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# Graphs of Linear Equations

## Graph lines presented in ax+by = c form

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Using Tables to Graph Functions

The students of the local high school are selling potted plants to raise funds for their soccer team to buy new uniforms. Each six-pack of potted plants will sell for 6.50. A new sports store has agreed to match the money raised by the students as a donation to the team. The money raised by the students will be displayed on a poster-size Cartesian graph and presented to the sports store. How can the students create such a graph? In this concept, you will learn to use tables to graph functions. ### Graphing Functions Consider the following Cartesian graph that represents the equation y=3x+4\begin{align*}y=3x+4\end{align*}. License: CC BY-NC 3.0 The equation y=3x+4\begin{align*}y=3x+4\end{align*} is written in function form and can be used to create a table of values that will make the statement of equality true. Remember an equation written in function form can be used to determine values for the output ‘y\begin{align*}y\end{align*}’ based on the different input ‘x\begin{align*}x\end{align*}’ values substituted into the equation. Using x\begin{align*}x\end{align*}-values of -4, -2, 0, 2 and 4, create a table of values to represent the equation y=3x+4\begin{align*}y=3x+4\end{align*}.  x\begin{align*}x\end{align*} y\begin{align*}y\end{align*} -4 -8 -2 -2 0 4 2 10 4 16 xyyyy=====43x+43(4)+4Substitute x=4 into the equation.12+4 Perform the multiplication to clear the parenthesis.8Simplify the right side of the equation.\begin{align*} \begin{array}{rcl} x&=&-4\\ y&=&3x+4\\ y&=&3(-4)+4 \qquad \text{Substitute } x=-4 \text{ into the equation}. \\ y&=&-12+4 \qquad \ \ \ \text{Perform the multiplication to clear the parenthesis}. \\ y&=&-8 \qquad \qquad \quad \text{Simplify the right side of the equation}. \end{array}\end{align*} Use this process to calculate the values of the output variable for each of the given input values. Given input value xyyyThe output value is y=====23x+43(2)+46+42\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&-2\\ y&=&3x+4\\ y&=&3(-2)+4\\ y&=&-6+4\\ \text{The output value is }y&=&-2 \end{array}\end{align*} Given input value xyyyThe output value is y=====03x+43(0)+40+44\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&0\\ y&=&3x+4\\ y&=&3(0)+4\\ y&=&0+4\\ \text{The output value is }y&=&4 \end{array}\end{align*} Given input value xyyyThe output value is y=====23x+43(2)+46+410\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&2\\ y&=&3x+4\\ y&=&3(2)+4\\ y&=&6+4\\ \text{The output value is }y&=&10 \end{array}\end{align*} Given input value xyyyThe output value is y=====43x+43(4)+412+416\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&4\\ y&=&3x+4\\ y&=&3(4)+4\\ y&=&12+4\\ \text{The output value is }y&=&16 \end{array}\end{align*} The input value associated with the corresponding output value can be written as an ordered pair (x,y)\begin{align*}(x,y)\end{align*} such that (4,8),(2,2),(0,8),(2,10) and (4,16)\begin{align*}(-4,-8), (-2,-2), (0,8), (2,10) \text{ and } (4,16)\end{align*} are the ordered pairs that can be plotted to represent the equation y=3x+4\begin{align*}y=3x+4\end{align*}. The ordered pairs are plotted on the Cartesian graph and are shown as red points. These points were then joined by a smooth straight line to draw the graph. The graph is a straight line such that the equation that produced this line was a linear function. The highest exponent of the variables of a linear function is one. There are two special linear functions that produce a straight line graph. One of the straight lines is a vertical line that is parallel to the y\begin{align*}y\end{align*}-axis and the other is a horizontal line that is parallel to the x\begin{align*}x\end{align*}-axis. Let’s graph each of these special lines. A line having x=5\begin{align*}x=5\end{align*} as its equation will pass through the point (5,0)\begin{align*}(5,0)\end{align*} such that it will be parallel to the y\begin{align*}y\end{align*}-axis. License: CC BY-NC 3.0 A line having y=4\begin{align*}y=4\end{align*} as its equation will pass through the point (0,4)\begin{align*}(0,4)\end{align*} such that it will be parallel to the x\begin{align*}x\end{align*}-axis. License: CC BY-NC 3.0 ### Examples #### Example 1 Earlier, you were given a problem about the plotted plants and the soccer uniforms. The students need to create a poster size graph to show how much money was raised. How can the students create this graph? They can create a table of values and plot the ordered pairs from the table. First, write an equation in function form to represent the sale of potted plants. Let y\begin{align*}y\end{align*} represent the money raises and let x\begin{align*}x\end{align*} represent the number of potted plants sold. y=6.50x\begin{align*}y=6.50x\end{align*} Next, create a table of values and use the equation expressed in function form to calculate the output value for each input value.  x\begin{align*}x\end{align*} y\begin{align*}y\end{align*} 50 100 150 200 250 xyyy====506.50x6.50(50)Substitute x=50 into the equation.325.00 Perform the multiplication to clear the parenthesis.\begin{align*} \begin{array}{rcl} x&=&50\\ y&=&6.50x\\ y&=&6.50(50) \qquad \text{Substitute } x=50 \text{ into the equation}. \\ y&=&\325.00 \qquad \ \text{Perform the multiplication to clear the parenthesis}. \end{array}\end{align*}

Repeat the same process for the remaining input values.

Given input value xyyThe output value is y====1006.50x6.50(100)650.00\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&100\\ y&=&6.50x\\ y&=&6.50(100)\\ \text{The output value is }y&=&\650.00 \end{array}\end{align*} Given input value xyyThe output value is y====1506.50x6.50(150)975.00\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&150\\ y&=&6.50x\\ y&=&6.50(150)\\ \text{The output value is }y&=&\975.00 \end{array}\end{align*}

Given input value xyyThe output value is y====2006.50x6.50(200)1300.00\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&200\\ y&=&6.50x\\ y&=&6.50(200)\\ \text{The output value is }y&=&\1300.00 \end{array}\end{align*} Given input value xyyThe output value is y====2506.50x6.50(250)1625.00\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&250\\ y&=&6.50x\\ y&=&6.50(250)\\ \text{The output value is }y&=&\1625.00 \end{array}\end{align*}

Next, write the calculated ‘y\begin{align*}y\end{align*}’ values in the table.

 x\begin{align*}x\end{align*} y\begin{align*}y\end{align*} 50 325 100 650 150 975 200 1300 250 1625

Then, plot the ordered pairs shown in the table on a Cartesian grid.

The sports store will have to match the \$1,625.00 raised by the students.

#### Example 2

For the following linear function written in function form, complete the table of values and plot the graph.

y=3x4\begin{align*}y=3x-4\end{align*}

 x\begin{align*}x\end{align*} y\begin{align*}y\end{align*} -3 -13 -1 -7 1 -1 3 5 5 11

First, use the equation to calculate the output values ‘y\begin{align*}y\end{align*}.’

xyyyy=====33x43(3)4Substitute x=3 into the equation.94 Perform the multiplication to clear the parenthesis.13  Simplify the right side of the equation.\begin{align*} \begin{array}{rcl} x&=&-3\\ y&=&3x-4\\ y&=&3(-3)-4 \qquad \text{Substitute } x=-3 \text{ into the equation}. \\ y&=&-9-4 \qquad \quad \ \text{Perform the multiplication to clear the parenthesis}. \\ y&=&-13 \qquad \qquad \ \ \text{Simplify the right side of the equation}. \end{array}\end{align*}

Repeat the process to calculate the values for the variable ‘y\begin{align*}y\end{align*}.’

Given input value xyyyThe output value is y=====13x43(1)4347\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&-1\\ y&=&3x-4\\ y&=&3(-1)-4\\ y&=&-3-4\\ \text{The output value is }y&=&-7 \end{array}\end{align*}

Given input value xyyyThe output value is y=====13x43(1)4341\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&1\\ y&=&3x-4\\ y&=&3(1)-4\\ y&=&3-4\\ \text{The output value is }y&=&-1 \end{array}\end{align*}

Given input value xyyyThe output value is y=====33x43(3)4945\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&3\\ y&=&3x-4\\ y&=&3(3)-4\\ y&=&9-4\\ \text{The output value is }y&=&5 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&5\\ y&=&3x-4\\ y&=&3(5)-4\\ y&=&15-4\\ \text{The output value is }y&=&11 \end{array}\end{align*}

Write the calculated ‘\begin{align*}y\end{align*}’ values in the table.

Plot the ordered pairs on the Cartesian grid and join the plotted points with a smooth, straight line.

#### Example 3

For the following linear function create a table of values and plot the points to draw the graph:

\begin{align*}y=2x-3\end{align*}

First, use the equation to calculate the output values ‘\begin{align*}y\end{align*}.’

\begin{align*} \begin{array}{rcl} x&=&-2\\ y&=&2x-3\\ y&=&2(-2)-3 \qquad \text{Substitute } x=-2 \text{ into the equation}. \\ y&=&-4-3 \qquad \quad \ \text{Perform the multiplication to clear the parenthesis}. \\ y&=&-7 \qquad \qquad \quad \text{Simplify the right side of the equation}. \end{array}\end{align*}

Repeat the process to calculate the values for the variable ‘\begin{align*}y\end{align*}.’

\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&-1\\ y&=&2x-3\\ y&=&2(-1)-3\\ y&=&-2-3\\ \text{The output value is }y&=&-5 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&0\\ y&=&2x-3\\ y&=&2(0)-3\\ y&=&0-3\\ \text{The output value is }y&=&-3 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&1\\ y&=&2x-3\\ y&=&2(1)-3\\ y&=&2-3\\ \text{The output value is }y&=&-1 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} \text{Given input value }x&=&2\\ y&=&2x-3\\ y&=&2(2)-3\\ y&=&4-3\\ \text{The output value is }y&=&1 \end{array}\end{align*}

Write the calculated ‘\begin{align*}y\end{align*}’ values in the table.

 \begin{align*}x\end{align*} \begin{align*}y\end{align*} -2 -7 -1 -5 0 -3 1 -1 2 1

Plot the ordered pairs on the Cartesian grid and join the plotted points with a smooth, straight line.

#### Example 4

Plot the graph of the line having \begin{align*}x=-2\end{align*} as its equation.

First, remember this is the graph of one of the special lines.

Next, describe what the graph will look like.

A vertical line passing through the point \begin{align*}(-2,0)\end{align*} and parallel to the \begin{align*}y\end{align*}-axis.

Then, graph the line on the Cartesian grid.

#### Example 5

Plot the graph of the line having \begin{align*}y=-3\end{align*} as its equation.

First, remember this is the graph of one of the special lines.

Next, describe what the graph will look like.

A horizontal line passing through the point \begin{align*}(0,-3)\end{align*} and parallel to the \begin{align*}x\end{align*}-axis.

Then, graph the line on the Cartesian grid.

### Review

Create a table of values for each equation and then graph it on the coordinate plane.

1. \begin{align*}y = 2x + 1\end{align*}
2. \begin{align*}y = 3x + 2\end{align*}
3. \begin{align*}y = -4x\end{align*}
4. \begin{align*}y = -2x\end{align*}
5. \begin{align*}y =-3x + 3\end{align*}
6. \begin{align*}y = 2x + 3\end{align*}
7. \begin{align*}y = 3x- 2\end{align*}
8. \begin{align*}y =-8x\end{align*}
9. \begin{align*}y = 3x + 1\end{align*}
10. \begin{align*}y = 4x\end{align*}
11. \begin{align*}y = -2x + 2\end{align*}
12. \begin{align*}y = 2x- 2\end{align*}
13. \begin{align*}y = x- 1\end{align*}
14. \begin{align*}x = 4\end{align*}
15. \begin{align*}y = -2\end{align*}

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### Vocabulary Language: English

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 $y$.

linear equation

A linear equation is an equation 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 $m$

Standard Form

The standard form of a line is $Ax + By = C$, where $A, B,$ and $C$ are real numbers.

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