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

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Rates of change in linear functions

What is a linear function?

A linear function is a function whose graph is a straight line
It can be written in the form $f (x) = b + m x$ , where:
- $m$ is the slope (the rate of change of the function)
- $b$ is the initial value (the output value when $x = 0$ , i.e. the $y$ -intercept)
E.g. $f (x) = 3 + 2 x$ is a linear function with slope $2$ and initial value $3$

What is the average rate of change of a linear function?

For a linear function, the average rate of change over any interval is constant
- It doesn't matter which two points you choose, or how wide or narrow the interval is
- The average rate of change will always equal the slope $m$
This is the defining property of linear functions
- They change at a constant rate
E.g. for $f (x) = 3 + 2 x$
- Average rate of change over $[0, 1]$ : $\frac{f (1) - f (0)}{1 - 0} = \frac{5 - 3}{1} = 2$
- Average rate of change over $[1, 4]$ : $\frac{f (4) - f (1)}{4 - 1} = \frac{11 - 5}{3} = 2$
- Average rate of change over $[10, 50]$ : $\frac{f (50) - f (10)}{50 - 10} = \frac{103 - 23}{40} = 2$
- No matter which interval is chosen, the average rate of change is always $2$

How does a linear function appear in a table of values?

If a linear function is represented in a table of values
- Then if the input values in a table are equally spaced
  - the differences between consecutive output values will be the same
- These constant differences correspond to the slope of the linear function
E.g. consider this table:

$x$	0	2	4	6	8
$f (x)$	5	11	17	23	29

The input values increase by $2$ each time (equally spaced)
- and the output values increase by $6$ each time (constant difference)
The average rate of change is $\frac{6}{2} = 3$ for every interval
So this function could be modelled by a linear function with slope $3$

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

The average rate of change of a linear function is constant
- This means the average rates of change over consecutive equal-length intervals will all have the same value
- So the average rates of change themselves can be described by a constant function

The rate of change of the average rates of change of a linear function is zero
- I.e. if you ask "how quickly are the average rates of change changing?"
  - the answer is, they are not changing at all
This is a key distinction between linear functions and other function types:
- For a linear function, the rate of change is constant
  - The average rates of change are changing at a rate of zero
- For other function types, the rate of change itself changes

Worked Example

$x$	$h (x)$
0	79
1	73
2	67
3	61
4	55

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 non-zero and 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 non-zero and 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:

$79 73 67 61 55 - 6 - 6 - 6 - 6$

Those values are constant, so the rate of change over consecutive intervals is constant
And the change in the average rates of change over consecutive equal-length input-value intervals is zero
- Because they don't change
- This rules out options (B) and (D)

You should know that a constant rate of change over consecutive equal-length input-value intervals is a key property of a linear function

This rules out option (C)

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

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