Learning Objectives
- Find the Slope of Horizontal and Vertical Lines
- Find the slope of the lines [latex]x=a[/latex] and [latex]y=b[/latex]
- Recognize that horizontal lines have slope = 0
- Recognize that vertical lines have slopes that are undefined
Finding the Slopes of Horizontal and Vertical Lines
So far you’ve considered lines that run “uphill” or “downhill.” Their slopes may be steep or gradual, but they are always positive or negative numbers. But there are two other kinds of lines, horizontal and vertical. What is the slope of a flat line or level ground? Of a wall or a vertical line?
Using the form [latex]y=0x+3[/latex], you can see that the slope is 0. You can also use the slope formula with two points on this horizontal line to calculate the slope of this horizontal line. Using [latex](−3,3)[/latex] as Point 1 and (2, 3) as Point 2, you get:
[latex] \displaystyle \begin{array}{l}m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\\\\m=\frac{3-3}{2-\left(-3\right)}=\frac{0}{5}=0\end{array}[/latex]
The slope of this horizontal line is 0.
Let’s consider any horizontal line. No matter which two points you choose on the line, they will always have the same y-coordinate. So, when you apply the slope formula, the numerator will always be 0. Zero divided by any non-zero number is 0, so the slope of any horizontal line is always 0.
The equation for the horizontal line [latex]y=3[/latex] is telling you that no matter which two points you choose on this line, the y-coordinate will always be 3.
How about vertical lines? In their case, no matter which two points you choose, they will always have the same x-coordinate. The equation for this line is [latex]x=2[/latex].
There is no way that this equation can be put in the slope-point form, as the coefficient of y is [latex]0\left(x=0y+2\right)[/latex].
So, what happens when you use the slope formula with two points on this vertical line to calculate the slope? Using [latex](2,1)[/latex] as Point 1 and [latex](2,3)[/latex] as Point 2, you get:
[latex] \displaystyle \begin{array}{l}m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\\\\m=\frac{3-1}{2-2}=\frac{2}{0}\end{array}[/latex]
But division by zero has no meaning for the set of real numbers. Because of this fact, it is said that the slope of this vertical line is undefined. This is true for all vertical lines—they all have a slope that is undefined.
Example
What is the slope of the line that contains the points [latex](3,2)[/latex] and [latex](−8,2)[/latex]?
In this video, you will see more examples of how to find the slope of horizontal and vertical lines:
Summary
Slope describes the steepness of a line. The slope of any line remains constant along the line. The slope can also tell you information about the direction of the line on the coordinate plane. Slope can be calculated either by looking at the graph of a line or by using the coordinates of any two points on a line. There are two common formulas for slope: [latex] \displaystyle \text{Slope }=\frac{\text{rise}}{\text{run}}[/latex] and [latex] \displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[/latex] where [latex]m=\text{slope}[/latex] and [latex] \displaystyle ({{x}_{1}},{{y}_{1}})[/latex] and [latex] \displaystyle ({{x}_{2}},{{y}_{2}})[/latex] are two points on the line.
The images below summarize the slopes of different types of lines.