B4

Functions

Average Rate of Change

The average rate of change (AROC) of a function between two points measures how much the function's output changes per unit change in input. It is a fundamental concept in College Algebra, connecting the idea of slope for linear functions to more general functions.

• Definition: The average rate of change of a function $f(x)$ from $x=a$ to $x=b$ is given by:

$\text{AROC} = \frac{f(b) - f(a)}{b - a}$

• Interpretation: This formula calculates the slope of the secant line connecting the points $(a, f(a))$ and $(b, f(b))$ on the graph of the function.

• Units: The units of AROC are the units of $f(x)$ per unit of $x$ (e.g., meters per second).

Average Rate of Change for Linear Functions

For linear functions, the average rate of change is constant and equals the slope of the line.

• Example: For $y = -2x + 3$, the slope $m = -2$ is the average rate of change everywhere.

• Graphical Representation: The graph of a linear function is a straight line, and the rate of change does not vary.

Average Rate of Change for Nonlinear Functions

For nonlinear functions, the average rate of change varies depending on the interval chosen.

• Example: For $y = x^2$, the rate of change increases as $x$ increases.

• Graphical Representation: The graph is curved, and the slope of the secant line between two points changes depending on the interval.

Equations & Inequalities

Calculating Average Rate of Change: Step-by-Step

To find the average rate of change between two points $(x_1, f(x_1))$ and $(x_2, f(x_2))$:

1. Evaluate the function at both points: $f(x_1)$ and $f(x_2)$.

2. Subtract the outputs: $f(x_2) - f(x_1)$.

3. Subtract the inputs: $x_2 - x_1$.

4. Divide the difference in outputs by the difference in inputs:

$\text{AROC} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Examples

• Example 1: Given $f(x) = x^2 + 2x - 4$, find the average rate of change from $x=2$ to $x=4$.

$f(2) = 2^2 + 2(2) - 4 = 4 + 4 - 4 = 4$ $f(4) = 4^2 + 2(4) - 4 = 16 + 8 - 4 = 20$ $\text{AROC} = \frac{20 - 4}{4 - 2} = \frac{16}{2} = 8$

• Example 2: For $g(x)$ given in a table, calculate AROC between $x=2$ and $x=13$:

x	g(x)
2	32
13	267

$\text{AROC} = \frac{267 - 32}{13 - 2} = \frac{235}{11} \approx 21.36$

Interpreting Positive and Negative AROC

• Positive AROC: The function is increasing over the interval.

• Negative AROC: The function is decreasing over the interval.

• Zero AROC: The function does not change over the interval (horizontal segment).

Graphs of Equations

Secant Line and Slope

The secant line between two points on a graph represents the average rate of change over that interval.

• Secant Line: A straight line connecting two points on the curve.

• Slope of Secant: Equal to the average rate of change.

Graphical Examples

• On a parabola, the secant line between two points may be steeper or shallower depending on the interval.

• On a linear graph, all secant lines are parallel to the graph itself.

Applications

Physical Interpretation

Average rate of change is widely used in real-world contexts, such as velocity, growth rates, and economics.

• Example: A rocket's height $h(t)$ after $t$ seconds is given by $h(t) = 120t - 16t^2$. The average rate of change between $t=3$ and $t=4$ seconds is:

$h(3) = 120(3) - 16(3)^2 = 360 - 144 = 216$ $h(4) = 120(4) - 16(4)^2 = 480 - 256 = 224$ $\text{AROC} = \frac{224 - 216}{4 - 3} = \frac{8}{1} = 8$ This represents the average velocity between 3 and 4 seconds.

Tabular Data

Average rate of change can be calculated from tables of values.

x	f(x)
2	32
13	267
15.5	354
21	754

Example: Between $x=15.5$ and $x=21$: $\text{AROC} = \frac{754 - 354}{21 - 15.5} = \frac{400}{5.5} \approx 72.73$

Summary Table: Average Rate of Change

Function Type	AROC Formula	Graphical Meaning
Linear	$m = \frac{y_2 - y_1}{x_2 - x_1}$	Constant slope; same everywhere
Nonlinear	$\frac{f(b) - f(a)}{b - a}$	Slope of secant line; varies by interval
Tabular	$\frac{f(x_2) - f(x_1)}{x_2 - x_1}$	Difference in outputs divided by difference in inputs

Practice Problems

• Find the average rate of change for $f(x) = 2x + 1$ between $x=1$ and $x=4$.

• Given $f(x) = x^2 - 2x + 3$, find the AROC between $x=2$ and $x=5$.

• Given a table of values, calculate the AROC between two specified points.

• Interpret the meaning of positive, negative, and zero AROC in context.

Key Takeaways

• The average rate of change generalizes the concept of slope to all functions.

• It is calculated as the change in output divided by the change in input.

• Graphically, it is the slope of the secant line between two points.

• Applications include velocity, economics, and any context where change over time or input is measured.