Graphs of Linear Functions: Learn It 2

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].
image

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.

imageWe 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.M equals change of output divided by change of input. The numerator is a non-zero real number, and the change of input is zero.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].

horizontal and vertical lines

Lines can be horizontal or vertical.

  • 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 vertical line is a line defined by an equation of the form [latex]x=a[/latex] where [latex]a[/latex] is a constant.
Write the equation of the line graphed below.Graph of x = 7.

Write the equation of the line graphed below.Graph of two functions where the baby blue line is y = -2/3x + 7, and the blue line is y = -x + 1.