# Properties of the Vertical Lines

A vertical line is a line perpendicular to another surface or line that is considered as the base. Here you get familiarized with the properties of the vertical line, its equation, and the slope of a vertical. A vertical line is always a straight line that goes up to down or down to up. Vertical lines are also known as standing lines. The lines that join the bases of a square or rectangle are the vertical lines that we usually draw.

## Step by Step guide to properties of the vertical line

A vertical line is a line on the coordinate plane where all the points on the line have the same $$x$$-coordinate. When we draw the points of the function $$x = a$$, on a coordinate plane, we find that a vertical line is obtained on joining the coordinates. In the image below, $$L1$$ and $$L2$$ are the two vertical lines. All the points in the $$L1$$ have only $$a$$ as the $$x$$-coordinate for all the values of $$y$$, and all the points in the $$L2$$ have only $$-a$$ as the $$x$$-coordinate for all the values of $$y$$.

### Vertical line equation

The equation of a vertical line is $$x = a$$ or $$x = -a$$, where $$x$$ is the coordinate of $$x$$ at any point on the line and $$a$$ is where the line crosses the $$x$$-intercept.

### Slope of a vertical line

A vertical line has an undefined slope. According to the definition of slope, we calculate the slope as follows:

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

Now, since the $$x$$-coordinate remains constant on a vertical line, therefore we have $$x_2=x_1=x$$. So, the slope of the vertical line is $$m=\frac{(y_2-y_1)}{(x-x)}=\frac{(y_2-y_1)}{0}$$, which is not defined as the denominator is zero.

The $$x$$ coordinates remain the same for all the points on the vertical line and there is no run horizontally. Thus the slope of a vertical line is undefined.

### Vertical line test

A vertical line is used to find if a given graph is a function. A relation is said to be a function only when a vertical line drawn intersects the graph only at one point. A function can have only one output for every input. So if a vertical line intersects a graph at more than one point, it is interpreted as a function that has more than one output that indicates it cannot be a function.

In the image below, we can see that a vertical line is drawn to the function $$y = f(x)$$ is a function, because the vertical line intersects only at a point on the curve, while in the circle, the vertical line touches at two points. Thus the circle is not a function.

### Properties of the vertical line

• The vertical line equation does not have a $$y$$-intercept, because the line is parallel to the $$y$$-axis.
• The equation of a vertical line always takes the form $$x = a$$, where a is the $$x$$-intercept.
• The slope of a vertical line is not defined. Since there are no changes in the $$x$$ coordinates, the denominator of the slope is zero.

### Vertical line of symmetry

A vertical line of symmetry is a straight line that goes from top to bottom and divides the shape into two halves. Below is the isosceles trapezoid that has only one vertical line of symmetry.

### Properties of the Vertical Line – Example 1:

What is the slope of the line $$x=-7$$?

Solution:

The slope of a vertical line is undefined. So the slope of the vertical line $$x=-8$$ is undefined.

## Exercises for Properties of the Vertical Line

• What is the equation of the vertical line passing through $$(-11,3)$$?
• Find the equation of the vertical line in the graph?
• $$\color{blue}{x+11=0}$$
• $$\color{blue}{x+4=0}$$

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