# Nonlinear Function: Definition & Examples

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• 0:00 What are Linear Functions?
• 1:53 Nonlinear Functions
• 3:40 Linear or Nonlinear?
• 4:39 Example
• 5:22 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will familiarize ourselves with linear functions in order to define and understand what nonlinear functions are. We will become comfortable determining if a function is linear or nonlinear through definitions and examples.

## What are Linear Functions?

The easiest way to define a nonlinear function is as a function that is not a linear function. Therefore, in order to understand what a nonlinear function is, it is essential to understand what a linear function is. Graphically, a linear function is a function whose graph is a line. Algebraically, a linear function can be defined as a polynomial with highest exponent equal to 1 or a horizontal line (y = c where c is a constant).

Linear functions can also be described as a function with a constant slope (rate of change of y with respect to x). In other words, the slope of the line between any two points of the function is the same.

For example, y = 2x + 3 is a linear function. Notice it is a polynomial with highest exponent equal to 1. Also, if we consider some random points that satisfy the equation, say (-1, 1), (0, 3), and (7, 17), we see that the slope of the line between any two pairs of these is the same.

• (-1, 1) and (0, 3): Slope: (3 - 1) / (0 - (-1)) = 2 / 1 = 2
• (0, 3) and (7, 17): Slope: (17 - 3) / (7 - 0) = 14 / 7 = 2
• (-1, 1) and (7, 17): Slope: (17 - 1) / (7 - (-1)) = 16 / 8 = 2

The slope of the line between any two of these points is 2, and this is true for any two points that satisfy the equation y = 2x + 3. Thus, the slope of the function is constant. The graph of y = 2x + 3 is shown below, and we see that the graph is a graph of a line.

## Nonlinear Functions

Now that we understand what a linear function is, let's define a nonlinear function. As we stated earlier, nonlinear functions are functions that are not linear functions. Therefore, they have the opposite properties of a linear function.

The graph of a linear function is a line. Thus, the graph of a nonlinear function is not a line. Linear functions have a constant slope, so nonlinear functions have a slope that varies between points. Algebraically, linear functions are polynomials with highest exponent equal to 1 or of the form y = c where c is constant. Nonlinear functions are all other functions.

An example of a nonlinear function is y = x^2. This is nonlinear because, although it is a polynomial, its highest exponent is 2, not 1. Also, if we consider some random points that satisfy the equation, say (-3, 9), (-1, 1), and (4, 16), we see that when we calculate the slope of the line between these points, we get different results.

(-3, 9) and (-1, 1): Slope: (1 - 9) / (-1 - (-3)) = -8 / 2 = -4

(-3, 9) and (4, 16): Slope: ((16 - 9) / (4 - (-3)) = 7 / 7 = 1

The slope of the line between different points that satisfy the function is different for different points considered, so the slope varies. Lastly, when we look at the graph of y = x^2, shown below, it is obvious that this is not the graph of line. Therefore, we see that this is not a linear function. It is a nonlinear function.

## Linear or Nonlinear

Based on all this information, if we want to determine if a function is a nonlinear function, we can do this in a few different ways.

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