# Vertical Line Test

The **vertical line test** is a method that is used to determine whether a given relation is a function or not. The approach is rather simple. Draw a vertical line cutting through the graph of a relation, and then observe the points of intersections.

The vertical line test supports the definition of a function. That is, every [latex]x[/latex]-value of a function must be paired to a single [latex]y[/latex]-value. If a vertical line intersects the graph of a relation at exactly one point, it implies that a single [latex]x[/latex]-value is only paired to a unique value of [latex]y[/latex].

On the other hand, if the vertical line intersects the graph more than once, this suggests that a single [latex]x[/latex]-value is being associated with more than one value of [latex]y[/latex]. This condition causes the relation to be “disqualified” as a function.

So here’s the deal!

If a vertical line intersects the graph in all places **at exactly one point**, then the relation is a **function**.

If a vertical line intersects the graph in some places **more than once**, then the relation is **NOT** a function.

Here are some examples of relations that are also functions because they pass the vertical line test.

## Cutting or Hitting the Graph at **Exactly One Point**

Graph of the line [latex]f\left( x \right) = x + 1[/latex]

Graph of the quadratic function (parabola) [latex]f\left( x \right) = {x^2} – 2[/latex]

Graph of the cubic function [latex]f\left( x \right) = {x^3}[/latex]

If a vertical line intersects the graph in some places at **more than one point**, then the relation is **NOT** a function.

Here are some examples of relations that are NOT functions because they fail the vertical line test.

## Cutting or Hitting the Graph in **More Than One Point**

Graph of the “sideway” parabola [latex]x = {y^2}[/latex]

Graph of the circle [latex]{x^2} + {y^2} = 9[/latex]

Graph of the relation [latex]x = {y^3} – y + 2[/latex]

**You might also be interested in:**