Linha vertical

(Video) Horizontal vs. Vertical Lines

by mathman1024

This video by mathman1024 explores horizontal and vertical lines. He goes into detail about the slope of each type of line.

$\large{\text{Summary}}$

The video starts with the slope of a horizontal line, but the second example covers vertical lines. He uses the line $x=-7$.

We can take any two points on the line and the $x$-value will always be $-7$ this means that the run value in the slope formula will always be $0$. Let's try this with the points $(-7,2)$ and $(-7,0)$.

$$\begin{align} m&=\frac{\text{rise}}{\text{run}}=\frac20\\ m&=\frac{y_2 - y_1}{x_2-x_1}\\ m&=\frac{0 - 2}{-7 - (-7)}\\ m&=\frac{-2}{0} \end{align}$$

Remember, the denominator of a fraction cannot be $0$, because we cannot divide by $0$. The value of a fraction with $0$ in the denominator is undefined. Therefore, the slope of any vertical line is undefined.

Slope and Vertical Lines

Vertical lines have undefined slopes. This is because no matter how much their height (position on the $y$-axis) changes, their position on the $x$-axis never does. This means that even if their rise is infinite, their run will always be $0$. Here are some examples with random values for rise:

$$\frac{\text{rise}}{\text{run}} = \frac{1}{0} = \text{undefined}$$ $$\frac{\text{rise}}{\text{run}} = \frac{451}{0} = \text{undefined}$$ $$\frac{\text{rise}}{\text{run}} = \frac{22}{0} = \text{undefined}$$

It's impossible to divide by $0$. A fraction with $0$ as its denominator will always be undefined.

Let's look at the graph for $x=1$ and consider its slope. On the line below, no matter what the value of $y$, $x$ is always equal to $1$. Since $x$ never changes, the run will always be $0$. This is true for any vertical line... the slope of the line will be undefined!

The equation for this line is $x = 1$.

Slope of Vertical Lines is Undefined

Vertical lines do not have positive slopes or negative slopes. They have undefined slopes.

Let's recall the definition of slope:

$$\text{Slope}=\frac{\text{rise}}{\text{run}}=m$$

Now, let's write this out with variables. We can use points $(x_1, y_1)$ and $(x_2, y_2)$ to figure out the slope. The $y$-values dictate the vertical position, or rise. The $x$-values dictate horizontal position, or run.

$$m = \frac{y_2-y_1}{x_2-x_1}$$

The $x$ value of any two points doesn't change for any vertical line. Let's use two points from the line $x=0$ and calculate slope.

$$(0,1),(0,-3)$$ $$\begin{align} m&=\frac{-3 - 1}{0 - 0}\\ m&=\frac{-4}{0}\end{align}$$

This gives us a fraction with a $0$ in the denominator. We cannot divide by zero, so our slope is undefined.

The picture above shows how every vertical line, no matter where it is on the $x$-axis, has an undefined slope.