Welcome to 1.2: Rates of Change!

In our last lesson (1.1 Change in Tandem), we looked at how two variables change together. Now, we are going to get specific. Instead of just saying "as x goes up, y goes up," we want to know exactly how much y changes for every unit of x. This is what we call the Average Rate of Change.

Whether you are tracking the speed of a car, the growth of a bank account, or the spread of a viral video, you are using rates of change. Let’s dive in!

What is the Average Rate of Change?

The Average Rate of Change (AROC) of a function \(f\) over an interval \([a, b]\) is the ratio of the change in the output (\(y\)) to the change in the input (\(x\)).

If you remember "slope" from Algebra 1, you already know the basics! The formula looks like this:

The Formula:
\( \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \)

Think of it as: \( \frac{\text{Change in Output}}{\text{Change in Input}} \) or \( \frac{\Delta y}{\Delta x} \).

Wait, what does \([a, b]\) mean?

In AP Precalculus, we use interval notation. The interval \([a, b]\) simply means "all the x-values from a to b." To find the rate of change, we only care about the starting point (\(a\)) and the ending point (\(b\)).

Three Ways to See Rates of Change

The AP Exam will ask you to find the Average Rate of Change using different representations. Here is how to handle each one:

1. From a Table (Numerical)

Suppose you have a table showing the height of a plant over several weeks:

Weeks (x): 2, 5, 8
Height (y): 10, 19, 31

To find the rate of change from week 2 to week 8:
1. Identify your points: \( (2, 10) \) and \( (8, 31) \).
2. Apply the formula: \( \frac{31 - 10}{8 - 2} = \frac{21}{6} = 3.5 \).
3. The Takeaway: The plant grew at an average rate of \( 3.5 \) units per week.

2. From a Graph (Graphical)

On a graph, the Average Rate of Change is the slope of the secant line. A secant line is just a straight line that connects two points on a curve.

Visual Trick: If the secant line is tilted upward, the rate of change is positive. If it’s tilted downward, the rate of change is negative.

3. From an Equation (Analytical)

If you are given \( f(x) = x^2 + 3 \) and asked for the rate of change on \([1, 4]\):
1. Find \( f(4) \): \( (4)^2 + 3 = 19 \).
2. Find \( f(1) \): \( (1)^2 + 3 = 4 \).
3. Calculate: \( \frac{19 - 4}{4 - 1} = \frac{15}{3} = 5 \).

Applying Units in Context

In the Free Response Questions (specifically Question 2), the AP Exam will ask you to interpret the rate of change in a "real-world" scenario. You must include units to get full credit!

The Golden Rule for Units:
The units of a rate of change are always (Output Units) per (Input Unit).

Example: If \( f(t) \) represents the amount of water in a tank (gallons) and \( t \) represents time (minutes), the rate of change is measured in gallons per minute.

Quick Review: If the question asks for the "meaning" of your answer, you could say: "On the interval from \( t=a \) to \( t=b \), the [Output Variable] changed by an average of [Your Answer] [Units] per [Input Unit]."

Common Pitfalls to Avoid

Don't worry if this seems tricky at first! Here are the most common mistakes students make:

  • Mixing up the order: If you use \( f(b) - f(a) \) on top, you must use \( b - a \) on the bottom. Don't swap them!
  • Forgetting the negative sign: If the function value decreases, your rate of change must be negative.
  • Input vs. Output: Always remember that the interval \([a, b]\) refers to the \(x\)-values (inputs), not the \(y\)-values.

Key Takeaways for Topic 1.2

Summary Checklist:

  • Average Rate of Change = \( \frac{f(b) - f(a)}{b - a} \).
  • It represents the slope of the secant line between two points.
  • Always include units in context: (Output Unit) / (Input Unit).
  • It can be found using tables, graphs, or equations.

Up next: In Topic 1.3, we will see how these rates of change behave specifically in linear and quadratic functions. (Hint: In linear functions, the rate of change is always the same!)