# Graphs of Linear Functions

## Graph f(x) = ax +b

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Graphs of Linear Functions

Credit: Jessica Spengler
Source: https://www.flickr.com/photos/wordridden/173689222/in/photolist-gmcKL-6M8NTT-crhk7U-67EARN-x5w4j-dNPG7R-bVikvg-4G2Ugn-axjBPK-4i6gqj-4xcmzR-axjCPk-qPuCFa-qwWpEA-pSHBX8-4CST6b-4Cw5KL-ne1yvC-4CSSTy-2413wT-4i6gHL-pSuzb3-7KXdd6-apQ4bm-5SnkWS-245qFA-7USx1T-4rczpR-9SwhjN-9ZHMod-6jeGFL-6YVagV-tDEfDB-9Bu2ky-6McY1J-6M8N5i-6McYmJ-6M8Nbi-6McYM1-6McZtY-6M8NhZ-6M8P7M-6McZnw-6McYdG-6M8Nck-6M8NLK-6M8Nvi-3kVdPf-673moZ-nB1KEB

It is Reggie’s first day on the job at The Taco Shop near his house. Reggie is looking over the menu when he notices that the restaurant now offers family or group combo orders of 12 tacos. If a customer orders 1 combo, he will get 12 tacos. If the customer orders 2 combos, he will get 24 tacos. Because his math class recently studied functions, Reggie realizes the number of tacos is a function of the number of combos ordered, as demonstrated below.

 Combos Tacos 1 12 2 24 3 36 4 48 5 60

How can this data can be displayed visually with a graph?

In this concept, you will learn how to graph linear functions.

### Guidance

A function is a set of data that has a specific relationship. One variable in the data set is related to or depends on a different variable in the same data set. Each input matches with only one output.

This is why graphing functions is important. A graph of a function can show the relationship between the  value and the  value.

Take a look at the following table.

 0 2 1 4 2 6 3 8

In this table, the letters  and  are used instead of input and output. They mean the same thing, but in mathematics as you work with functions, you will use  and  more often.

Here,  is the input value and  is the output value. The  value depends on the  value. You can see that each value of the  column matches with only one value of the  column. This means that this table forms a function.

Here is another table.

 1 5 1 7 3 9 4 13

In this table, the  value of 1 is connected to two different “ values (5 and 7) at the same time. This is NOT a function. Remember, to be a function each value of the  column can match with only ONE value of the  column.

Let’s look at functions in a real life situation.

#### Example 3

According to the graph in Example 1, what is the largest number of wind chimes sold?

First, locate “wind chimes sold” on the graph. It is represented on the  axis.

Next, locate the largest -coordinate.

5

The answer is 5 wind chimes.

Credit: Luca Nebuloni

Remember Reggie and his first day at The Taco Shop?

A customer walks up to the counter and asks Reggie how many tacos come in the family combo. Reggie tells the customer that each combo contains 12 tacos. The customer tells Reggie that he needs to order enough tacos for 11 people. He tells Reggie that 5 tacos should be enough for each person, which means he needs to get at least 55 tacos. Reggie does the math in his head and figures out that 5 combo packs will be 60 tacos because the number of tacos is a function of the number of combos the man orders. Reggie walks over to the kitchen and tells the cooks to get busy.

The following chart represents this function.

 Combos Tacos 1 12 2 24 3 36 4 48 5 60

How can Reggie display this function and data on a graph?

To graph this linear function, first identify the ordered pairs.

Next, plot those five points on the coordinate plane by locating the -coordinate then going up or down to the -coordinate. Then connect them as shown below.

Here is a graph representing the data from the table.

Notice that this is a linear graph showing the relationship between number of combo packs and the number of tacos.

### Explore More

Graph each function represented by the data in the table then determine which represent linear graphs and which ones do not.

1.

 Input Output 1 4 2 5 3 6 4 7

2.

 Input Output 2 4 3 6 4 8 5 10

3.

 Input Output 1 3 2 6 4 12 5 15

4.

 Input Output 9 7 7 5 5 3 3 1

5.

 Input Output 8 12 9 13 11 15 20 24

6.

 Input Output 3 21 4 28 6 42 8 56

7.

 Input Output 2 5 3 7 4 9 5 11

8.

 Input Output 4 7 5 9 6 11 8 15

9.

 Input Output 5 14 6 17 7 20 8 23

10.

 Input Output 4 16 5 20 6 24 8 32

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Function A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.