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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

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?

x (width in cm)23456y (length in cm)246810\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.

x-4-2024y531-1-3\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 {(4,5),(2,3),(0,1),(2,1),(4,3)}\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.

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 y=2x1\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 x\begin{align*}x\end{align*}-values, some positive x\begin{align*}x\end{align*}-values and 0. For example, you can create a table to find the values of y\begin{align*}y\end{align*} when x\begin{align*}x\end{align*} is equal to -2, -1, 0, 1, and 2.

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

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

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?

x (width in cm)23456y (length in cm)246810\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 {(2,2),(3,4),(4,6),(5,8),(6,10)}\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 x\begin{align*}x\end{align*} increases by 1, y\begin{align*}y\end{align*} increases by 2. So, you know that 2x\begin{align*}2x\end{align*} is involved in the equation. But y\begin{align*}y\end{align*} is not quite 2x\begin{align*}2x\end{align*}. It is 2x2\begin{align*}2x -2\end{align*}.

So the equation that models this information is 2x2\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?

x1234y5101520\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 x\begin{align*}x\end{align*}-coordinate is the first value and the y\begin{align*}y\end{align*}-coordinate is the second value.

{(1,5),(2,10),(3,15),(4,20)}\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 x\begin{align*}x\end{align*}-value by 5 you get the y\begin{align*}y\end{align*}-value. The rule is multiply x\begin{align*}x\end{align*} by 5 to get y\begin{align*}y\end{align*}. You can write this as an equation.

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

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

#### Example 3

Is the function above increasing or decreasing?

Notice that as x\begin{align*}x\end{align*} increases y\begin{align*}y\end{align*} increases. Notice that every time you increase x\begin{align*}x\end{align*} by 1, y\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

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### Vocabulary Language: English

TermDefinition
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 $m$