Linear Functions, Slope, and Applications

Linear Functions

A linear function is a function that can be written in the form:

where m is the slope and b is the y-intercept. Special cases include:

• Identity Function: (where , )

• Constant Function: (where )

Linear functions graph as straight lines. The value of m determines the steepness and direction of the line, while b determines where the line crosses the y-axis.

Horizontal and Vertical Lines

• Horizontal lines: Equations of the form . These lines have a slope of 0 and are not functions if written as .

• Vertical lines: Equations of the form . These lines have an undefined slope and are not functions.

Slope of a Line

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

The slope measures the rate of change of with respect to .

• If , the line rises from left to right.

• If , the line falls from left to right.

• If , the line is horizontal.

• If the denominator is zero, the line is vertical (undefined slope).

Average Rate of Change

The slope can also be interpreted as the average rate of change between two points on a function:

• Average rate of change =

This concept is useful for analyzing how a quantity changes over an interval.

The Slope-Intercept Equation

The slope-intercept form of a line is:

Where:

• m is the slope

• b is the y-intercept (the value of when )

This form is useful for quickly graphing lines and identifying their properties.

Examples

• Find the slope and y-intercept: For , the slope is and the y-intercept is .

• Find the slope and y-intercept: For , first solve for to get . The slope is and the y-intercept is .

• Graph: To graph , plot the y-intercept at and use the slope to find another point.

Applications

• Cost Problems: For example, a cable company charges a C(t)tC(t) = 12.50t + 40$.

• Business Applications: Fixed and variable costs can be modeled with linear functions, such as , where is the variable cost per unit and is the fixed cost.

Practice Problems

• Find the slope of the line containing the points and .

• Find the slope of the line containing the points and .

• Determine the slope and y-intercept of the graph of .

• Find the total cost for 18 months of service for a plan with a $40 per month.

Additional info: These notes cover foundational concepts in linear functions, including their equations, graphical representations, and applications in real-world contexts such as cost analysis.