Explainer: Linear and Nonlinear Functions

In this explainer, we will learn how to determine whether a function is linear or nonlinear.

A linear function is a special type of function whose graphs are straight lines (as their name suggests). Let’s look at a straight line and find out what this particular shape of graph tells us about the relationship between the input π‘₯ and the output 𝑦 of a linear function.

As shown in the diagram, for any two points lying on a straight line, the difference in their 𝑦-coordinates is proportional to the difference in their π‘₯-coordinates. In other words, the rate of change, which is defined as the ratio of these differences, is constant; it is the slope of the line. Note that it can be either positive or negative (or zero, when the line is horizontal, representing a constant function of the form 𝑦=𝑏).

Let’s see what a constant rate of change means algebraically for the relationship between π‘₯ and 𝑦. Let’s assume that point 𝐴 of coordinates (π‘₯,𝑦) lies on a given line. What do we know about any other point 𝑀(π‘₯,𝑦) lying on the same line?

We know that the rate of change, which is the ratio of the difference in the 𝑦-coordinates between 𝑀 and 𝐴 to the difference in their π‘₯-coordinates (i.e., π‘¦βˆ’π‘¦π‘‹βˆ’π‘₯), is constant, so let’s write π‘¦βˆ’π‘¦π‘‹βˆ’π‘₯=π‘š.

Multiplying both sides of the equation by (π‘₯βˆ’π‘₯), we get π‘¦βˆ’π‘¦=π‘š(π‘₯βˆ’π‘₯).

Adding π‘¦οŒ  to each side then gives us 𝑦=π‘š(π‘₯βˆ’π‘₯)+𝑦.

We can expand the brackets and find 𝑦=π‘šπ‘₯βˆ’π‘šπ‘₯+𝑦.

As π‘š, π‘₯, and π‘¦οŒ  are constants, we can replace βˆ’π‘šπ‘₯+π‘¦οŒ οŒ  with a constant 𝑏. Also, this equation is true for any point of the line. Therefore, we can write that the π‘₯- and 𝑦-coordinates of any point on the line verify the equation 𝑦=π‘šπ‘₯+𝑏, where 𝑏 is a constant.

This is called the equation of a line.

Let’s summarize what we have just learned.

Characteristics of a Linear Function

The graph of a linear function is a straight line.

A linear function has a constant rate of change, which means that the difference in the 𝑦-coordinates of any two points on the straight line representing the linear function is proportional to the difference in their π‘₯-coordinates.

The rate of change is the slope of the line.

The equation of a line is generally written in the form 𝑦=π‘šπ‘₯+𝑏, where π‘š is the slope of the line and 𝑏 is a constant.

Example 1: Determining Whether a Function Is Linear or Nonlinear from a Table of Values

Determine whether the table below represents a linear or nonlinear function.

π‘₯36912
𝑦32251811

Answer

If the values of π‘₯ and 𝑦 given above represent a linear function, then the rate of change must be constant. We observe that, between any two adjacent columns in the table, π‘₯ increases by 3 (from left to right). So, if the function is linear, then the change in the 𝑦-value from one column to the next, to the right, must be always the same.

We observe that it decreases by 7 from the first to the second column. The same is true from the second to the third column and from the third to the fourth. Hence, we conclude that the table represents a linear function whose rate of change is βˆ’73.

Example 2: Determining Whether a Function Is Linear or Nonlinear from a Table of Values

Determine whether the given table of values must be from a nonlinear function, or could be from a linear function.

π‘₯-value0246
𝑦-value13919

Answer

If the given function is linear, then the rate of change must be constant.

We observe that, between any two adjacent columns in the table, the π‘₯-value increases by 2 (from left to right). So, if the function is linear, then the change in the 𝑦-value from one column to the next, to the right, must always be the same.

Between the first and second columns, the 𝑦-value increases by 3βˆ’1=2, while from the second column to the third, it increases by 9βˆ’3=6. Hence, this table must be from a nonlinear function.

Example 3: Determining Whether a Function Is Linear or Nonlinear from Its Equation

Which of the following equations represents a nonlinear function?

  1. 𝑦=5(π‘₯βˆ’3)
  2. 9π‘₯𝑦=4
  3. 𝑦=π‘₯2
  4. 𝑦=5π‘₯+6

Answer

We need to recognize from the equations of four different functions which of them is nonlinear.

We know that the equation of a linear function is generally written in the form 𝑦=π‘šπ‘₯+𝑏. The first equation given (A) is 𝑦=5(π‘₯βˆ’3). By expanding the brackets, we get 𝑦=5π‘₯βˆ’15. We recognize the form 𝑦=π‘šπ‘₯+𝑏 with π‘š=5 and 𝑏=βˆ’15.

The second equation (B) is 9π‘₯𝑦=4. By dividing both sides of the equation by 9π‘₯, we get 𝑦=49π‘₯. Here, we do not recognize the form 𝑦=π‘šπ‘₯+𝑏. Hence, this function is nonlinear.

We can quickly check that the other two functions are indeed linear. The third one (C) is 𝑦=π‘₯2. We could rewrite it as 𝑦=12π‘₯, seeing that π‘š=12 and 𝑏=0. The fourth one (D) is 𝑦=5π‘₯+6. This is clearly the equation of a linear function, with π‘š=5 and 𝑏=6.

The equation that represents a nonlinear function is 9π‘₯𝑦=4 (B).

Example 4: Determining Whether a Function Is Linear or Nonlinear from Its Graph

Determine whether the given figure represents a linear or nonlinear function.

Answer

Here, we are given the graph of a function and we need to decide whether this is the graph of a linear or nonlinear function.

We know that the graph of a linear function is a straight line. The graph given is not a straight line. Hence, it represents a nonlinear function.

Example 5: Determining Whether a Function Is Linear or Nonlinear from Its Graph

Does the shown figure represent a linear function?

Answer

The graph of a linear function is a straight line. Although this graph is made of two sections, which are straight lines, the whole graph is not a straight-line graph. Therefore, the function is not linear. Additionally, we could consider the rate of change for π‘₯>0. Notice that as π‘₯ changes by 1, 𝑦 also changes by 1. This rate of change is constant and equal to 1 for all positive values of π‘₯. On the other hand, for π‘₯<0, we have that as π‘₯ increases by 1, 𝑦 decreases by 1.

Hence, the rate of change for negative π‘₯ is βˆ’1. This confirms that the function does not have a constant rate of change for all values of π‘₯. This demonstrates once again that the function is not linear.

Key Points

  1. The graph of a linear function is a straight line.
  2. A linear function has a constant rate of change, which means that the difference in the y-coordinates of any two points on the straight line representing the linear function is proportional to the difference in their x-coordinates.
  3. The rate of change is the slope of the line.
  4. The equation of a line is generally written in the form 𝑦=π‘šπ‘₯+𝑏, where π‘š is the slope of the line and 𝑏 is a constant.

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