# 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 x-value of a function must be paired to a single y-value. If a vertical line intersects the graph of a relation at exactly one point, it implies that a single x-value is only paired to a unique value of y.

On the other hand, 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” 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 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

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Horizontal Line Test