Linear Functions

Definition and General Form

A linear function is a function that can be written in the form , where m and b are constants. The graph of a linear function is always a straight line.

• Example: or

• Non-example: is not a function because it cannot be written as

Application Example: A car initially located 30 miles north of the Texas border, traveling north at 60 miles per hour, is represented by .

• Interpretation: means the car is 30 miles north at time zero.

• The graph of this function is increasing, indicating a constant rate of travel.

Constant Rate of Change and Slope

In a linear function, the slope is constant. This means the value of always changes by an equal amount for each unit increase in .

• Slope Formula: The slope of the line passing through points and is given by:

• Interpretation: The slope indicates how much changes for each unit increase in .

Types of Slope

• Positive Slope: Line rises as increases.

• Negative Slope: Line falls as increases.

• Zero Slope: Line is horizontal.

Four Representations of a Function

Functions can be represented in four main ways:

• Verbal: Describes the relationship in words.

• Symbolic: Uses an equation, e.g., .

• Numerical: Table of values showing input-output pairs.

• Graphical: Graph of the function on the coordinate plane.

Zeros and Intercepts of a Function

• Zero of a Function: A value such that . This is the x-intercept of the graph.

• Y-intercept: The value of , where the graph crosses the y-axis.

Example: For :

• Y-intercept:

• Zero:

Nonlinear Functions

Definition and Characteristics

A nonlinear function is any function that is not linear. These functions do not have a constant rate of change, and their graphs are not straight lines.

• No constant rate of change

• Graph is not a line

• Can increase and decrease over different intervals

• No single slope value

Examples:

Increasing and Decreasing Functions

Definitions

Let be defined on an interval :

• Increasing on : If implies for all in .

• Decreasing on : If implies for all in .

Example: Rock music sales decreased from 36% to 24% between 1990 and 2001 (decreasing function), then increased from 24% to 34% between 2001 and 2006 (increasing function).

Average Rate of Change

Definition and Calculation

For nonlinear functions, the average rate of change over an interval is calculated using the slope formula:

• This measures how much the function changes on average per unit increase in over the interval.

Example: For , the average rate of change from to is:

Difference Quotient

Definition

The difference quotient is a formula that expresses the average rate of change of a function over an interval of length :

, where

• This is foundational for calculus and measures the average rate of change as approaches zero.

Example: For :

• Difference quotient:

Applications and Group Work

Interpreting Slope and Intervals of Increase/Decrease

• Example 1: Advertising spending increased from $95 billion in 2005. The slope is billion per year.

• Example 2: For the function graphed below, identify intervals where the function is increasing or decreasing.

• Example 3: For , . The slope is , meaning the tank loses 5 gallons per minute.

• Example 4: If and , the average rate of change from 1 to 4 is .

• Example 5: For , the difference quotient is .