Mathematics » Introducing Graphs » Graphing Linear Equations

# Graphing Vertical and Horizontal Lines

## Graphing Vertical and Horizontal Lines

Can we graph an equation with only one variable? Just $$x$$ and no $$y,$$ or just $$y$$ without an $$x?$$ How will we make a table of values to get the points to plot?

Let’s consider the equation $$x=-3.$$ The equation says that $$x$$ is always equal to $$-3,$$ so its value does not depend on $$y.$$ No matter what $$y$$ is, the value of $$x$$ is always $$-3.$$

To make a table of solutions, we write $$-3$$ for all the $$x$$ values. Then choose any values for $$y.$$ Since $$x$$ does not depend on $$y,$$ you can chose any numbers you like. But to fit the size of our coordinate graph, we’ll use $$1,2,$$ and $$3$$ for the $$y$$-coordinates as shown in the table.

$$x=-3$$
$$x$$$$y$$$$\left(x,y\right)$$
$$-3$$$$1$$$$\left(-3,1\right)$$
$$-3$$$$2$$$$\left(-3,2\right)$$
$$-3$$$$3$$$$\left(-3,3\right)$$

Then plot the points and connect them with a straight line. Notice in the figure below that the graph is a vertical line. ### Definition: Vertical Line

A vertical line is the graph of an equation that can be written in the form $$x=a.$$

The line passes through the $$x$$-axis at $$\left(a,0\right)$$.

## Example

Graph the equation $$x=2.$$ What type of line does it form?

### Solution

The equation has only variable, $$x,$$ and $$x$$ is always equal to $$2.$$ We make a table where $$x$$ is always $$2$$ and we put in any values for $$y.$$

$$x=2$$
$$x$$$$y$$$$\left(x,y\right)$$
$$2$$$$1$$$$\left(2,1\right)$$
$$2$$$$2$$$$\left(2,2\right)$$
$$2$$$$3$$$$\left(2,3\right)$$

Plot the points and connect them as shown. The graph is a vertical line passing through the $$x$$-axis at $$2.$$

What if the equation has $$y$$ but no $$x$$? Let’s graph the equation $$y=4.$$ This time the $$y$$-value is a constant, so in this equation $$y$$ does not depend on $$x.$$

To make a table of solutions, write $$4$$ for all the $$y$$ values and then choose any values for $$x.$$

We’ll use $$0,2,$$ and $$4$$ for the $$x$$-values.

$$y=4$$
$$x$$$$y$$$$\left(x,y\right)$$
$$0$$$$4$$$$\left(0,4\right)$$
$$2$$$$4$$$$\left(2,4\right)$$
$$4$$$$4$$$$\left(4,4\right)$$

Plot the points and connect them, as shown in the figure below. This graph is a horizontal line passing through the $$y\text{-axis}$$ at $$4.$$ ### Definition: Horizontal Line

A horizontal line is the graph of an equation that can be written in the form $$y=b.$$

The line passes through the $$y\text{-axis}$$ at $$\left(0,b\right).$$

## Example

Graph the equation $$y=-1.$$

### Solution

The equation $$y=-1$$ has only variable, $$y.$$ The value of $$y$$ is constant. All the ordered pairs in the table have the same $$y$$-coordinate, $$-1$$. We choose $$0,3,$$ and $$-3$$ as values for $$x.$$

$$y=-1$$
$$x$$$$y$$$$\left(x,y\right)$$
$$-3$$$$-1$$$$\left(-3,-1\right)$$
$$0$$$$-1$$$$\left(0,-1\right)$$
$$3$$$$-1$$$$\left(3,-1\right)$$

The graph is a horizontal line passing through the $$y$$-axis at $$–1$$ as shown. The equations for vertical and horizontal lines look very similar to equations like $$y=4x.$$ What is the difference between the equations $$y=4x$$ and $$y=4?$$

The equation $$y=4x$$ has both $$x$$ and $$y.$$ The value of $$y$$ depends on the value of $$x.$$ The $$y\text{-coordinate}$$ changes according to the value of $$x.$$

The equation $$y=4$$ has only one variable. The value of $$y$$ is constant. The $$y\text{-coordinate}$$ is always $$4.$$ It does not depend on the value of $$x.$$ The graph shows both equations. Notice that the equation $$y=4x$$ gives a slanted line whereas $$y=4$$ gives a horizontal line.

## Example

Graph $$y=-3x$$ and $$y=-3$$ in the same rectangular coordinate system.

### Solution

Find three solutions for each equation. Notice that the first equation has the variable $$x,$$ while the second does not. Solutions for both equations are listed. The graph shows both equations. 