Dennis, the scorekeeper for the boys’ basketball team, has been keeping track of the team’s scoring during the weekend tournament. During the first two games, Dennis tracked the team’s shooting performance from different distances from the goal. He found that the team made approximately 50 percent of their shots when they were less than 10 feet from the goal. He also recorded their shooting averages from within 15, 20, and 25 feet of the goal. He displayed his results in the table below.

\begin{align*}y\end{align*} | |

10 | 50 |

15 | 40 |

20 | 30 |

25 | 20 |

Dennis wants to use this data to show that the team’s shooting average \begin{align*}(y)\end{align*} is dependent on the shooter’s distance from the goal .

In this concept, you will learn to distinguish between linear and non-linear functions by displaying these functions on a graph.

### Distinguishing Between Linear and Non-Linear Functions

A **linear function** is a systematic or sequential increase or decrease represented by a straight line.

Take a look at this table.

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

0 | 2 |

1 | 4 |

2 | 6 |

3 | 8 |

Notice that as the \begin{align*}x\end{align*} value increases the dependent or \begin{align*}y\end{align*} value increases in a sequential way or by 2 each time. This graph will form a straight line. To graph the equation, first arrange the \begin{align*}x\end{align*} and values into ordered pairs.

(0, 2) (1, 4) (2, 6) (3, 8)

Then, plot the points on the graph to see if they connect in a straight line.

Here is the graph of this function.

Now, let’s take a look at the other type of function.

A **non-linear function** is a function where the data does not increase or decrease in a systematic or sequential way.

In short, a non-linear function does not form a straight line when it is graphed.

Let’s look at a non-linear function in a table.

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

1 | 3 |

2 | 5 |

3 | 4 |

4 | 9 |

In this table, the data does not move in a sequential way. The

value increases (3 to 5), decreases (5 to 4) then increases again (4 to 9). This graph will not form a straight line.Let’s graph this function to be sure. Here is the graph of a non-linear function.

You could connect these points, but it does not change the fact that this is a non-linear function.

### Examples

#### Example 1

Earlier, you were given a problem about the basketball team’s shooting percentage.

Dennis wants to present his results to the coach and team. Here is the table with Dennis’s results.

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

10 | 50 |

15 | 40 |

20 | 30 |

25 | 20 |

How can Dennis use this information to determine if the relationship between shooting distance and shooting average is a linear or non-linear function?

First, look at the chart and determine whether the dependent

value increases or decreases sequentially as the value increases or decreases.Yes, the

value decreases by 10 whenever the value increases by 5. This is in a sequence.The answer is this pattern follows a linear rule. It will form a straight line when graphed. It is a linear function.

#### Example 2

Identify whether the table represents a linear or a non-linear function.

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

2 | 4 |

4 | 6 |

6 | 8 |

10 | 12 |

First, look at the chart and determine whether the dependent \begin{align*}x\end{align*} value increases or decreases.

value increases or decreases sequentially as theYes, the \begin{align*}x\end{align*} value increases. This is in a sequence.

value increases by 2 whenever theThe answer is this is a linear function. It will form a straight line when graphed.

#### Example 3

Identify whether the graph represents a linear or a non-linear function.

First, look at the points on the graph.

Next, ask yourself “Can all the points be connected by one straight line?”

No, so this is a non-linear function.

The answer is this is a non-linear function.

#### Example 4

Identify whether the table represents a linear or a non-linear function.

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

1 | 10 |

2 | 8 |

3 | 6 |

4 | 4 |

First, look at the chart and determine whether the dependent \begin{align*}y\end{align*} value increases or decreases sequentially as the \begin{align*}x\end{align*} value increases or decreases.

Yes, the \begin{align*}x\end{align*} value increases by 1. This is in a sequence.

value decreases by 2 whenever theThe answer is this pattern follows a linear rule. It will form a straight line when graphed. It is a linear function.

#### Example 5

Identify whether the table represents a linear or a non-linear function.

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

1 | 8 |

2 | 6 |

3 | 4 |

4 | 2 |

First, look at the chart and determine whether the dependent \begin{align*}y\end{align*} value increases or decreases sequentially as the value increases or decreases.

Yes, the \begin{align*}x\end{align*} value increases. This is in a sequence.

value decreases by 2 whenever theThe answer is this is a linear function.

### Review

Look at each table and determine whether the function is linear or non-linear.

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

0 | 2 |

1 | 3 |

2 | 5 |

4 | 4 |

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

1 | 3 |

2 | 5 |

3 | 7 |

4 | 9 |

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

2 | 6 |

3 | 9 |

5 | 15 |

6 | 18 |

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

2 | 3 |

3 | 4 |

6 | 7 |

8 | 9 |

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

8 | 4 |

6 | 12 |

2 | 8 |

0 | 0 |

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

0 | 3 |

1 | 4 |

2 | 5 |

6 | 9 |

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

5 | 11 |

4 | 9 |

3 | 7 |

2 | 5 |

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

1 | 7 |

3 | 4 |

2 | 9 |

5 | 8 |

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

1 | 3 |

2 | 6 |

4 | 12 |

6 | 18 |

\begin{align*}x\end{align*} |
\begin{align*}y\end{align*} |
---|---|

4 | 2 |

5 | 3 |

6 | 5 |

7 | 1 |

Now use each table in 1 – 10 and graph each function. You should have 10 graphs for this section. Number these graphs 11 – 20. If the graph is a linear graph, then please connect the points with a line.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.13.

### Resources