What Is the Vertical Line Test?

Vertical Line Test

Before, we had to use tables and mappings to figure out if a relation was a function. Now, we can easily figure this out graphically.

This is a test used to tell if a graph is a function. Draw a vertical line and see how many points it touches on the graph. If it only ever touches one point, the graph is a function.

Function Example

Let's look at the graph $y=x^3$ and show it's a function. This is cubic.

All we have to do is sketch some vertical lines and see how many points they pass through.

Drawing some vertical lines, we see they only pass through one point on the graph. Therefore, $y=x^3$ is a function!

Not a Function Example

Let's look at an example where the graph doesn't pass the vertical line test. Let's look at $y^2=x+1$.

Again, we draw vertical lines and see how many points on our graph they pass through.

Even though one of lines passes through only one point, the other line we drew intersects the graph twice. Therefore, our graph is a relation but not a function.

Solution

Graphing this, we get:

Drawing a vertical line, we see the vertical line intersects at two points.

Therefore, this is not a function.

This graphed equation also fails the horizontal line test, which you can learn more about in the next lesson!

(Video) Introduction to Functions

by mathman1024

This video by mathman1024 explains the basics of functions and goes over the vertical line test.

$\large{\text{Summary}}$

What about our graph above? Is that a function? Well one way to check is the vertical line test. This test says that a graph is a function if, when you draw any vertical line, the line only goes through at most one point.

As you see above, if we draw vertical lines on this graph they go through only one point. Therefore, this graph is a function!

The vertical line test is a great way to test whether a graphed equation is a function in a quick way.

Vertical Line Test

The vertical line test is the best way to graphically determine whether a relation is a function.

While there are other ways to determine whether a relation is a function, such as finding the domain and range, it is often easier to graph the relation and draw vertical lines.