# 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 each vertical line as an *x*-value, then intersecting the graph of a relation at exactly one point implies that a single *x*-value is 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 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 *(*x*)* = x + *1

Graph of the parabola *f *(*x*)* = x*^{2 }*− *2

Graph of the cubic function *f *(*x*)* = 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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