# 4.2: Graphs of Linear Equations

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Graph a linear function using an equation.
- Write equations for, and graph, horizontal and vertical lines.
- Analyze graphs of linear functions and read conversion graphs.

## Vocabulary

Terms introduced in this lesson:

- formula
- equation

## Teaching Strategies and Tips

In Example 1:

- The coefficient of \begin{align*}x\end{align*} represents the taxi’s rate or cost per mile. Ask students to consider how the graph would change if the taxi charged more per mile, then less per mile.
- The constant in the equation is the taxi’s base fee. Ask students to interpret the \begin{align*}y-\end{align*}intercept (the taxi’s base fee).
- Include units on the axes labels; i.e., \begin{align*}x\end{align*}
*(in miles)*, \begin{align*}y\end{align*}*(in dollars)*. - Do the same for Example 2. Ask students to also interpret the \begin{align*}x-\end{align*}intercept (number of years when the debt will be fully paid).

In Examples 1 and 2:

- Students make estimates and predictions from their graphs. Answers will be approximate, even though the equations can be solved exactly. Emphasis should be placed on reading and interpreting the graphs and not solving equations.
- Students make graphs by constructing tables from the given equations. Have them plot at least \begin{align*}3\end{align*} points for accuracy.
- Teachers may find it useful to show how the equations derive from the given information.

Use Examples 3 and 4 to motivate the equations of horizontal and vertical lines.

- The equations \begin{align*}x =\mathrm{constant}\end{align*} and \begin{align*}y = \mathrm{constant}\end{align*} are the simplest equations possible.
- They are two-variable equations despite only one variable being present. The coordinate plane must be used to graph them. Often, students will see an equation like \begin{align*}x = 3\end{align*} and plot a single point on the \begin{align*}x-\end{align*}axis at \begin{align*}(3,0)\end{align*}.
- The equation \begin{align*}y = 1\end{align*}, for example, signifies that the \begin{align*}y\end{align*} quantity is fixed at \begin{align*}1\end{align*} and \begin{align*}x\end{align*} is free to take on any value.
- Given a horizontal line, the equation is \begin{align*}y =\end{align*} the \begin{align*}y-\end{align*}value of any point on the line. Analogously, vertical lines have the equation \begin{align*}x =\end{align*} the \begin{align*}x-\end{align*}value of any point on the line. This can help with Example 5.

Additional Examples:

*What is the equation of the vertical line passing through \begin{align*}(4, 5)\end{align*}?*

Solution: \begin{align*}x = 4\end{align*}.

*Given that a line passes through \begin{align*}(2, -1)\end{align*} and is parallel to the \begin{align*}x-\end{align*}axis, what is its equation?*

Solution: \begin{align*}y = -1\end{align*}.

## Error Troubleshooting

Allow students to distinguish between the input and output variables in Examples 6 and 7 *before estimating.* This way, they will know whether to begin correctly along the horizontal axis or the vertical axis.

In the *Review Questions*, it may help students to rewrite \begin{align*}y = 6 - 1.25 x\end{align*} in Problem 1c as \begin{align*}y = -1.25 x + 6\end{align*}.