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Graphs of Linear Equations

Graph lines presented in ax+by = c form

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Graphs of Linear Equations
Credit: CK-12, Larame Spence
Source: Desmos Graphing Calculator
License: CC BY-NC 3.0

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?

\begin{align*}\begin{array}{|c|c|} \hline x \text{ (width in cm)} & y \text{ (length in cm)} \\\hline 2 & 2 \\\hline 3 & 4 \\\hline 4 & 6 \\\hline 5 & 8 \\\hline 6 & 10 \\\hline \end{array}\end{align*} 
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.

\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline \text{-}4 & 5 \\\hline \text{-}2 & 3 \\\hline 0 & 1 \\\hline 2 & \text{-}1 \\\hline 4 & \text{-}3 \\\hline \end{array}\end{align*} 

You can represent the information in this table as a set of ordered pairs \begin{align*}\{(-4,5),(-2,3),(0,1),(2,-1),(4,-3)\}\end{align*}.

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

License: CC BY-NC 3.0

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 \begin{align*}y = 2x-1\end{align*} 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 \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.

 \begin{align*}y=2x-1\end{align*}
 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
 \begin{align*}-2\end{align*}  \begin{align*}-5\end{align*}  \begin{align*}2(-2)-1=-5\end{align*}
 \begin{align*}-1\end{align*}  \begin{align*}-3\end{align*}  \begin{align*}2(-1)-1=-3\end{align*}
   \begin{align*}-1\end{align*}  \begin{align*}2(0)-1=-1\end{align*}
 \begin{align*}1\end{align*}  \begin{align*}1\end{align*}  \begin{align*}2(1)-1=1\end{align*}
 \begin{align*}2\end{align*}  \begin{align*}3\end{align*}  \begin{align*}2(2)-1=3\end{align*}

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

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

License: CC BY-NC 3.0

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?

\begin{align*}\begin{array}{|c|c|} \hline x \text{ (width in cm)}& y \text{ (length in cm)} \\ \hline 2 & 2 \\ \hline 3 & 4 \\ \hline 4 & 6 \\ \hline 5 & 8 \\ \hline 6 & 10 \\ \hline \end{array}\end{align*} 
First, represent this information as a set of ordered pairs so that you can plot the points \begin{align*}\{(2,2),(3,4),(4,6),(5,8),(6,10)\}\end{align*}.

Now, can you see a pattern in the table and then write the rule that describes it?

Notice that as \begin{align*}x\end{align*} increases by 1, \begin{align*}y\end{align*} increases by 2. So, you know that \begin{align*}2x\end{align*} is involved in the equation. But \begin{align*}y\end{align*} is not quite \begin{align*}2x\end{align*}. It is \begin{align*}2x -2\end{align*}.

So the equation that models this information is \begin{align*}2x -2\end{align*}.

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?

\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 1 & 5 \\\hline 2 & 10 \\\hline 3 & 15 \\\hline 4 & 20 \\\hline \end{array}\end{align*} 

You can extract information from the table and represent the same information as a set of ordered pairs. The \begin{align*}x\end{align*}-coordinate is the first value and the \begin{align*}y\end{align*}-coordinate is the second value.

\begin{align*}\{(1,5),(2,10),(3,15),(4,20)\}\end{align*}

Next, looking at the information in the table, you can see that when you multiply the \begin{align*}x\end{align*}-value by 5 you get the \begin{align*}y\end{align*}-value. The rule is multiply \begin{align*}x\end{align*} by 5 to get \begin{align*}y\end{align*}. You can write this as an equation.

\begin{align*}y=5x\end{align*}

You can graph plot the coordinates \begin{align*}\{(1,5),(2,10),(3,15),(4,20)\}\end{align*} and draw a line through them to see the graph.

Answer the following questions about functions and coordinates.

Example 3

License: CC BY-NC 3.0

Is the function above increasing or decreasing?

Notice that as \begin{align*}x\end{align*} increases \begin{align*}y\end{align*} increases. Notice that every time you increase \begin{align*}x\end{align*} by 1, \begin{align*}y\end{align*} will always increase. In this case, \begin{align*}y\end{align*} increases by two every time \begin{align*}x\end{align*} increases by 1.

The answer is the function is increasing.

Example 4

In the point \begin{align*}(-3, 4)\end{align*} is the \begin{align*}x\end{align*}-value positive or negative?

The \begin{align*}x\end{align*}-value is the first value in the coordinate. It is a negative number.

The answer is the \begin{align*}x\end{align*}-value is negative.

Example 5

In \begin{align*}(-6, -7)\end{align*}, which value is \begin{align*}y\end{align*}-value?

The \begin{align*}y\end{align*}-value is the second value in a coordinate, and it is equal to -7.

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

  1.   
Input Output
1 4
2 5
3 6
4 7
  1.   
Input  Output
2 4
3 6
4 8
5 10
  1.   
Input Output
1 3
2 6
4 12
5 15
  1.   
Input Output
9 7
7 5
5 3
3 1
  1.   
Input Output
8 12
9 13
11 15
20 24
  1.   
Input  Output
3 21

4

28
6 42
8 56
  1.   
Input Output
2 5
3 7
4 9
5 11
  1.  
Input Output
4 7
5 9
6 11
8 15
  1.   
Input Output
5 14
6 17
7 20
8 23
  1.   
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.

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Vocabulary

Cartesian Plane

The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.

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.

Function Rule

A function rule describes how to convert an input value (x) into an output value (y) for a given function. An example of a function rule is f(x) = x^2 + 3.

Linear Function

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

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the y over the change in the x.” The symbol for slope is m

Image Attributions

  1. [1]^ Credit: CK-12, Larame Spence; Source: Desmos Graphing Calculator; License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0

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