Rates of change in linear and quadratic functions

AP Precalculus· Unit 1 · Polynomial & Rational Functions· 13 min read

1. Average Rate of Change = Slope of the Secant Line★☆☆☆☆⏱ 2 min

Recall from Topic 1.2: the average rate of change of $f$ over $[a, b]$ is the change in output over the change in input. Topic 1.3 names it precisely — it is the slope of the secant line joining the two endpoints $(a, f (a))$ and $(b, f (b))$ .

📘 Definition

Average rate of change (secant slope)

The average rate of change of $f$ over $[a, b]$ is the slope of the line (the secant line) through $(a, f (a))$ and $(b, f (b))$ .

\text{average rate of change} = \frac{f(b) - f(a)}{b - a}

The whole topic asks a simple follow-up question: as you slide along the function, how does this rate behave? The answer is different — and revealing — for lines versus parabolas.

2. Linear Functions: a Constant Rate★★☆☆☆⏱ 3 min

For a linear function, the average rate of change is the same over every interval, no matter its length — it is just the slope.

📐 Worked Example

Show that $f (x) = 2 x + 1$ has the same average rate of change over $[0, 3]$ and over $[5, 9]$ .

1
Over $[0, 3]$ :
2
$\frac{f(3) - f(0)}{3 - 0} = \frac{7 - 1}{3} = 2$
3
Over $[5, 9]$ :
4
$\frac{f(9) - f(5)}{9 - 5} = \frac{19 - 11}{4} = 2$
5
Both give $2$ — the slope. A line's average rate of change is constant everywhere.

Because the rate never changes, we say the rate of change of a linear function is changing at a rate of zero.

3. Quadratic Functions: the Rates Form a Linear Pattern★★★☆☆⏱ 4 min

For a quadratic function, the average rates of change over consecutive equal-length intervals are not constant — but they change in a very regular way: they form a linear pattern.

📐 Worked Example

Find the average rate of change of $f (x) = x^{2}$ over each consecutive unit interval from 0 to 4.

interval	$[0, 1]$	$[1, 2]$	$[2, 3]$	$[3, 4]$
avg rate	?	?	?	?

1
Compute each as $\frac{f ( b ) - f ( a )}{b - a}$ (each interval has width 1):
2
$1,\quad 3,\quad 5,\quad 7$
3
The rates are not constant — but each one is 2 more than the last. The sequence $1, 3, 5, 7$ is a linear pattern (an arithmetic sequence).
4
So a quadratic's average rates over equal intervals can be described by a linear function, and the rate is changing at a constant rate (here, up by 2 each step).

4. Concavity — Read from the Rates★★★☆☆⏱ 4 min

The direction those average rates move tells you the function's concavity.

📘 Definition

Concave up / concave down

If the average rates of change over equal-length intervals are increasing (getting bigger) for all small intervals, the graph is concave up (it cups upward). If those rates are decreasing, the graph is concave down (it caps downward).

This is the point students miss: concavity is not the same as increasing/decreasing. Increasing/decreasing is about the output (does $f$ rise or fall — the sign of the rate). Concavity is about the rate itself (does the rate rise or fall). A function can be decreasing and concave up at once — falling, but falling less and less steeply, because its rate is still increasing (say, from $- 4$ to $- 2$ to $0$ ).

📐 Worked Example

A function's average rates over four consecutive equal intervals are $8, 5, 3, 2$ . Is it concave up or concave down here?

1
Look at how the rates change: $8 \to 5 \to 3 \to 2$ — they are decreasing.
2
Decreasing average rates ⇒ concave down. (The outputs are still rising — all rates are positive — but each step rises less than the last.)

5. AP-Style Practice★★☆☆☆⏱ 4 min

✓ Quick check

Test your understanding with this multiple-choice question:

A function's average rates of change over consecutive equal intervals are $2, 2, 2, 2$ . What kind of function is it?
- Quadratic
- Linear
- Concave up
- Concave down
Reveal answer
1 —
Constant average rates over equal intervals is the signature of a linear function (its rate never changes). A quadratic's rates would form a non-constant linear pattern like $1, 3, 5, 7$ .

📐 Worked Example

The table gives $g$ . (a) Find the average rate of change over each unit interval. (b) Is $g$ linear or quadratic? (c) Is it concave up or concave down?

$x$	$0$	$1$	$2$	$3$
$g (x)$	$2$	$3$	$6$	$11$

1
(a) Average rates over $[0, 1], [1, 2], [2, 3]$ :
2
$3-2 = 1, \quad 6-3 = 3, \quad 11-6 = 5$
3
(b) The rates $1, 3, 5$ are not constant but go up by 2 each time — a linear pattern — so $g$ is quadratic.
4
(c) The rates are increasing ( $1 < 3 < 5$ ), so $g$ is concave up.

6. Common Pitfalls

Wrong move:

Thinking a linear function's rate of change speeds up or slows down.

Why:

A line has the same slope everywhere, so its average rate of change is constant over every interval — it changes by zero.

Correct move:

For a linear function, expect the same average rate on every interval; only non-linear functions have a changing rate.

Wrong move:

Calling a quadratic 'linear' because its rates form a linear pattern.

Why:

The pattern of average rates ( $1, 3, 5, 7$ ) is linear, but the function itself ( $x^{2}$ ) is quadratic. Two different levels.

Correct move:

Constant rates ⇒ the function is linear; rates that form a (non-constant) linear pattern ⇒ the function is quadratic.

Wrong move:

Treating concave up as the same thing as increasing.

Why:

Increasing/decreasing is the sign of the rate (does the output rise or fall); concavity is whether the rate itself rises or falls. A function can be decreasing yet concave up (falling less and less steeply).

Correct move:

Judge concavity only from whether the average rates are increasing (up) or decreasing (down) — never from whether the function itself is going up or down.

Wrong move:

Reaching for calculus to decide concavity.

Why:

Calculus tools are beyond this course. Concavity here is defined entirely by average rates over equal intervals.

Correct move:

Compare the average rates over equal steps: increasing ⇒ concave up, decreasing ⇒ concave down.

Wrong move:

Deciding concavity from only two points.

Why:

Two points give a single average rate; you need several equal intervals to see whether the rates are increasing or decreasing.

Correct move:

Use at least three consecutive equal-length intervals so you can compare consecutive average rates.

7. Quick Reference Cheatsheet

Concept	Rule	Key Notes
Average rate of change over $[a, b]$	$\frac{f ( b ) - f ( a )}{b - a}$ = slope of secant line	the line through $(a, f (a))$ and $(b, f (b))$
Linear function	average rate of change is constant	same on every interval = the slope; rate changes by 0
Quadratic function	average rates form a linear pattern	e.g. $1, 3, 5, 7$ ; the rate changes at a constant amount
Concave up	average rates over equal steps are increasing	graph cups upward
Concave down	average rates over equal steps are decreasing	graph caps downward
Concavity vs increasing/decreasing	concavity = the rate's direction, not the output's	a function can fall and still be concave up

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2024 · AP Precalculus
Classify a function from a table of average rates of change
2024 · AP Precalculus
Determine concavity from increasing/decreasing rates

What's Next

You can now read a function's family and its concavity straight from a table of average rates: constant rates mean linear, a linear pattern of rates means quadratic, and increasing versus decreasing rates give concave up versus concave down. Next, Topic 1.4 extends these rate-of-change ideas to polynomial functions of any degree.