Nonlinear function
Also known as: non-linear function
A nonlinear function is any function whose graph is not a straight line — it cannot be written in the form y = mx + b. Its rate of change varies, producing curves such as parabolas, exponential growth curves, and hyperbolas.
A nonlinear function is any function that is not linear — its graph is not a straight line, and it cannot be written in the form y = mx + b. Where a linear function changes by the same amount for every equal step in x, a nonlinear function's rate of change varies: the graph bends, curves, accelerates, or levels off.
The most familiar examples are quadratics like y = x², whose graph is a U-shaped parabola; exponential functions like y = 2ˣ, which grow slowly and then explode upward; absolute value functions like y = |x|, which form a V; square root, cubic, and reciprocal functions like y = 1/x, whose graph is a hyperbola. Any equation where x appears squared, cubed, under a radical, in an exponent, or in a denominator produces a nonlinear function.
You can spot nonlinearity in a table of values, too. In a linear table, equal steps in x always produce equal steps in y. If x increases by 1 while y jumps by 1, then 3, then 5, the differences aren't constant — the function is nonlinear. This first-differences test is a fast check on test day, and it works even when no equation is given.
The ACT math section leans on this skill directly: questions ask you to classify functions from equations, tables, or graphs, and to match nonlinear equations — parabolas, exponentials, absolute value — to their curves. Achievable's ACT course covers nonlinear functions and graphs as part of coordinate geometry.
Key takeaways
- A nonlinear function's graph is not a straight line, and its equation cannot be written as y = mx + b.
- Common examples include quadratic (y = x²), exponential (y = 2ˣ), absolute value, cubic, square root, and reciprocal functions.
- Nonlinear functions have a changing rate of change — equal steps in x do not produce equal steps in y.
- In a table, constant first differences mean linear; changing differences mean nonlinear.