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Quadratic Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Rates of change in quadratic functions

What is a quadratic function?

A quadratic function can be written in the form $f (x) = a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$
The graph of a quadratic function is a parabola
- If $a > 0$ , the parabola opens upward ( $\cup$ -shaped)
- If $a < 0$ , the parabola opens downward ( $\cap$ -shaped)

How do the average rates of change of a quadratic function behave?

The average rate of change of a quadratic function is not constant (i.e. unlike a linear function)
- It changes depending on which interval you look at
For a quadratic function, the average rates of change over consecutive equal-length input-value intervals can be described by a linear function
- This means the average rates of change increase (or decrease) by the same amount from one interval to the next
E.g. consider $f (x) = x^{2} + 1$ , evaluated at equally spaced inputs

$x$	0	1	2	3	4
$f (x)$	1	2	5	10	17

The consecutive intervals each have length 1, so the average rates of change equal the differences in output values
- From $x = 0$ to $x = 1$ , average rate of change $= 1$
- From $x = 1$ to $x = 2$ , average rate of change $= 3$
- From $x = 2$ to $x = 3$ , average rate of change $= 5$
- From $x = 3$ to $x = 4$ , average rate of change $= 7$
The average rates of change are $1, 3, 5, 7$
- These increase by $2$ each time, forming a linear pattern

What is the rate of change of the average rates of change of a quadratic function?

For a quadratic function, the average rates of change over consecutive equal-length intervals form a linear pattern
- therefore those average rates of change are changing at a constant rate
The "second differences" (i.e. the differences between consecutive average rates of change) are constant
This is a key property that distinguishes quadratic functions:
- For a linear function, the average rates of change are constant (changing at a rate of zero)
- For a quadratic function, the average rates of change are changing at a constant (non-zero) rate

Examiner Tips and Tricks

If the equal-length input-value intervals in a table have a length of 1, then the average rates of change over the intervals will just be equal to the differences between the output-value values.

Because

$\frac{f (x_{n + 1}) - f (x_{n})}{x_{n + 1} - x_{n}} = \frac{f (x_{n + 1}) - f (x_{n})}{1} = f (x_{n + 1}) - f (x_{n})$

Worked Example

$x$	$h (x)$
0	99
1	76
2	58
3	45
4	37

The table shows values for a function $h$ at selected values of $x$ . Which of the following claim and explanation statements best fits the data?

(A) $h$ is best modeled by a linear function, because the rate of change over consecutive equal-length input-value intervals is constant.

(B) $h$ is best modeled by a linear function, because the change in the average rates of change over consecutive equal-length input-value intervals is constant.

(C) $h$ is best modeled by a quadratic function, because the rate of change over consecutive equal-length input-value intervals is constant.

(D) $h$ is best modeled by a quadratic function, because the change in the average rates of change over consecutive equal-length input-value intervals is constant.

Answer:

The consecutive $x$ intervals all have a length of 1

so the average rate of change over each interval will just be equal to the difference in the $h (x)$ values

Calculate the differences of the $h (x)$ values:

$99 76 58 45 37 - 23 - 18 - 13 - 8$

Those values are not constant, so the rate of change over consecutive intervals is not constant
This rules out options (A) and (C)

Calculate the difference of those differences

$99 76 58 45 37 - 23 - 18 - 13 - 8 5 5 5$

Those are the same, so the change in the average rates of change over consecutive equal-length input-value intervals is constant

I.e. the average rates of change are changing at a constant rate
This is the property of a quadratic function, which rules out option (B)

(D) $h$ is best modeled by a quadratic function, because the
change in the average rates of change over consecutive
equal-length input-value intervals is constant

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