---
title: "AP Precalculus 1.3: Rates of Change in Linear and Quadratic Functions"
description: "Review AP Precalculus 1.3 rates of change in linear and quadratic functions, including average rate of change, secant line slope, equal-length intervals, and concavity."
canonical: "https://fiveable.me/ap-pre-calc/unit-1/rates-change-linear-quadratic-functions/study-guide/8cCFDC3VHLyBZGbA"
type: "study-guide"
subject: "AP Pre-Calculus"
unit: "Unit 1 – Polynomial and Rational Functions"
lastUpdated: "2026-06-09"
---

# AP Precalculus 1.3: Rates of Change in Linear and Quadratic Functions

## Summary

Review AP Precalculus 1.3 rates of change in linear and quadratic functions, including average rate of change, secant line slope, equal-length intervals, and concavity.

## Guide

The [average rate of change](/ap-pre-calc/unit-1/function-model-construction-application/study-guide/n3ZaYWJqkvxnoJEt "fv-autolink") of a function over an interval is the slope of the secant line connecting the two endpoints. For a linear function this rate is always constant, and for a [quadratic function](/ap-pre-calc/key-terms/quadratic-function "fv-autolink") the average rates of change over equal-length intervals follow a linear pattern, which means those rates themselves change at a constant rate.

## Why This Matters for the AP Precalculus Exam

This topic builds the rate-of-change thinking you use across all of AP Precalculus. You will calculate average rates of change from equations, tables, and graphs, then interpret what those values say about a function's behavior. Recognizing that linear functions have constant average rates while quadratic functions have linearly changing average rates helps you tell function types apart and connect rates to [concavity](/ap-pre-calc/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh "fv-autolink"). On both the multiple-choice and free-response sections, you may need to compute an average rate, read it as the slope of a secant line, and explain what it means in context with correct units.

## Key Takeaways

- The average rate of change over the [closed interval](/ap-pre-calc/key-terms/closed-interval "fv-autolink") [a, b] equals the slope of the secant line from (a, f(a)) to (b, f(b)), computed as (f(b) - f(a)) / (b - a).
- A linear function has a constant average rate of change over any interval, so its rates of change are changing at a rate of [zero](/ap-pre-calc/unit-1/polynomial-functions-complex-zeros/study-guide/Ex6Y5wBlobCpxdVr "fv-autolink").
- A quadratic function's average rates of change over consecutive equal-length intervals can be described by a linear function, so those rates change at a constant rate.
- Increasing average rates over small equal-length intervals mean the graph is [concave up](/ap-pre-calc/key-terms/concave-up "fv-autolink"); decreasing average rates mean [concave down](/ap-pre-calc/key-terms/concave-down "fv-autolink").
- Always pair a rate value with its meaning and units when a context is given.

## Average Rates of Change

A **linear function** produces a straight-line graph. Its average rate of change over any input-value interval is **constant**, because the slope stays the same everywhere along the line.

A **quadratic function** has a squared term in the input, so its graph is a curve. Its average rate of change is not constant. Instead, when you compute the average rates of change over consecutive equal-length input intervals, those values can be given by a **linear function**.

The average rate of change over a closed interval [a, b] measures how much the output changes on average per unit change in input over that interval.

You can picture it as a line connecting two points on the graph, **(a, f(a)) and (b, f(b))**. This line is the **secant line**, and its slope is the average rate of change over [a, b]:

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

![secant-line-vs-tangent-line.png](https://storage.googleapis.com/static.prod.fiveable.me/images/secant-line-vs-tangent-line.png-1692823156347-26327)

###### A graph showing a secant line and a tangent line. Image Courtesy of Calc Workshop

## Change in Average Rates of Change

A linear function has a [constant rate of change](/ap-pre-calc/unit-2/change-arithmetic-geometric-sequences/study-guide/TjmiwbtDpN420iuL "fv-autolink") across its whole [domain](/ap-pre-calc/key-terms/domain "fv-autolink"). So the average rate of change over consecutive equal-length input intervals is given by a **constant function**, which means these average rates are **changing at a rate of zero**. The function is not speeding up or slowing down.

For a quadratic function, the rate of change is not constant. But the average rates of change over consecutive equal-length input intervals can be given by a **linear function**. That means the **average rates of change for a quadratic are changing at a constant rate**, even though the rate of change itself is not constant. Whether the function speeds up or slows down depends on the sign of the leading coefficient.

### A Note on Concavity

When the average rate of change over equal-length input intervals is **increasing** for all small intervals, the graph is curving upward, or **concave up**.

When the average rate of change over equal-length input intervals is **decreasing** for all small intervals, the graph is curving downward, or **concave down**.

Concavity tells you a lot about how a function behaves, including where its high and low points tend to be.

## How to Use This on the AP Precalculus Exam

### MCQ

- Read tables carefully. To find an average rate of change between two [rows](/ap-pre-calc/unit-4/matrices/study-guide/V3FaFXSlBTaJW9k0 "fv-autolink"), use (f(b) - f(a)) / (b - a). The interval does not need to start at zero.
- If the average rates over equal-length intervals are all equal, the function is linear. If those average rates change by a constant amount each step, the function is quadratic.
- Use the sign of the average rate to describe [direction](/ap-pre-calc/unit-4/vectors/study-guide/E38atN4oigqKq7in "fv-autolink"): a positive rate means output increases as input increases, and a negative rate means output decreases as input increases.

### Free Response

- Show the secant slope setup, not just the final number. Writing (f(b) - f(a)) / (b - a) with the actual values makes your work clear.
- When a context is given, state what the rate means and include units, such as "the average rate of change is 0.4 cubic meters per minute."
- If asked about concavity, connect it to whether average rates of change are increasing or decreasing over small equal-length intervals.

### Common Trap

- Average rate of change is the slope of the secant line, not the value of the function at a point. Avoid plugging in a single x-value when the question asks for a rate over an interval.

## Common Misconceptions

- A quadratic does not have a constant average rate of change. What is constant is the *rate at which its average rates change*, because those average rates follow a linear pattern.
- The average rate of change over [a, b] is not f(b) - f(a). You must divide by the change in input, b - a.
- Concavity is about whether the rate of change is increasing or decreasing, not about whether the function itself is increasing or decreasing. A function can be decreasing while still concave up.
- A secant line and a [tangent](/ap-pre-calc/unit-3/tangent-function/study-guide/MmhWdpovNDRCpyBgCc0x "fv-autolink") line are different. The secant connects two points and gives an average rate over an interval, while a tangent touches at one point.
- A larger average rate of change does not always mean larger output values. It describes how fast the output is changing over the interval, not how big the output is.

## Related AP Precalculus Guides

- [1.9 Rational Functions and Vertical Asymptotes](/ap-pre-calc/unit-1/rational-functions-vertical-asymptotes/study-guide/lNBy0zlDvb8r6tn1)
- [1.11 Equivalent Representations of Polynomial and Rational Expressions](/ap-pre-calc/unit-1/equivalent-representations-polynomial-rational-expressions/study-guide/NRzwc7vjmULoqIyP)
- [1.5 Polynomial Functions and Complex Zeros](/ap-pre-calc/unit-1/polynomial-functions-complex-zeros/study-guide/Ex6Y5wBlobCpxdVr)
- [1.8 Rational Functions and Zeros](/ap-pre-calc/unit-1/rational-functions-zeros/study-guide/tLF1Ul5EL1rbz8Ye)
- [1.12 Transformations of Functions](/ap-pre-calc/unit-1/transformations-functions/study-guide/6S5lhzaXAYrpVwQz)
- [1.4 Polynomial Functions and Rates of Change](/ap-pre-calc/unit-1/polynomial-functions-rates-change/study-guide/tQN39nNwYGsKoKj1)

## Vocabulary

- **average rate of change**: The change in the output of a function divided by the change in the input over a specified interval, calculated as (f(b) - f(a))/(b - a) for the interval [a, b].
- **concave down**: A characteristic of a graph where the rate of change is decreasing, creating a curve that opens downward.
- **concave up**: A characteristic of a graph where the rate of change is increasing, creating a curve that opens upward.
- **constant rate**: A rate of change that remains the same across all intervals; for quadratic functions, the rate at which average rates of change are changing.
- **equal-length input-value intervals**: Consecutive intervals along the input axis that have the same width, used to compare average rates of change.
- **linear function**: A polynomial function of degree 1 with the form f(x) = mx + b, representing a constant rate of change.
- **quadratic function**: A polynomial function of degree 2 with the form f(x) = ax² + bx + c, creating a parabolic graph.
- **secant line**: A line that intersects a curve at two points, used to represent the average rate of change between those points.
- **sequence**: A function from the whole numbers to the real numbers, producing a list of ordered values.
- **slope**: The rate of change of a line, representing how much the output changes for each unit change in the input.

## FAQs

### What is average rate of change?

Average rate of change measures how much a function's output changes per unit change in input over an interval. Graphically, it is the slope of the secant line through the endpoints.

### How do you calculate average rate of change on [a, b]?

Use the formula (f(b) - f(a)) / (b - a). This compares the change in output to the change in input over the interval from a to b.

### How are rates of change different for linear and quadratic functions?

A linear function has the same average rate of change over every interval. A quadratic function has average rates of change over equal-length intervals that follow a linear pattern.

### What do consecutive equal-length intervals show?

Consecutive equal-length intervals help reveal whether average rates stay constant or change by a constant amount. Constant rates suggest a linear function, while rates that change linearly suggest a quadratic function.

### How is concavity related to average rates of change?

If average rates of change over small equal-length intervals are increasing, the graph is concave up. If they are decreasing, the graph is concave down.

### How is AP Precalculus 1.3 tested?

AP Precalculus 1.3 is tested with equations, tables, graphs, and contexts. Be ready to compute secant slopes, compare equal-length intervals, identify linear or quadratic patterns, and explain rates with units.

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