Linear Functions

Definition

A linear function is a function that can be expressed in the form $f(x) = mx + b$, where $m$ is the slope (constant rate of change) and $b$ is the $y$-intercept. Its graph is a straight line.

• How to read: “The function f of x equals m x plus b.”
• Meaning: Output changes at a constant rate $m$; starts at height $b$ when $x=0$.

Why It Matters

Constant change is the simplest form of predictability. Linear functions are the “straight-line” models of growth and decay, providing the essential baseline against which all more complex, non-linear behaviors are measured.

Core Concepts

• Slope ($m$): The ratio of the change in $y$ to the change in $x$. $m = \frac{\Delta y}{\Delta x}$.

  • How to read: “The slope m equals delta y divided by delta x.”
  • Meaning / when to use: Rise over run—steepness of the line; positive climbs, negative falls, zero is horizontal.

• Constant Average Rate of Change: The defining characteristic of a linear function is that its average rate of change is the same between any two points.

• Directionality:

  • $m > 0$: The function is increasing.
  • $m < 0$: The function is decreasing.
  • $m = 0$: The function is constant (a horizontal line).

• Line of Best Fit: A linear model derived from data using regression, where the correlation coefficient $r$ indicates the strength and direction of the linear relationship.

Connected Concepts

• Slope
• Lines
• Linear Equations, Problem Solving in Mathematics
• Quadratic Functions
• Average Rate of Change and Secant Lines
• Linear Programming
• Map-Territory Relation