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Graphs of Linear Functions

Graph f(x) = ax +b

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Graphs of Linear Functions

Let’s Think About It

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
License: CC BY-NC 3.0

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.

\begin{align*}x\end{align*} Combos \begin{align*}y\end{align*} 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.


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 \begin{align*}x\end{align*} value and the \begin{align*}y\end{align*} value.

Take a look at the following table.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
0 2
1 4
2 6
3 8

In this table, the letters \begin{align*}x\end{align*} and \begin{align*}y\end{align*} are used instead of input and output. They mean the same thing, but in mathematics as you work with functions, you will use \begin{align*}x\end{align*} and \begin{align*}y\end{align*} more often.

Here, \begin{align*}x\end{align*} is the input value and \begin{align*}y\end{align*} is the output value. The \begin{align*}y\end{align*} value depends on the \begin{align*}x\end{align*} value. You can see that each value of the \begin{align*}x\end{align*} column matches with only one value of the \begin{align*}y\end{align*} column. This means that this table forms a function.

Here is another table.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
1 5
1 7
3 9
4 13

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

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

Felix has a job cutting grass in the summer time. He earns $10.00 per lawn that he cuts. The amount of money that Felix makes is related to the number of lawns that he cuts. Let’s look at some data about Felix and then show how this forms a function.

Felix cut the following lawns on four different days. 

\begin{align*}\text{Day } 1 = 1 \text{ lawn} = \$10.00\end{align*}

\begin{align*}\text{Day } 2 = 2 \text{ lawns} = \$20.00\end{align*} 

\begin{align*}\text{Day } 3 = 3 \text{ lawns} = \$30.00\end{align*}

\begin{align*}\text{Day } 4 = 4 \text{ lawns} = \$40.00\end{align*}

This data can be organized in a table with the number of lawns represented by the \begin{align*}x\end{align*} value and the amount of money earned represented by the \begin{align*}y\end{align*} value. The \begin{align*}x\end{align*} is the value that can be counted on or depended on and the \begin{align*}y\end{align*} value changes depending on the \begin{align*}x\end{align*} value.

This is represented in the following table.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
1 $10
2 $20
3 $30
4 $40

You can say that the amount of money that Felix earns is a function of the number of lawns that he mows.

You can also graph functions on the coordinate grid by using the values in each column to form ordered pairs.

Notice that in an ordered pair you have an \begin{align*}x\end{align*} value and a \begin{align*}y\end{align*} value. Let’s write this data as ordered pairs. 

\begin{align*}(1, 10)\end{align*}

\begin{align*}(2, 20)\end{align*}

\begin{align*}(3, 30)\end{align*}

\begin{align*}(4, 40)\end{align*}

Now you can graph the data.

Create a graph by plotting the \begin{align*}x\end{align*} values (the number of lawns mowed) on the \begin{align*}x\end{align*} axis and the \begin{align*}y\end{align*} values (the amount of money earned) on the \begin{align*}y\end{align*} axis.

License: CC BY-NC 3.0

This graph forms a linear function. Anytime a graph forms a line like this, it is called a graph of a linear function. If the line rises from left to right, the graph of the function increases. If the line slopes downward from left to right, the graph of the linear function decreases.

Guided Practice

On a coordinate plane, graph the linear function that is represented by the ordered pairs in the table below.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
-4 5
-2 3
0 1
2 -1
4 -3

First, identify the ordered pairs by pairing the \begin{align*}x\end{align*} value with its \begin{align*}y\end{align*} value.

The ordered pairs shown in the table are \begin{align*}(-4, 5), (-2, 3), (0, 1), (2, -1)\end{align*} and \begin{align*}(4, -3)\end{align*}.

Next, plot those five points on the coordinate plane by locating the \begin{align*}x\end{align*}-coordinate then going up or down to the \begin{align*}y\end{align*}-coordinate.

Then connect them as shown below.

License: CC BY-NC 3.0

Notice that the graph of this function is a straight line that decreases from left to right. That is because this function is a linear function.


Answer the following questions about graphs and linear functions.

Example 1

Does this graph show an increase or a decrease?

License: CC BY-NC 3.0

First, look at the line representing this function. The line rises as it goes from left to right. Therefore, the function increases.

The answer is the function increases.

Example 2

In the graph in Example 1, is the amount of money earned represented on the \begin{align*}x\end{align*} axis or the \begin{align*}y\end{align*} axis?

First, look at the graph. The money earned is in increments of $5 \begin{align*}(5, 10, 15, 20, 25)\end{align*}. This data is on the \begin{align*}y\end{align*} axis.

The answer is the \begin{align*}y\end{align*} axis.

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 \begin{align*}x\end{align*} axis.

Next, locate the largest \begin{align*}x\end{align*}-coordinate.


The answer is 5 wind chimes.

Follow Up

Credit: Luca Nebuloni
Source: https://www.flickr.com/photos/nebulux/9055162205/in/photolist-eNb5fi-6WzHSk-5gxP2U-3WRUZ-8JCshJ-rAzXiK-7GUZDQ-yEtg-dqHqdq-3AfYWu-axx5EZ-2JMpsc-7omaGv-6YVagV-74s3L-2WUtH-bZmfGQ-8JCrWN-aDCg5o-8vnnwH-k5XpjT-nv27LV-8nEWAA-mXuusQ-diZoKC-4C8Nkz-bi8Nx6-dCpu4i-noGeK5-MH5VW-8nEWFb-mXswrH-bTb6Ug-aB51V-aBfoBr-7NveWo-gxPkS-4EuEqv-axZPN5-8nEWA9-8oms99-fm8aLD-7mpNpT-bqLy9p-aRgdme-mucPKK-9Bu2ky-fEsuZ5-nTygrv-ntfcTg
License: CC BY-NC 3.0

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.

\begin{align*}x\end{align*} Combos \begin{align*}y\end{align*} 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.

\begin{align*}(1, 12), (2, 24), (3, 36), (4, 48), (5, 60)\end{align*}

Next, plot those five points on the coordinate plane by locating the \begin{align*}x\end{align*}-coordinate then going up or down to the \begin{align*}y\end{align*}-coordinate. Then connect them as shown below.

Here is a graph representing the data from the table.

License: CC BY-NC 3.0

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

Video Review


Explore More

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


Input Output
1 4
2 5
3 6
4 7


Input Output
2 4
3 6
4 8
5 10


Input Output
1 3
2 6
4 12
5 15


Input Output
9 7
7 5
5 3
3 1


Input Output
8 12
9 13
11 15
20 24


Input Output
3 21
4 28
6 42
8 56


Input Output
2 5
3 7
4 9
5 11


Input Output
4 7
5 9
6 11
8 15


Input Output
5 14
6 17
7 20
8 23


Input Output
4 16
5 20
6 24
8 32

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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.

Linear Function

A linear function is a relation between two variables that produces a straight line when graphed.

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