# 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 the relation, and then observe the points of intersection.

Why does this work?

The vertical line test supports the definition of a function. That is, every x-value of a function must be paired to a single y-value. If we think of a vertical line as an infinite set of x-values, then intersecting the graph of a relation at exactly one point by a vertical line implies that a single x-value is only paired to a unique value of y.

In contrary, if the vertical line intersects the graph more than once this suggests that a single x-value is being associated with more than one value of y. This condition causes the relation to be “disqualified” or not considered 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.

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 f\left( x \right) = x + 1

Graph of the quadratic function (parabola) f\left( x \right) = {x^2} - 2

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

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 x = {y^2}

Graph of the circle {x^2} + {y^2} = 9

Graph of the relation x = {y^3} - y + 2

You might also be interested in:﻿

Horizontal Line Test