Linear vs. Nonlinear Functions

In a table: equal changes

If the $x$-values go up by $1$ each row, look at the changes in $y$. All equal — linear. Not all equal — nonlinear.

In the table below, $x$ climbs by $1$ each row and $y$ climbs by the same $4$ every time. That constant change is what makes the function linear.

If the $x$-values skip around, do not compare raw $y$-changes. Divide each change in $y$ by the matching change in $x$ — the rate between two rows is $\displaystyle \frac{y_2 - y_1}{x_2 - x_1}$ — and check whether that ratio stays constant.

In an equation: does it fit y = mx + b?

A function is linear exactly when its equation can be rearranged into $y = mx + b$. Watch for anything that breaks that form: an exponent on $x$ (as in $y = x^2 - 3$), an $x$ in a denominator (as in $y = \dfrac{6}{x}$), or an $x$ in an exponent (as in $y = 2^x$).

The written form can hide the problem. $y = x(x + 5)$ has no visible exponent, but distributing gives $y = x^2 + 5x$ — nonlinear. Simplify or rearrange before you judge.

On a graph: straight or bent

A constant rate of change draws a straight line, so on a graph the answer is visible at a glance: a straight line at any tilt is linear, and any bend at all means nonlinear.

Below, the black line is linear — it climbs by the same amount over every step. The blue curve is nonlinear: it is nearly flat at the bottom and much steeper at the sides, so its rate of change is not constant. Be careful not to confuse increasing with linear — a graph can rise the whole time and still be a curve.

Worked examples

Try one yourself

Common questions