Slope of Horizontal Line - Definition & Examples

Horizontal lines do not have positive slopes or negative slopes. The slope of a horizontal line will always be $0$. This is because it never changes height: if we use the rise over run approach, rise will always be $0$, no matter what run is. For example, inserting some random values for run: $$\frac{\text{rise}}{\text{run}} = \frac{0}{1} = 0$$ $$\frac{\text{rise}}{\text{run}} = \frac{0}{292} = 0$$ $$\frac{\text{rise}}{\text{run}} = \frac{0}{4} = 0$$

No matter what the (nonzero) denominator is, if the numerator of the fraction is $0,$ the simplified result will always be $0.$

As another example, try looking at the graph below. The height of the line is always $2$, which means that the difference in height $($even between two points on either side of the graph, $y_1$ and $y_2)$ will always be $0$. So: $$\text{slope}=\frac{y_2-y_1}{x_2-x_1} = \frac{0}{x_2-x_1} = 0$$

In summary, horizontal lines have only one value in the range: their unchanging $y$-value. This means that their change in $y$-value is constant (i.e., the rise is $0$), so the slope is $0.$

(Video) Horizontal vs. Vertical Lines

by mathman1024

Here's a video by mathman1024 where he looks at vertical and horizontal lines and determines their slopes.

$\large{\text{Summary}}$

The first graph he looks at is $y=5$ which is a horizontal line.

Take any two points on this line, say $(0,5)$ and $(5,5)$. To get from one point to the other how much do you have to rise? Zero! And how much do you have to run? Five. So: $$m=\frac{\text{rise}}{\text{run}}=\frac05=0$$

If you have a straight line that is horizontal, the slope of the line will be zero.

Slope of Horizontal Lines is Zero

Horizontal lines have a slope of zero no matter where the line is graphed. This is because the rise of any horizontal line will always be zero, regardless of the $x$ values. Let's use two points on the line $y=3$, $(-2,3)$ and $(4,3)$, to show this. We will use the formula for the slope of a line, $m=\frac{y_2-y_1}{x_2-x_1},$ to help us. $$\begin{align}\text{Slope}&=\frac{\text{rise}}{\text{run}}\\ \\ m&=\frac{y_2-y_1}{x_2-x_1}\\ \\ m&=\frac{3-3}{4-(-2)}\\ \\ m&=\frac{0}{6} \end{align}$$ No matter which points are picked on any horizontal line, the rise will be $0$.