Horizontal and Vertical Lines

# Horizontal and Vertical Lines

## Equations of Horizontal Lines

Horizontal lines have equations in the form $y = a$ where $a$ is some number. For example, the line $y = 2$ is a horizontal line to which every point on the line has the $y$-coordinate of 2. The following image is the graph of the line $y = 2$:

We should note some important properties of horizontal lines. First off, any horizontal line $y = a$ **always* has a $y$-intercept since the line must pass through the vertical $y$-axis at some point, namely at the point $(0, a)$. Furthermore, we note that a horizontal line never has an $x$-intercept unless the line is $y = 0$. Furthermore, we note that the equation $y = 0$ is the equation of the $x$-axis, which has infinitely many $x$-intercepts.

## Equations of Vertical Lines

Vertical lines have equations in the form $x = a$ where $a$ is some number. For example, the line $x = -3$ is a vertical line to which every point on the line has the same $x$-coordinate of -3. The following image is the graph of the line $x = -3$:

Once again, there are some important properties to note regarding vertical lines. Any vertical line $x = a$ always has an $x$-intercept since the line must pass through the horizontal $x$-axis at some point, namely at the point $(a, 0)$. Furthermore, a vertical line never has a $y$-intercept unless the line is $x = 0$. If our line is $x = 0$, then we have infinitely many $y$-intercepts since our line is essentially the $y$-axis.

## Example Questions

• 1. Graph the line $y = 5$. Write the coordinates of any x-intercepts and any y-intercepts.
• 2. Graph the line $y = -4$. Write the coordinates of any x-intercepts and any y-intercepts.
• 3. Graph the line $x = 2$. Write the coordinates of any x-intercepts and any y-intercepts.
• 4. Graph the line $x = -6$. Write the coordinates of any x-intercepts and any y-intercepts.
• 5. Explain why the line $y = 0$ has infinitely many solutions.
• 6. Suppose that a box is defined by the lines $x = 1$, $x = 3$, $y = 2$ and $y = 7$. What is the area of this box?