How to Recognize Linear Functions vs Non-Linear Functions

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  • 0:01 Linear vs. Nonlinear
  • 0:20 Graphing Linear and…
  • 1:01 What Makes a Function Linear?
  • 2:39 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

In this lesson, you'll learn about two different groups of functions: linear and nonlinear. We'll cover how they look on a graph, and how you can tell them apart when they're written as equations.

Linear vs. Non-linear

You can divide up functions using all kinds of criteria:

Functions You Don

But some distinctions are more important than others, and one of those is the difference between linear and non-linear functions. In this lesson, you'll learn all about the two different types, how to tell them apart, and what they look like on a graph.

Graphing Linear and Non-linear Functions

We'll start with a graph because graphing makes it easiest to see the difference. The word 'linear' means something having to do with a line.

On a Cartesian Plane, a linear function is a function where the graph is a straight line. The line can go in any direction, but it's always a straight line.

A non-linear function has a shape that is not a straight line. Here are some examples:

Linear and Non-linear function examples

But why are some functions straight lines, while other functions aren't? Look at these graphs of some linear and non-linear functions. Can you guess what the important difference is that makes the functions on the left linear, and the functions on the right non-linear?

What Makes a Function Linear?

Did you guess the difference between linear and non-linear functions? The important difference is all in the exponents.

Image functions

Take a look at all these functions. The linear functions don't have any exponents higher than 1. Some of them don't have variables at all, and if they do, they have just plain x, which is equal to x to the first power.


When a linear function is written in its simplest form, it looks like y = a + bx, where a and b are both constants. For example, here are some linear functions:

y = 3 + 5x

y = 2 - 6x

(Note that - 6x is the same thing as + -6x)

y = 4x

(Note that this is equivalent to y = 0 + 4x)

y = 4

(Note that this equals y = 4 + 0x)

On the other hand, non-linear functions all have at least one variable raised to the power of two or more. There's no one formula for non-linear functions because they're all different. Here are some examples:

y = 3x^3 + x^2 -7

5 = y^2 + x^2

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