Describing Horizontal and Vertical Lines

There are two special cases of lines on a graph—horizontal and vertical lines.

A horizontal line is a line defined by an equation of the form [latex]f\left(x\right)=b[/latex] where [latex]b[/latex] is a constant. A horizontal line indicates a constant output or [latex]y[/latex]-value.

In the graph and table, we see that the output has a value of [latex]2[/latex] for every input value. The change in outputs between any two points is [latex]0[/latex]. In the slope formula, the numerator is [latex]0[/latex], so the slope is [latex]0[/latex].If we use [latex]m = 0[/latex] in the equation [latex]f(x)=mx+b[/latex], the equation simplifies to [latex]f(x)=b[/latex].
[latex]\\[/latex]
In other words, the value of the function is a constant. This graph represents the function [latex]f(x)=2[/latex].

A vertical line is a line defined by an equation of the form [latex]x=a[/latex] where [latex]a[/latex] is a constant. A vertical line indicates a constant input or [latex]x[/latex]-value.

We can see that the input value for every point on the line is [latex]2[/latex], but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.Notice that a vertical line has an [latex]x[/latex]-intercept but no [latex]y[/latex]–intercept unless it’s the line [latex]x= 0[/latex]. This graph represents the line [latex]x= 2[/latex].

Write the equation of the line graphed below.

Show Answer

For any [latex]x[/latex]-value, the [latex]y[/latex]-value is [latex]–4[/latex], so the equation is [latex]y=–4[/latex].

Write the equation of the line graphed below.

Show Solution

The constant [latex]x[/latex]-value is [latex]7[/latex], so the equation is [latex]x=7[/latex].