Dana is collecting information about caterpillars for science class. She’s comparing the lengths and widths of several caterpillars. Dana puts the data she has so far into a table. Dana is convinced there is a pattern. Can organize this information as a set of ordered pairs, graph it on a coordinate plane and write an equation that could model this?
In this concept, you will learn to graph linear functions on the coordinate plane.
Graphing Linear Functions
A linear function is a specific type of 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 can represent a function as a set of ordered pairs, through a table, and as an equation. You can also take the information in ordered pairs or in a table and represent a function as a graph.
Let’s look at an example.
The table of values below represents a function on a coordinate plane. On a coordinate plane, graph the linear function that is represented by the ordered pairs in the table below.
You can represent the information in this table as a set of ordered pairs
Plot those five points on the coordinate plane. Then, connect them as shown below.
Notice that the graph of this linear function is a straight line.
You can also graph a linear function if you are given an equation for that function. This will involve a few more steps. When you have an equation, you can use the equation to create a table. Then, plot several of the ordered pairs in the table and connect them with a line.
Here is another example.
The equation
is a linear function. Graph that function on a coordinate plane.First, use the equation to create a table and find several ordered pairs for the function. It is a good idea to use some negative


















The ordered pairs shown in the table are
and .Plot those five points on the coordinate plane. Then connect them as shown below.
Examples
Example 1
Earlier, you were given a problem about Dana’s project, which was comparing the lengths and widths of caterpillars.
She’s put the data collected so far in a table (shown below). Can you plot these points and write the equation that models this information?
First, represent this information as a set of ordered pairs so that you can plot the points .
Now, can you see a pattern in the table and then write the rule that describes it?
Notice that as
So the equation that models this information is
.Next, plot the points on the coordinate plane and draw a line through them. The graph is shown below.
Example 2
The table below represents inputs and outputs of a linear function. Can you represent this information as ordered pairs, figure out the equation for this function, and then graph the function?
You can extract information from the table and represent the same information as a set of ordered pairs. The
coordinate is the first value and the coordinate is the second value.Next, looking at the information in the table, you can see that when you multiply the
value by 5 you get the value. The rule is multiply by 5 to get . You can write this as an equation.You can graph plot the coordinates
and draw a line through them to see the graph.
Answer the following questions about functions and coordinates.
Example 3
Is the function above increasing or decreasing?
Notice that as
increases increases. Notice that every time you increase by 1, will always increase. In this case, increases by two every time increases by 1.The answer is the function is increasing.
Example 4
In the point
is the value positive or negative?The
value is the first value in the coordinate. It is a negative number.The answer is the
value is negative.Example 5
In
, which value is value?The
value is the second value in a coordinate, and it is equal to 7.The answer is the
value is 7.Review
The information in the table represents points from a linear function. Plot the points in the table on a coordinate plane, and then draw a straight line through them to graph each function. Then identify the rule (equation) for the function.
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 
Review (Answers)
To see the Review answers, open this PDF file and look for section 7.17.