As the input values increase, the output values remain constant.

C) As the input values increase, the output values always increase.

D) As the input values decrease, the output values always increase.

Correct Answer: C

This question directly tests the definition provided: 'A function is increasing over an interval of its domain if, as the input values increase, the output values always increase.'

If a function's rate of change is decreasing over an interval, what can be concluded about the graph of the function on that interval?

A) The graph is a straight line.

B) The graph is concave up.

C) The graph is increasing.

D) The graph is concave down.

Correct Answer: D

The provided content states: 'The graph of a function is concave down on intervals in which the rate of change is decreasing.'

Let f be a function. If f(5) = 10 and f(8) = 20, which of the following statements must be true for f to be considered increasing on the interval [5, 8]?

A) For any a and b in [5, 8] where a < b, it must be that f(a) < f(b).

B) The graph of f must be concave up on [5, 8].

C) f(x) = (10/3)x - 20/3 for all x in [5, 8].

D) The rate of change of f must be constant on [5, 8].

Correct Answer: A

The definition of an increasing function is that as input values increase, output values always increase. Option A is the formal way of stating this. The function could be increasing but concave down, or not be a line at all, so B, C, and D are not necessarily true.

A new startup company's total profit is growing each month. However, the amount of new profit gained each month is getting larger over time. If P(t) represents the total profit after t months, which statement correctly describes the graph of P(t)?

A) The graph is increasing and concave down.

B) The graph is increasing and concave up.

C) The graph is decreasing and concave up.

D) The graph is decreasing and concave down.

Correct Answer: B

The total profit is growing, so the function P(t) is increasing. The amount of new profit gained each month represents the rate of change. Since this rate is getting larger, the rate of change is increasing. A function with an increasing rate of change is concave up.

What does it mean for the graph of a function to be concave up on an interval?

A) The function's output values are increasing.

B) The function's rate of change is increasing.

C) The function's output values are positive.

D) The function's rate of change is positive.

Correct Answer: B

The definition provided is: 'The graph of a function is concave up on intervals in which the rate of change is increasing.' An increasing function has a positive rate of change, but concavity is about how the rate of change itself is changing.

A runner is completing a marathon. In the first hour, they run 8 miles. In the second hour, they run 6 miles. In the third hour, they run 5 miles. Let D(t) be the total distance run after t hours. Which pair of properties describes the graph of D(t) for t from 0 to 3?

A) Increasing and concave up

B) Decreasing and concave up

C) Increasing and concave down

D) Decreasing and concave down

Correct Answer: C

The total distance D(t) is always getting larger, so the function is increasing. The rate of change is the speed (miles per hour). The speed is 8 mph, then 6 mph, then 5 mph. Since the rate of change is decreasing, the graph is concave down.

Consider a function g(x). We know that g(1)=5, g(2)=10, and g(3)=14. How does the average rate of change from x=1 to x=2 compare to the average rate of change from x=2 to x=3?

A) The rate of change is increasing.

B) The rate of change is decreasing.

C) The rate of change is constant.

D) The relationship cannot be determined.

Correct Answer: B

The average rate of change from x=1 to x=2 is (10-5)/(2-1) = 5. The average rate of change from x=2 to x=3 is (14-10)/(3-1) = 4. Since 4 < 5, the rate of change is decreasing over these intervals.

A function f(x) is increasing on the interval [0, 10], but it is concave down on the same interval. Which of the following statements comparing function values could be true?

A) f(2) - f(1) = 3 and f(8) - f(7) = 4

B) f(2) - f(1) = 3 and f(8) - f(7) = 2

C) f(1) = 10 and f(2) = 5

D) f(2) - f(1) = -3 and f(8) - f(7) = -2

Correct Answer: B

The function is increasing, so the output values must increase as input values increase. This eliminates C and D. The function is concave down, which means its rate of change is decreasing. The change from x=1 to x=2 is 3. The change from x=7 to x=8 is 2. Since the rate of change has decreased (from 3 to 2), this is consistent with being concave down. Option A shows an increasing rate of change.

A pot of water is heated on a stove. Let T(m) be the temperature of the water after m minutes. The water heats up quickly at first, but as it approaches boiling, the temperature increases more slowly. Which statement best describes the graph of T(m) before boiling begins?

A) The graph is concave up because the temperature is rising.

B) The graph is concave down because the rate of temperature increase is decreasing.

C) The graph is a straight line because the stove provides constant heat.

D) The graph is concave up because the rate of temperature increase is increasing.

Correct Answer: B

The temperature is rising, so the function is increasing. The phrase 'increases more slowly' means that the rate of change is decreasing. A graph with a decreasing rate of change is concave down.

Which of the following relations does NOT represent a function?

A) Input 1 -> Output A, Input 2 -> Output B, Input 3 -> Output C

B) Input 1 -> Output A, Input 2 -> Output A, Input 3 -> Output A

C) Input 1 -> Output A, Input 1 -> Output B, Input 2 -> Output C

D) Input 1 -> Output B, Input 2 -> Output C, Input 3 -> Output D

Correct Answer: C

A function maps each input to exactly one output. In option C, the input value '1' is mapped to two different output values, 'A' and 'B', which violates the definition of a function.

The value of a rare collectible is tracked over time. From 2010 to 2012, its value increased by $500. From 2012 to 2014, its value increased by $800. Let V(t) be the value of the collectible, where t is the number of years since 2010. Assuming this trend continues, how would you describe the graph of V(t)?

A) The graph is increasing and concave down.

B) The graph is constant.

C) The graph is increasing and concave up.

D) The graph is decreasing and concave up.

Correct Answer: C

The value is increasing in each period, so the function is increasing. The rate of change over the first two-year period is $500. The rate of change over the second two-year period is $800. Since the rate of change is increasing, the graph is concave up.

A function h(t) is used to model the height of a plant. If h(t) is both increasing and concave up over an interval, what must be true about the plant's growth?

A) The plant is growing, but its growth rate is slowing down.

B) The plant is shrinking, but its rate of shrinking is slowing down.

C) The plant is growing, and its growth rate is speeding up.

D) The plant is growing at a constant rate.

Correct Answer: C

'Increasing' means the height is getting larger, so the plant is growing. 'Concave up' means the rate of change is increasing. In this context, the rate of change is the growth rate. Therefore, the plant is growing and its growth rate is speeding up (accelerating).

For a function f(x) on an interval [a, c], we observe that for any x1 and x2 in the interval such that a < x1 < x2 < c, the value of (f(x2)-f(x1))/(x2-x1) is less than the value of (f(c)-f(x2))/(c-x2). What can be concluded about the graph of f(x) on this interval?

A) The graph is concave up.

B) The graph is concave down.

C) The graph is a line with a positive slope.

D) The graph is a line with a negative slope.

Correct Answer: A

The expression (f(y)-f(x))/(y-x) represents the average rate of change between two points. The condition states that the average rate of change over a later interval is greater than the average rate of change over an earlier interval. This indicates that the rate of change is increasing, which is the definition of a concave up function.

A car starts from rest and gradually increases its speed to 60 mph. Let d(t) be the distance the car has traveled at time t. Which of the following best describes the relationship between the quantities d and t?

A) As t increases, d increases, and the rate of change of d is decreasing.

B) As t increases, d increases, and the rate of change of d is increasing.

C) As t increases, d decreases, and the rate of change of d is increasing.

D) As t increases, d increases, and the rate of change of d is constant.

Correct Answer: B

The distance traveled, d, is always increasing as time, t, increases. The rate of change of distance is speed. Since the car is starting from rest and increasing its speed, the rate of change of d is increasing. This corresponds to an increasing, concave up function.