Identifying Linear vs. Nonlinear Functions from a Table
Identifying Linear vs. Nonlinear Functions from a Table is a Grade 7 math skill in Big Ideas Math Advanced 2, Chapter 6: Functions. Students learn to determine linearity by calculating the rate of change (delta y / delta x) between consecutive rows in a table — a constant rate indicates a linear function, while a changing rate indicates a nonlinear function.
Key Concepts
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!
Common Questions
How do you tell if a function is linear or nonlinear from a table?
Calculate the rate of change (change in y divided by change in x) between every pair of consecutive rows. If the rate is always the same, the function is linear. If the rate changes at any point, the function is nonlinear.
What is a common mistake when checking tables for linearity?
Only looking at how much the y-values change without dividing by the change in x. You must always compute delta y divided by delta x for every step.
Give an example of a linear table.
x values (0, 1, 2, 3), y values (2, 5, 8, 11). The rate of change is always 3/1 = 3, so this is a linear function.
Give an example of a nonlinear table.
x values (0, 1, 2, 3), y values (0, 1, 4, 9). The rates of change are 1, 3, and 5 — they change, so this represents a nonlinear (curved) function.