Identifying Linear & Nonlinear Functions Using Graphs & Tables

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  • 0:04 Linear & Nonlinear Functions
  • 0:58 Using Graphs for…
  • 1:50 Using Tables for…
  • 3:19 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.

All functions are either linear or nonlinear. In this lesson, we will learn how to identify linear and nonlinear functions using graphs and tables. We will use an example to illustrate this process.

Linear & Nonlinear Functions

Suppose two people, Fermat and Sophie, go out for a jog. Sophie is planning on ending her jog at a park, so she is getting further and further from her house as she jogs. Her distance from her house can be modeled by the function y = 4x, where x is the number of hours she has been jogging for. On the other hand, Fermat is planning on running an out-and-back course, starting and ending at his house. His distance from his house can be modeled by the function y = -x 2 + 4x.

Each of the functions described represents a different type of function. In mathematics, Sophie's function is called a linear function, and Fermat's function is called a nonlinear function.

A linear function is a function in which the rate at which y is changing with respect to x is constant. A nonlinear function is--like the name suggests--a function that is not linear. That is, the rate at which y is changing with respect to x varies in a nonlinear function.

Using Graphs for Identification

Believe it or not, we can determine whether a function is linear or nonlinear simply by looking at its graph! Because the rate at which y is changing with respect to x is constant in a linear function, the graph of a linear function is a line, as the name implies. For example, observe the graph of Sophie's function.


Linear Function
linnon1


We see that the graph of the function y = 4x is the graph of a line, so this is a linear function.

Since a linear function's graph is a line, can you guess what the graph of a nonlinear function looks like? Let's look at the facts! A nonlinear function is a function that is not linear, and a linear function's graph is a line. It makes sense, then, that the graph of a nonlinear function is not a line, as the name implies. We can see this by looking at the graph of Fermat's function described in our opening example.


Nonlinear Function
linnon2


It's pretty clear that the graph of the function y = -x 2 + 4x isn't a line, so the function is a nonlinear function.

Using Tables for Identification

Another way to identify linear and nonlinear functions is by viewing the function in tabular form. That is, we observe a table that displays the inputs and outputs of the function.

Again, because the rate at which y is changing with respect to x is constant in linear functions, it follows that the change in y for each unit of change in x is the same. For example, in a linear function, if our input goes up by 1, our output changes by a value of n, where n is constant. Therefore, in a table representing a linear function, when the inputs are evenly spaced out, the jumps in the outputs are constant. To clarify this, let's look at Sophie's function represented in a table.


Linear Function
linnon3


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