Rate of Change

Understanding rate of change is one of the most important foundations in Precalculus because it leads directly into Calculus, where the derivative measures instantaneous rate of change.
In Precalculus, we focus on average rate of change, interpreting change in graphs, and connecting rate to real-world situations.

What Is Rate of Change?

Rate of change describes how much one quantity changes in relation to another.

For functions, it measures how much the output (y) changes as the input (x) changes.

General formula:

Rate of change = $$\frac{\Delta{y}}{\Delta{x}}=\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}$$​

This is called the average rate of change over the interval $$[x_{1} , x_{2}]$$.

Rate of Change as Slope

For linear functions:

$$f(x)=mx+b$$

The rate of change is constant and equal to the slope $m$.

• Positive slope → function increases

• Negative slope → function decreases

• Zero slope → function is constant

• Steeper slope → changes faster

Diagram idea: rising line (positive), falling line (negative), flat line (zero).

Rate of Change in Nonlinear Functions

Nonlinear functions do not have a constant rate of change.

Example:

$$f(x)=x^{2}$$

Compute the rate of change from x=1 to x=3:

$$\frac{f(3)-f(1)}{3-1}=\frac{9-1}{2}=4$$

But from x=3 to x=5:

$$\frac{f(5)-f(3)}{5-3}=\frac{25-9}{2}=8$$

The rate doubled because $$x^{2}$$ gets steeper as x increases.

Key Idea:
For nonlinear functions, the rate of change depends on the interval.

Interpreting Rate of Change from a Graph

To estimate rate of change:

1. Choose two points on the graph

2. Find the rise (change in y)

3. Find the run (change in x)

4. Compute slope = rise/run

Increasing vs. Decreasing:

• Upward trend → positive rate

• Downward trend → negative rate

• Flat segment → rate = 0

This skill is essential for modeling real-world situations.

Rate of Change in Real-Life Contexts

A. Speed

If distance changes with time:

Rate of change = $$\frac{\Delta{distance}}{\Delta{time}}$$ = average speed

Example:
A car travels from 20 km to 80 km in 1 hour.

average speed = $$\frac{80-20}{1}$$ = 60km/h

B. Economics

If cost changes with production:

$$\frac{\Delta{cost}}{\Delta{items}}$$​

Represents marginal cost (approximate rate of cost increase).

C. Population Change

$$\frac{\Delta{population}}{\Delta{years}}$$​

Measures growth rate over time.

Average vs. Instantaneous Rate of Change

In Precalculus, we compute average rate of change.
But we introduce the idea of instantaneous rate, which is explored fully in Calculus.

Average (Precalculus):

$$\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}$$​

Instantaneous (Calculus):

Slope of the tangent line

$$\lim_{\Delta{x}\to{0}}\frac{f(x_{\Delta{x}})-f(x)}{\Delta{x}}$$​

Why this matters:
Precalculus prepares you for the idea that slopes of secant lines approximate the slope of the tangent line.

Example 1: Average Rate of Change

$$f(x)=3x^{2}-2x+1$$

Find rate from x=2 to x=5:

$$\frac{f(5)-f(2)}{5-2}=\frac{66-9}{3}=\frac{57}{3}=19$$

Example 2: Interpreting Rate

If a population increases from 12,000 to 13,800 in 3 years:

Rate=$$=\frac{13800-12000}{3}=\frac{1800}{3}$$=600people/year