What is a Linear Function? - Definition & Examples

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  • 0:05 Definition of a Linear…
  • 0:24 Identifying Linear Functions
  • 1:32 Working with Linear Functions
  • 3:34 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you will learn the key identifying mark of linear functions. You'll also learn how to graph them with just two points and how you can use the graph to easily find your answers.

Definition of a Linear Function

A linear function is any function that graphs to a straight line. What this means mathematically is that the function has either one or two variables with no exponents or powers. If the function has more variables, the variables must be constants or known variables for the function to remain a linear function.

Identifying Linear Functions

To identify linear functions, you can create a checklist of several items the function must meet.

  1. The first item the function must satisfy is that it must have either one or two real variables. If another variable is present, it must be a known variable or constant. For example, the function C = 2 * pi * r is a linear function because only the C and r are real variables, with the pi being a constant.
  2. The second item is that none of the variables can have an exponent or power to them. They cannot be squared, cubed, or anything else. All variables must be in the numerator.
  3. The third item is that the function must graph to a straight line. Any kind of a curve disqualifies the function.

So, linear functions all have some kind of straight line when graphed. The line could be going up and down, left and right, or slanted but the line is always straight. It doesn't matter where on the graph the function is plotted as long as the line comes out straight.

Example of a linear function

Working with Linear Functions

If you know that a function is linear, you can plot the graph using just two points. If you are unsure, you can use three or four points to double check.

To figure out your points to plot them, set up a T-chart and start plugging in values for one of the variables. Plug the values into the function to calculate the other variable and note it on the T-chart. When you have filled the T-chart, go ahead and plot the points on the graph. Then, take a ruler and make a straight line through them. All linear functions will have points that are lined up nicely.

Let's try graphing y = 2x.

First, we fill up a T-chart. I'm going to do four points so you can see how the points line up and how, if you know that the function is linear, just two points are sufficient.

x y
0 0
1 2
2 4
3 6

Now, that I've filled up my T-chart, I can go ahead and plot them. Let's see what we get.

Plotting points from the T-chart

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