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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
License: CC BY-NC 3.0

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 \begin{align*}y=3x+4\end{align*}.

License: CC BY-NC 3.0

The equation \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 ‘\begin{align*}y\end{align*}’ based on the different input ‘\begin{align*}x\end{align*}’ values substituted into the equation.

Using \begin{align*}x\end{align*}-values of -4, -2, 0, 2 and 4, create a table of values to represent the equation \begin{align*}y=3x+4\end{align*}.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
-4 -8
-2 -2
0 4
2 10
4 16

\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.

\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*}

\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*}

\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*}

\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 \begin{align*}(x,y)\end{align*} such that \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 \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 \begin{align*}y\end{align*}-axis and the other is a horizontal line that is parallel to the \begin{align*}x\end{align*}-axis.

Let’s graph each of these special lines.

A line having \begin{align*}x=5\end{align*} as its equation will pass through the point \begin{align*}(5,0)\end{align*} such that it will be parallel to the \begin{align*}y\end{align*}-axis.

License: CC BY-NC 3.0

A line having \begin{align*}y=4\end{align*} as its equation will pass through the point \begin{align*}(0,4)\end{align*} such that it will be parallel to the \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 \begin{align*}y\end{align*} represent the money raises and let \begin{align*}x\end{align*} represent the number of potted plants sold.

\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.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
50
100
150
200
250

\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.

\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*}

\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*}

\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*}

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

\begin{align*}x\end{align*} \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.

License: CC BY-NC 3.0

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.

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

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

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

\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 ‘\begin{align*}y\end{align*}.’ 

\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*}

\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*}

\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.

License: CC BY-NC 3.0

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.

License: CC BY-NC 3.0

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.

License: CC BY-NC 3.0

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.

License: CC BY-NC 3.0

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*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.5.  

 

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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.

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
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  9. [9]^ License: CC BY-NC 3.0

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