Section 1.3: Linear Functions, Slope, and Applications

Objectives

• Determine the slope of a line given two points on the line.

• Solve applied problems involving slope or average rate of change.

• Find the slope and the y-intercept of a line given the equation or .

• Graph a linear equation using the slope and the y-intercept.

• Solve applied problems involving linear functions.

Linear Functions

Definition and Forms

A linear function is a function that can be written in the form , where m and b are constants.

• If and , the function is the identity function: .

• If , the function is a constant function: .

Examples:

• General linear function:

• Identity function:

• Constant function:

Horizontal and Vertical Lines

• Horizontal lines have equations of the form . These are functions, and their graphs are parallel to the x-axis.

• Vertical lines have equations of the form . These are not functions, as they fail the vertical line test.

Slope of a Line

Definition

The slope of a line passing through points and is given by:

• Formula:

• This represents the rise over run, or the change in divided by the change in .

Types of Slopes

• Positive slope: Line rises from left to right ().

• Negative slope: Line falls from left to right ().

• Zero slope: Horizontal line ().

• Undefined slope: Vertical line (division by zero; not a function).

Examples

• Given , to find the slope, identify .

• For points and , .

Special Cases: Horizontal and Vertical Lines

• Horizontal line: ; slope .

• Vertical line: ; slope is undefined.

Applications of Slope

Grade of a Road

• The grade of a road is the slope expressed as a percentage.

• Formula:

• Example: A 4% grade means the road rises 4 ft for every 100 ft of horizontal distance.

Average Rate of Change

• The average rate of change of a function between two points is the slope of the line connecting those points.

• Formula:

• Example: If the number of participants in a program increases from 17.2 million in 2000 to 47.6 million in 2013, the average rate of change is million per year.

Slope-Intercept Form of a Line

Definition

• The slope-intercept form of a linear equation is .

• m is the slope, and b is the y-intercept (the point where the line crosses the y-axis).

Finding Slope and Y-Intercept

• Given , slope , y-intercept is .

• Given , solve for to get ; slope , y-intercept is .

Graphing Linear Equations

• Plot the y-intercept .

• Use the slope to find another point.

• Draw a straight line through the points.

Applications of Linear Functions

Estimating Adult Height

• For a male child: , where is the sum of the parents' heights in inches.

• For a female child: .

• Example: If parents' combined height is 135 in, in (5 ft 5 in).

Summary Table: Types of Linear Equations