Marc and Kara have been having a terrific time at the amusement park. For them it has been a wonderful way to end their month with their Grandparents.
In the Function Rules for Input-Output Tables Concept, we used tables. Well, we can use this data to create a visual display of the information. This information is not necessary for Marc and Kara to have a great time at the amusement park, but let’s say that you were someone who worked at the amusement park and you were trying to figure out how to organize packs of tickets for people to purchase. If you wanted to create a group of tickets to be purchased then you could figure out how many rides someone could go on for the number of tickets in the booklet.
If the booklet had 6 tickets, then the person could go on two rides. If the booklet had 12 tickets in it, then the person could go on 4 rides. You can see how 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
Remember what a function is?
A function is a set of ordered pairs where one element in the domain is paired with exactly one element in the range. There is a relationship identified with a function rule between the values in the domain and the values in the range.
In this Concept, we will begin by focusing on a specific type of function called a linear function. You may notice that the word “line” is part of the word “linear”. That fact can help you remember that when a linear function is graphed on a coordinate plane, its graph will be a straight line.
You have already learned how to represent functions through a set of ordered pairs and through a table. We can also take the information in ordered pairs or in a table and represent a function in a graph.
How do we graph a linear function?
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 |
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.
You can also graph a linear function if you are given an equation that represents the function. This will involve a few more steps. When you have an equation, you can use the equation to create a function table. Then plot several of the ordered pairs in the table and connect them with a line.
The equation \begin{align*}y=2x-1\end{align*} represents a linear function. Graph that function on a coordinate plane.
First, use the equation to create a function table and find several ordered pairs for the function. It is a good idea to use some negative \begin{align*}x-\end{align*}values, some positive \begin{align*}x-\end{align*}values and 0. For example, you can create a table to find the values of \begin{align*}y\end{align*} when \begin{align*}x\end{align*} is equal to –2, –1, 0, 1, and 2. You may need to use what you know about computing with integers to help you find those \begin{align*}y-\end{align*}values.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} | |
---|---|---|
-2 | -5 | \begin{align*}\leftarrow 2x-1=2(-2)-1=-4-1=-4+(-1)=-5\end{align*} |
-1 | -3 | \begin{align*}\leftarrow 2x-1=2(-1)-1=-2-1=-2+(-1)=-3\end{align*} |
0 | -1 | \begin{align*}\leftarrow 2x-1=2(0)-1=0-1=0+(-1)=-1\end{align*} |
1 | 1 | \begin{align*}\leftarrow 2x-1=2(1)-1=2-1=1\end{align*} |
2 | 3 | \begin{align*}\leftarrow 2x-1=2(2)-1=4-1=3\end{align*} |
The ordered pairs shown in the table are (–2, –5), (–1, –3), (0, –1), (1, 1) and (2, 3). This is the first way to show a function by using a table.
Plot those five points on the coordinate plane. Then connect them as shown below.
Answer the following questions about functions.
Example A
Is the graph above a positive graph or negative?
Solution: Positive
Example B
In (-3, 4) is the \begin{align*}x\end{align*} value positive or negative?
Solution: Negative
Example C
In (-6, -7), which value is \begin{align*}y\end{align*} value?
Solution: \begin{align*}-7\end{align*}
Here is the original problem once again. Reread it and look at the graph at the end of it. Notice how the data from the table is represented in a visual way.
Marc and Kara have been having a terrific time at the amusement park. For them it has been a wonderful way to end their month with their Grandparents.
Think about the table from the last section. We can use this data to create a visual display of the information. This information is not necessary for Marc and Kara to have a great time at the amusement park, but let’s say that you were someone who worked at the amusement park and you were trying to figure out how to organize packs of tickets for people to purchase. If you wanted to create a group of tickets to be purchased then you could figure out how many rides someone could go on for the number of tickets in the booklet.
If the booklet had 6 tickets, then the person could go on two rides. If the booklet had 12 tickets in it, then the person could go on 4 rides. You can see how 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.
Vocabulary
- Function
- A pattern where one element of from the domain is paired with exactly one element from the range.
- Function Rule
- the pattern rule for a function.
- Linear Function
- a function that forms a straight line when graphed
Guided Practice
Here is one for you to try on your own.
Identify the ordered pairs in this function.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 5 |
2 | 10 |
3 | 15 |
4 | 20 |
Is it a linear function? Why or why not?
Answer
This is a linear function because the same rule applies to each input - output.
The rule of the function is \begin{align*}y = 5x\end{align*}.
Video Review
This is a James Sousa video on how to graph a linear function in the coordinate plane.
Practice
Directions: On your own graph each function in the coordinate plane. Identify each rule. There are two answers for each problem.
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 |