Question

What is the difference between a linear function and a nonlinear function? Explain what each looks like when represented as a table and as a graph.

Answer 1

Short answer
- A linear function has a constant rate of change and its graph is a straight line.
- A nonlinear function does not have a constant rate of change and its graph is not a straight line (it curves or has other shapes).

What each looks like in a table
- Linear: when x-values are equally spaced, the changes in y (first differences) are the same each step.
Example: y = 2x + 1
x: 0, 1, 2, 3
y: 1, 3, 5, 7
First differences: 2, 2, 2 (constant)
- Nonlinear: the first differences are not constant (they change). For some types (quadratic) the second differences may be constant.
Example (quadratic) y = x^2
x: 0, 1, 2, 3
y: 0, 1, 4, 9
First differences: 1, 3, 5 (not constant)
Second differences: 2, 2 (constant for a quadratic)

What each looks like on a graph
- Linear: a straight line. It can slope up (positive slope), slope down (negative slope), or be horizontal (slope 0). The line’s slope gives the constant rate of change; the y-intercept is where it crosses the y-axis.
- Nonlinear: not a straight line — could be a parabola, exponential curve, circle, etc. The steepness (rate of change) varies from point to point; the shape shows that the relationship between x and y is changing.

How to tell them apart quickly
- From a table with equally spaced x: check whether the first differences of y are constant. If yes → linear. If no → nonlinear.
- From a graph: see whether all plotted points lie on a single straight line. If yes → linear. If no → nonlinear.