# Rates of change in linear and quadratic functions

> AP Precalculus · Unit 1: Polynomial and Rational Functions
> Source: https://www.owlsprep.com/study/ap-precalculus-u1-rates-of-change-in-linear/

How the average rate of change behaves for two key families — linear functions (a constant rate) and quadratic functions (rates that form a linear pattern) — and how the direction those rates move gives you concavity. All from average rates, with no calculus.

**Prerequisites:** Topic 1.2 — average rate of change over an interval; Slope of a line; reading a table of values; Recognizing linear vs quadratic functions

## Learning objectives

- Find the average rate of change of a function as the slope of the secant line.
- Recognize that a linear function has a constant average rate of change.
- Recognize that a quadratic's average rates over equal intervals form a linear pattern.
- Determine concavity from whether the average rates are increasing or decreasing.

## Average Rate of Change = Slope of the Secant Line

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 $(b, f(b))$.

**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.

## Linear Functions: a Constant Rate

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 $f(x) = 2x + 1$ 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**.

> **tip**
>
> This is the defining feature of linear functions: over equal-length steps, the output goes up (or down) by the same amount every time.

## Quadratic Functions: the Rates Form a Linear Pattern

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).

> **warning**
>
> Don't confuse the two levels: the *function* $x^2$ is quadratic (not linear), but the *pattern of its average rates* ($1,3,5,7$) is linear. It's the **rates**, not the function, that are linear.

## Concavity — Read from the Rates

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

**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.)

> **tip**
>
> Going deeper (optional). The rate of change of $f$ is itself a function of $x$ — call it the rate function. Saying $f$ is concave up is exactly saying this rate function is increasing (its values go up); concave down means it's decreasing. For a quadratic, that rate function is a straight line (the $1,3,5,7$ pattern is it), so 'increasing' just means that line goes up. You'll meet this same idea, formalized, later in calculus — but here we get all of it from average rates, no calculus needed.

## AP-Style Practice

**Check your understanding**

Test your understanding with this multiple-choice question:

1. 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

   *Answer:* Linear

   *Why:* 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**.

## Common pitfalls

- **Wrong:** Thinking a linear function's rate of change speeds up or slows down.
  - Why it fails: A line has the same slope everywhere, so its average rate of change is constant over every interval — it changes by zero.
  - Correct: For a linear function, expect the same average rate on every interval; only non-linear functions have a changing rate.
- **Wrong:** Calling a quadratic 'linear' because its rates form a linear pattern.
  - Why it fails: The pattern of average rates ($1,3,5,7$) is linear, but the function itself ($x^2$) is quadratic. Two different levels.
  - Correct: Constant rates ⇒ the function is linear; rates that form a (non-constant) linear pattern ⇒ the function is quadratic.
- **Wrong:** Treating concave up as the same thing as increasing.
  - Why it fails: 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: 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:** Reaching for calculus to decide concavity.
  - Why it fails: Calculus tools are beyond this course. Concavity here is defined entirely by average rates over equal intervals.
  - Correct: Compare the average rates over equal steps: increasing ⇒ concave up, decreasing ⇒ concave down.
- **Wrong:** Deciding concavity from only two points.
  - Why it fails: Two points give a single average rate; you need several equal intervals to see whether the rates are increasing or decreasing.
  - Correct: Use at least three consecutive equal-length intervals so you can compare consecutive average rates.

## Cheatsheet

| Concept | Rule | Key Notes |
| --- | --- | --- |
| Average rate of change over $[a,b]$ | $\dfrac{f(b)-f(a)}{b-a}$ = 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 |

## 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.

- [Polynomial functions and rates of change](https://www.owlsprep.com/study/ap-precalculus-u1-polynomial-functions-and-rates-of/)
- [Rates of change (average and over equal intervals)](https://www.owlsprep.com/study/ap-precalculus-u1-rates-of-change/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ap-precalculus-u1-rates-of-change-in-linear/