Lesson 5-5: Nonlinear Functions

Property

The core difference between these two types of functions lies in their rate of change:

• Linear Function: Has a constant rate of change (a steady slope). Its equation can always be written in the form $y = mx + b$, and its graph is a straight line.

• Nonlinear Function: Has a variable rate of change (the steepness keeps changing). Its equation cannot be written as $y = mx + b$, and its graph forms a curve.

Property

To determine linearity from a table, you must calculate the rate of change ($\frac{\Delta y}{\Delta x}$) between consecutive points.

• If this ratio simplifies to the exact same number everywhere, the function is linear.
• If the ratio changes, the function is nonlinear.

Examples

• Linear Table: x values are (0, 1, 2, 3), y values are (2, 5, 8, 11). The rate of change is 3/1 = 3 between every single point.
• Nonlinear Table: x values are (0, 1, 2, 3), y values are (0, 1, 4, 9). The rate of change goes from 1/1 to 3/1 to 5/1. Because the rate keeps changing, it is a curve.

Explanation

When checking a table, a common trap is only looking at how much the y-values jump. You must always divide the jump in 'y' by the jump in 'x' for every single step. If that final fraction stays exactly the same, you have a straight line!

Property

A function is globally linear if and only if it graphs as a single, continuous straight line with one constant rate of change.

If a graph is composed of multiple straight-line segments with different slopes, the overall function is nonlinear.