Introduction to Nonlinear Functions

A linear function has a constant rate of change, meaning its graph is always a perfectly straight line. However, not all relationships in math are linear. A nonlinear function is a function where the rate of change varies, meaning its graph is not a straight line.

Let's explore how to identify nonlinear functions using equations, graphs, and tables.

Identifying from Equations

You can often spot a nonlinear function just by looking at its equation. If the variable $x$ has an exponent other than $1$, is in the denominator, or is acting as an exponent itself, the function is nonlinear.

• Linear Example: $y = 2x + 3$ (The exponent of $x$ is $1$).
• Nonlinear Example 1: $y = x^2$ (This is a quadratic function. Because $x$ is squared, the rate of change is not constant).
• Nonlinear Example 2: $y = 2^x$ (This is an exponential function. The variable $x$ is the exponent, causing the value to grow faster and faster).

Identifying from Graphs

The visual test for a nonlinear function is very straightforward: if the graph is not a single straight line, it is nonlinear.

• The graph of $y = x^2$ forms a U-shape called a parabola.
• The graph of $y = 2^x$ starts off relatively flat and curves upwards very steeply.
• Because these graphs bend and curve, their slopes (rates of change) are constantly changing.

Identifying from Tables

To check if a table of values represents a nonlinear function, look at the first differences—the changes in the $y$-values when the $x$-values increase at a constant rate.

Let's test the function $y = x^2$:

$x$	$y$	Change in $y$
$1$	$1$	-
$2$	$4$	$4 - 1 = 3$
$3$	$9$	$9 - 4 = 5$
$4$	$16$	$16 - 9 = 7$

Since the $x$-values increase by $1$ each time, we check the change in the $y$-values. The differences are $3$, $5$, and $7$. Because these differences are not constant, the rate of change is varying, proving that the function is nonlinear.