Vertical Line – Definition, Equation, Facts, Examples, FAQs

What Is a Vertical Line in Math?

Vertical line is a line perpendicular to a horizontal line or a surface. In a coordinate plane, the y-axis is a vertical line since it is perpendicular to the x-axis, the baseline. Thus, any line that is parallel to the y-axis is a vertical line. A vertical line is also called a standing line.

Examples of vertical lines:

Vertical Line Definition

A vertical line is a straight line that runs from the top to the bottom, parallel to the y-axis, on the Cartesian coordinate system. It has an undefined slope and an equation of the form x = c, where “c” represents a constant value, indicating that the x-coordinate remains constant for all points on the line. Vertical lines divide the coordinate plane into two halves and have no y-intercept.

Properties of Vertical Lines

1. Vertical lines are parallel to the y-axis. Thus, they do not intersect the y-axis. In other words, they do not have a y-intercept.

2. The equation of a vertical line is of the form of x = a, where a is the x intercept.

3. Since there is no change in the x-coordinates in a vertical line (x is constant), the denominator of the slope is always zero. Thus, the slope of a vertical line is undefined.

4. A vertical line intersects a horizontal line at a right angle.

Equation of a Vertical Line

The equation of a vertical line in math can be given by x = ± a, where x is the x-coordinate and (a,0) is the point where the line intersects the x axis, the x-intercept.

For example, consider the line which intersects the x-axis at (5,0) and is parallel to the y-axis. We get the equation of the line as x = 5.

Vertical Line on a Coordinate Plane

On a coordinate plane, the x-axis is considered as the horizontal axis, whereas the y-axis is the vertical axis. Vertical line in a coordinate plane is a line that is parallel to the y-axis. 

For a vertical line, all the points lying on the line have the same x-coordinate. Thus, the x-coordinate stays the same, only the y-coordinate changes. A vertical line never intersects the y-axis. 

Slope of a Vertical Line

Slope of a line tells us how steep a line is. If a line passes through two points having coordinates (x₁, y₁) and (x₂, y₂), then we can define the slope of a line as

Slope $= m = \frac{y_{2} \;-\; y_{1}}{x_{2}\;-\; x_{1}} = \frac{\Delta y}{\Delta x} = \frac{Change \;in\; y}{Change \;in\; x} = \frac{Rise}{Run}$

As mentioned earlier, the x-coordinate remains constant in the equation of a vertical line. Only the y-coordinate changes. Thus, the denominator in the formula of the slope becomes 0. Since the division by 0 is not defined, the slope of a vertical line is undefined.

Difference between Horizontal Line and Vertical Line

Horizontal Line	Vertical Line
It is parallel to the x-axis.	It is parallel to the y-axis.
It is perpendicular to the y-axis.	It is perpendicular to the x-axis.
The equation is of the form y = ± a.	The equation is of the form x = ± a.
The slope of a horizontal line is 0.	The slope of a vertical line is undefined.

Vertical Line Test

Vertical line test is used to assess whether a given graph represents a function or not. It is based on the fact that a function can only have one output for every input. This test ensures that for each input (x-value), there is only one corresponding output (y-value), satisfying the criteria of a function.

If in a graph, a vertical line has more than one intersection, it means that there is more than one output for a single input, which means it cannot be a function. 

Thus, any vertical line in the plane can only ever cross the graph of a function once.

Vertical line test examples:

Vertical Lines of Symmetry

When a vertical line goes through the middle of a shape from top to bottom dividing it into two identical halves, that vertical line is called a vertical line of symmetry.

Facts about Vertical Lines

Conclusion

In this article, we learned about vertical lines, their meaning and definition in geometry, their properties, equation and slope. Let’s look at a few examples. 

Solved Examples on Vertical Lines

1. Find the slope of the line x = – 3.

Solution:

Since x = – 3 is a line parallel to the y-axis. 

Thus, it is a vertical line.

The slope of this line is undefined.

2. Find the equation of the vertical line passing through the point (2, – 4).

Solution:

The equation of the vertical line is of the form x = ± a, where a is the x-intercept.

Since the line passes through (2, – 4), the x-intercept is 2.

Thus, the equation of the vertical line is x = 2.

We can also write it as x – 2 = 0.

3. Find the equation of the line given in the figure

Solution:

The line is parallel to the y axis and passes through the point (– 3, 0) on the x axis.

Thus, the x-intercept is – 3.

The equation of the given vertical line is x = – 3.

4. Plot the line x = 1.5 on a graph.

Solution:

x = 1.5 represents a vertical line passing through the point (1.5, 0). 

Practice Problems on Vertical Lines

Frequently Asked Questions about Vertical Lines