Graphs of Linear Equations

Graph lines presented in ax+by = c form

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Using Tables to Graph Functions

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.

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

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 $y$.
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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