Lesson 3: Compare Linear and Nonlinear Functions

Property

An equation of the form $y = mx + b$ describes a process: As the value of $x$ changes, the value of $y$ changes along with it. The slope is calculated as the quotient of the change in $y$ by the change in $x$ between any two points on the line:

so it is the rate of change of $y$ with respect to $x$, and the fact that the graph is a line tells us that the rate of change is constant. Since $b$ is the value of $y$ when $x = 0$, we also refer to $b$ as the initial value.

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 linear if its graph is a non-vertical straight line.
A function is nonlinear if its graph is not a straight line (e.g., it is a curve).

Examples