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Graphs of Linear Functions

Graph f(x) = ax +b

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Practice Graphs of Linear Functions
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Graphs of Linear Functions

The students have been having a fantastic time at the amusement park! Here is some information about tickets.

If the booklet had 6 tickets, then you can go on two rides. If the booklet had 12 tickets in it, then you could go on 4 rides. The number of rides is a function of the number of tickets. Using this chart can help the managers and the amusement park design new booklets of tickets.

\begin{align*}x\end{align*} Rides \begin{align*}y\end{align*} Tickets
1 3
2 6
3 9
4 12
7 21

We can create a visual display of this data. How do we do this?

This is where 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. In this Concept, you will learn about graphs. We will come back to this problem at the end of the Concept.

Guidance

Did you know that you learned about functions in an earlier Concept? We actually didn’t really call them “functions,” but we called them input/output tables. Let’s look at what it means for the data in an input/output table to be a function.

What is a function?

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.

Let’s look at a table to look at this.

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

Do you see something different in this table?

In this table we use the letters \begin{align*}x\end{align*} and \begin{align*}y\end{align*} 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. We can get used to seeing them in our work here.

Here the \begin{align*}x\end{align*} is the input value and the \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. They go together. 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

Do you see something different here?

The \begin{align*}x\end{align*} value of 1 is connected two different "y" values at the same time. This is NOT a function. Because at least one input has multiple outputs (1 corresponds to both 5 and 7), this is not a function.

What does it mean when real life data forms a function?

It means that one variable depends on or is a function of the other variable in the data.

Felix has a job cutting grass in the summer time. He earns $10.00 per lawn that he cuts. This is an example of a function. The amount of money that Felix makes is related to the number of lawns that he cuts. If Felix cuts 10 lawns, then he will make$100.00. The amount of money is a function of the number of lawns.

We can look at some data about Felix and then show how this forms a function.

Felix cut the following lawns on four different days.

Day 1 = 1 lawn = $10.00 Day 2 = 2 lawns =$20.00

Day 3 = 3 lawns = $30.00 Day 4 = 4 lawns =$40.00

How can we organize this data in a table?

Well, the number of lawns would be the \begin{align*}x\end{align*} value and the amount of money earned would be 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.

Here is our table.

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

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

We can also graph functions on the coordinate grid. We do this by using the values in each column to form our ordered pairs.

Notice that we have an \begin{align*}x\end{align*} value and a \begin{align*}y\end{align*} value. In an ordered pair we have an \begin{align*}x\end{align*} value and a \begin{align*}y\end{align*} value.

Let’s write this data as ordered pairs.

(1, 10)

(2, 20)

(3, 30)

(4, 40)

Now we can graph our data.

We create a graph by plotting the \begin{align*}x\end{align*} values (the number of dollars earned) on the \begin{align*}x\end{align*} axis and the \begin{align*}y\end{align*} values (the number of lawns) on the \begin{align*}y\end{align*} axis.

Wow! This graph forms a line!

Yes it does. This graph forms what we call a linear function . Anytime that a graph forms a line like this will it is called a linear graph-and a linear graph is a graph of a linear function.

Example A

Does the graph about cutting lawns show an increase or a decrease?

Solution: Increase

Example B

Is the amount of money earned represented on the \begin{align*}x\end{align*} axis or the \begin{align*}y\end{align*} axis?

Solution: \begin{align*}x\end{align*} axis

Example C

What is the greatest number of lawns shown on the graph?

Solution: 5 lawns

Now back to the dilemma about tickets at the amusement park. Here is the original problem once again.

The students have been having a fantastic time at the amusement park! Here is some information about tickets.

If the booklet had 6 tickets, then you can go on two rides. If the booklet had 12 tickets in it, then you could go on 4 rides. The number of rides is a function of the number of tickets. Using this chart can help the managers and the amusement park design new booklets of tickets.

\begin{align*}x\end{align*} Rides \begin{align*}y\end{align*} Tickets
1 3
2 6
3 9
4 12
7 21

We can create a visual display of this data. How do we do this?

Here is a graph representing the data from the table.

Notice that this is a linear graph showing the relationship between rides and tickets.

Guided Practice

Here is one for you to try on your own.

Let’s look at a table of values and see how we can represent the function on a coordinate grid.

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, let's identify the ordered pairs. The ordered pairs shown in the table are (–4, 5), (–2, 3), (0, 1), (2, –1) and (4, –3).

Plot those five points on the coordinate plane. Then connect them as shown below.

Notice that the graph of this function is a straight line. That is because this function is a linear function.

Video Review

Here is a video for review.

Explore More

Directions : On your own graph each function in the coordinate plane. Identify which tables 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

Vocabulary Language: English

Function

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 $y$.
Linear Function

Linear Function

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