AP PreCalculus Practice Quiz: Rates of Change in Linear and Quadratic Functions

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

Test your understanding with short quizzes. This quiz has 16 questions to check your progress.

Question 1 of 16

According to the provided content, which statement accurately describes the average rate of change for a linear function?

A)It increases at a constant rate.B)It is always positive.C)It is constant over any length input-value interval.D)It can be modeled by a linear function.

All Questions (16)

According to the provided content, which statement accurately describes the average rate of change for a linear function?

A) It increases at a constant rate.

B) It is always positive.

C) It is constant over any length input-value interval.

D) It can be modeled by a linear function.

Correct Answer: C

The content explicitly states that for a linear function, the average rate of change over any length input-value interval is constant.

The average rate of change of a function f(x) over the closed interval [a, b] is equivalent to which of the following?

A) The slope of the tangent line at x = (a+b)/2.

B) The slope of the secant line connecting the points (a, f(a)) and (b, f(b)).

C) The instantaneous rate of change at x = a.

D) The constant rate of change of the function.

Correct Answer: B

The provided text specifies that the average rate of change over the closed interval [a, b] is the slope of the secant line from the point (a, f(a)) to (b, f(b)).

For a quadratic function, the average rates of change are calculated over several consecutive, equal-length intervals. What property do these average rates of change exhibit?

A) They are constant.

B) They change at a constant rate.

C) They are always negative.

D) They cannot be determined without the function's equation.

Correct Answer: B

The content states that for a quadratic function, the average rates of change over consecutive equal-length input-value intervals can be given by a linear function. Because they form a linear pattern, these average rates of change are changing at a constant rate.

The average rate of change for a function is 2 over the interval [0, 1], 5 over the interval [1, 2], and 8 over the interval [2, 3]. What can be determined about the change in the average rates of change?

A) The change in the average rates of change is 0.

B) The change in the average rates of change is constant.

C) The change in the average rates of change is decreasing.

D) The function must be linear.

Correct Answer: B

The average rate of change increases from 2 to 5 (a change of +3), and then from 5 to 8 (a change of +3). Since the change in the average rates of change is a constant 3, this pattern is characteristic of a quadratic function.

If the average rate of change of a function over equal-length intervals is consistently increasing, what does this imply about the graph of the function?

A) The graph is a straight line.

B) The graph is concave down.

C) The graph is concave up.

D) The graph has a maximum point.

Correct Answer: C

The provided content states: 'When the average rate of change over equal-length input-value intervals is increasing for all small-length intervals, the graph of the function is concave up.'

The average rate of change of a quadratic function g(x) over the interval [2, 4] is 6. The average rate of change over the interval [4, 6] is 10. What is the average rate of change over the interval [6, 8]?

A) 10

B) 12

C) 14

D) 16

Correct Answer: C

For a quadratic function, the average rates of change over consecutive equal-length intervals change at a constant rate. The interval length is 2. The rate of change increased from 6 to 10, which is a change of +4. Therefore, the next rate of change over an equal-length interval will also increase by 4, making it 10 + 4 = 14.

A function f(x) is defined by the points (0, 5), (2, 9), and (4, 13). What is the average rate of change over the interval [0, 4]?

A) 2

B) 4

C) 8

D) 13

Correct Answer: A

The average rate of change is the slope of the secant line from (0, f(0)) to (4, f(4)). Using the points (0, 5) and (4, 13), the slope is (13 - 5) / (4 - 0) = 8 / 4 = 2. Notice the AROC from [0,2] is (9-5)/(2-0)=2 and from [2,4] is (13-9)/(4-2)=2, indicating a linear function.

The average rates of change for a function over consecutive intervals of length 1 can be described by the linear function A(x) = 4x - 1, where x is the start of the interval. What type of function is the original function?

A) Linear

B) Quadratic

C) Cubic

D) Cannot be determined

Correct Answer: B

The content states that for a quadratic function, the average rates of change over consecutive equal-length input-value intervals can be given by a linear function. Since the average rates of change are described by a linear function, the original function must be quadratic.

Which function type is characterized by a constant average rate of change over any interval?

A) Quadratic

B) Exponential

C) Linear

D) All function types

Correct Answer: C

This is a direct application of the rule provided: 'For a linear function, the average rate of change over any length input-value interval is constant.'

For a quadratic function f(x), the average rate of change over [x, x+h] is calculated. Then, the average rate of change over [x+h, x+2h] is calculated. What is true about the difference between these two average rates of change?

A) The difference is zero.

B) The difference depends on the value of x.

C) The difference is a constant value that does not depend on x.

D) The difference is a quadratic expression.

Correct Answer: C

The content states that for a quadratic function, the average rates of change over consecutive equal-length intervals (in this case, length h) change at a constant rate. This means the difference between any two such consecutive average rates of change is a constant.

A secant line passes through the points (1, 2) and (5, 18) on the graph of a function f(x). What information does the slope of this line provide?

A) The instantaneous rate of change at x=3.

B) The function is linear.

C) The average rate of change of f(x) over the interval [1, 5].

D) The graph of f(x) is concave up.

Correct Answer: C

By definition, the slope of the secant line between two points on a function's graph represents the average rate of change of the function over the interval defined by the x-coordinates of those points. The slope is (18-2)/(5-1) = 16/4 = 4, which is the AROC on [1, 5].

Consider the sequence 3, 6, 11, 18, 27. If we treat this as a function f(n) where n=1, 2, 3, 4, 5, what can be concluded about the change in the average rates of change over consecutive unit intervals?

A) It is zero, indicating a linear relationship.

B) It is constant, indicating a quadratic relationship.

C) It is increasing, indicating a relationship of a degree higher than two.

D) It is decreasing, indicating the graph is concave down.

Correct Answer: B

First, find the average rates of change (the first differences): 6-3=3, 11-6=5, 18-11=7, 27-18=9. The average rates of change are 3, 5, 7, 9. Now, find the change in these rates (the second differences): 5-3=2, 7-5=2, 9-7=2. Since the change in the average rates of change is a constant (2), the relationship is quadratic.

Which statement best contrasts the average rate of change (AROC) of linear and quadratic functions?

A) The AROC of a linear function is constant, while the AROC of a quadratic function is also constant.

B) The AROC of a linear function is constant, while the AROC of a quadratic function is not constant.

C) The AROC of a linear function is not constant, while the AROC of a quadratic function is constant.

D) The AROC for both function types can be modeled by a linear function.

Correct Answer: B

Based on the provided content, a key distinction is that a linear function's average rate of change is constant for any interval, whereas a quadratic function's average rate of change varies depending on the interval.

The average rate of change for a function g(x) over the interval [-2, 0] is -5, and over the interval [0, 2] is -1. What does this suggest about the graph of g(x) on the interval [-2, 2]?

A) The graph is likely concave down.

B) The function is likely linear.

C) The graph is likely concave up.

D) The function must have a maximum at x=0.

Correct Answer: C

The average rate of change increased from -5 to -1 over consecutive equal-length intervals. Since the average rate of change is increasing, the content suggests that the graph of the function is concave up.

For a certain function, the slope of the secant line from (x, f(x)) to (x+1, f(x+1)) is given by the expression 2x+3. What is the slope of the secant line from (3, f(3)) to (5, f(5))?

A) 9

B) 10

C) 11

D) 20

Correct Answer: B

The expression 2x+3 represents the average rate of change over a unit interval starting at x. Since this is a linear function, the original function is quadratic. The AROC over [3, 4] is 2(3)+3=9. The AROC over [4, 5] is 2(4)+3=11. The AROC over [3, 5] is the average of these two AROCs over consecutive unit intervals, which is (9+11)/2 = 10.

To determine the change in the average rates of change for a function from a table of values over consecutive, equal-length intervals, what would you calculate?

A) The ratio of the outputs (y-values).

B) The difference between consecutive average rates of change.

C) The slope of the function at a single point.

D) The average of all the calculated rates of change.

Correct Answer: B

The 'change in the average rates of change' is found by first calculating the average rate of change for each interval (often called 'first differences') and then finding the difference between those consecutive rates (often called 'second differences').