Summary of Linear Functions

Linear Functions

Identifying Linear Functions

• A linear function's graph is a nonvertical line.
• It can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Determining Linearity

• Graph: A straight, nonvertical line indicates a linear function.
• Equation: Can be written in the form $y = mx + b$.
• Table: Check if there is a constant rate of change (constant difference in $y$ for every constant difference in $x$).

Writing Linear Functions

• From a Graph/Table: Find the slope and y-intercept to write the equation in the form $y = mx + b$.

Applications

• Linear functions can model real-world situations, such as growth or earnings.
• Compare linear functions (equations, tables, or graphs) to determine rates of change or initial values.

Function Basics

• Function: A relation where each input has only one output.
• Domain: The set of input values ($x$).
• Range: The set of output values ($y$).

Function Rules

• A function rule describes the relationship between input and output.
• Example: "The output is one-third of the input" translates to $y = \frac{1}{3}x$.