1.2 Rates of Change

The average rate of change of a function over an interval is the ratio of the change in output to the change in input, which is the slope of the secant line connecting the two endpoints. To estimate the rate of change at a single point, shrink the interval around that point and compare average rates over smaller and smaller intervals.

Why This Matters for the AP Precalculus Exam

Rates of change are a core idea that runs through all of AP Precalculus. In this topic you build two skills the exam keeps testing: finding an average rate of change over an interval and approximating the rate of change at a point using smaller and smaller intervals. You will use these in graphical, numerical, analytical, and verbal forms, so being able to switch between a table, a graph, and a formula matters.

This topic also sets up later work on linear and quadratic rates of change, polynomial behavior, and modeling. On the free-response section you are expected to explain and justify your conclusions, not just give a number, so practice describing what a rate of change means in context and including correct units.

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Key Takeaways

• Average rate of change over $[a, b]$ is $\dfrac{f(b) - f(a)}{b - a}$, which is the slope of the secant line through $(a, f(a))$ and $(b, f(b))$.
• The rate of change at a point can be approximated by average rates of change over small intervals containing that point, when those values exist.
• Compare rates at two points by computing average rates over sufficiently small intervals around each point.
• A positive rate of change means the two quantities move in the same direction; a negative rate means they move in opposite directions.
• Always attach correct units to a rate, such as meters per minute, when a context is given.

Average Rate of Change: A Ratio of Changes

The average rate of change measures how much a function changes over a given interval. The interval can be any section of the function's domain, and the average rate of change is the ratio of the change in output values to the change in input values over that interval.

In plain terms, it tells you how fast or slow a function changes over a specific stretch of input. It is the slope of the straight line, called the secant line, that connects the two endpoints of the interval on the graph.

For a function $f(x)$ over the interval $[a, b]$, the average rate of change is

$\frac{f(b) - f(a)}{b - a}$

This compares how much the output changed to how much the input changed. A larger value means the output moved a lot relative to the input. A value near zero means the output barely changed across that interval.

Average rate of change is the same constant rate that would produce the same total change in output over that interval. That is why it acts like the slope of one straight line stretched across the interval, even when the actual function curves.

Rate of Change at a Point

The rate of change at a single point describes how quickly output values would change as the input changes right at that point. Think of it as how steep or flat the curve is at that exact spot. A steeper curve means a larger rate of change; a flatter curve means a smaller one.

You cannot read the rate at a point straight from the average rate formula, because that formula needs two different input values. Instead, you approximate it. Pick small intervals that contain the point and compute the average rate of change over those intervals. As the intervals get smaller, those average rates close in on the rate of change at the point, when such values exist.

Comparing Rates at Two Points

To compare how fast a function is changing at two different points, compute an average rate of change over a sufficiently small interval around each point. Then compare the two results.

If the average rate near one point is larger than the average rate near another, the function is changing faster at the first point. This works in tables and graphs too: a steeper secant line near a point signals a larger rate of change there.

Positive and Negative Rates of Change

Rates of change describe how two quantities vary together, and they can be positive or negative.

A positive rate of change means the two quantities move in the same direction. As the input increases, the output increases; as the input decreases, the output decreases. For example, as a car keeps accelerating, more time leads to more speed.

A negative rate of change means the two quantities move in opposite directions. As the input increases, the output decreases. For example, as a falling object's time in the air increases, its height above the ground decreases.

The sign of the rate of change connects directly to increasing and decreasing behavior. A positive rate over an interval matches an increasing function there, and a negative rate matches a decreasing function.

How to Use This on the AP Precalculus Exam

Problem Solving

• Identify the interval endpoints, then plug into $\dfrac{f(b) - f(a)}{b - a}$. Watch the order: the change in output goes on top, the change in input on the bottom.
• For a rate at a point, set up small intervals around the point and compute average rates. Smaller intervals give better approximations.
• When a context is given, state the meaning of your answer and include units, such as cubic meters per minute for a volume-versus-time situation.

Working Across Representations

• From a table, pick the two rows for your interval and use their input and output values directly.
• From a graph, find the two points and compute the slope of the secant line between them.
• From an equation, evaluate $f$ at both endpoints before subtracting.

Common Trap

• Do not flip the ratio. It is change in output over change in input, not the reverse.
• Do not confuse the average rate over an interval with the rate at a single point. They are different ideas, and an interval that is too wide can give a poor estimate of the rate at a point.

Common Misconceptions

• Average rate of change is not the average of the two output values. It is the change in output divided by the change in input.
• A rate of change does not have to be constant. Only linear functions have the same average rate over every interval; for most functions the rate varies from interval to interval.
• A zero rate of change over an interval does not mean the function never moved. It means the output ended where it started, even if it rose and fell in between.
• A negative rate of change does not mean the function is "smaller" or unimportant. It just means the output decreases as the input increases.
• Approximating a rate at a point requires small intervals around that point. A wide interval can hide how the function behaves close to the point.

Related AP Precalculus Guides

• 1.9 Rational Functions and Vertical Asymptotes
• 1.11 Equivalent Representations of Polynomial and Rational Expressions
• 1.5 Polynomial Functions and Complex Zeros
• 1.8 Rational Functions and Zeros
• 1.12 Transformations of Functions
• 1.4 Polynomial Functions and Rates of Change