Types of Functions and Their Rates of Change

Linear Functions

Linear functions are fundamental in algebra and are characterized by a constant rate of change. They are represented by the equation , where m is the slope and b is the y-intercept.

• Definition: A linear function is any function of the form .

• Graph: The graph of a linear function is always a straight line.

• Example: A car traveling north at 60 miles per hour, starting 30 miles north of the Texas border, is modeled by .

Rate of Change and Slope

The rate of change in a linear function is constant and is equal to the slope of the line. The slope measures how much the function's output changes for each unit increase in the input.

• Formula: The slope between two points and is given by:

• Interpretation: If , the function increases; if , it decreases; if , the function is constant.

Types of Slope

The slope of a line can be positive, negative, zero, or undefined, each with distinct graphical characteristics.

• Positive Slope: The line rises from left to right. Example: .

• Negative Slope: The line falls from left to right. Example: .

• Zero Slope: The line is horizontal.

• Undefined Slope: The line is vertical.

Calculating Slope from Two Points

To find the slope of a line given two points, use the slope formula. The sign and value of the slope indicate the direction and steepness of the line.

• Example: Find the slope between and :

• Interpretation: The slope means the line falls $5 units increase in .

Zero of a Function

A zero of a function is a value such that . This is where the graph crosses the x-axis.

• Definition: Any number for which is called a zero of the function .

Four Representations of a Linear Function

Linear functions can be represented in four ways: verbal, symbolic, numerical, and graphical. Understanding all forms is essential for interpreting and solving algebraic problems.

• Verbal: Describes the relationship in words.

• Symbolic: Uses equations, e.g., .

• Numerical: Shows input-output pairs in a table.

• Graphical: Plots the function on a coordinate plane.

Characteristics of Linear Functions

Linear functions have several defining properties:

• Graph is a straight line.

• Can be written as .

• Has a constant rate of change, equal to .

• Has exactly one zero if .

Nonlinear Functions

Definition and Characteristics

Nonlinear functions do not have a constant rate of change and their graphs are not straight lines. They can have any number of zeros and cannot be written in the form .

• Graph is not a straight line.

• Rate of change is not constant.

• Can have multiple zeros.

Common Nonlinear Functions

Examples of nonlinear functions include quadratic, square root, cubic, and absolute value functions.

• Square Function:

• Square Root Function:

• Cube Function:

• Absolute Value Function:

Increasing and Decreasing Functions

Definitions

A function is said to be increasing on an interval if its output rises as the input increases, and decreasing if its output falls as the input increases.

• Increasing: For , .

• Decreasing: For , .

Examples and Applications

Graphs can illustrate where functions are increasing or decreasing. Interval notation is used to specify these regions.

• Example: The function is decreasing on and increasing on .

Visualizing Increasing and Decreasing Functions

Graphs and real-world examples, such as sales data, can help visualize these concepts.

• Example: Rock music sales decreased from 1990 to 2001 and increased from 2001 to 2006.

• General Illustration: Graphs show uphill (increasing), downhill (decreasing), and mixed behavior.

Additional info: All explanations are expanded for clarity and completeness, suitable for college-level algebra study.