Vertical Line Test
In short: The vertical line test says a graph shows a function if no vertical line can be drawn that touches the graph at more than one point.
Algebra 1Functions and GraphsF-IF.A.1
What is Vertical Line Test?
A function gives each input exactly one output, and the vertical line test checks that on a graph. Picture a vertical line sweeping from left to right: at each x-value it marks one input.
If that line ever meets the graph twice, the same x has two different y-values, so the graph is not a function. If every vertical line meets it at most once, it is a function. A circle fails; a parabola opening upward passes.
How to spot it
- A graph is shown and the question asks whether y is a function of x.
- The picture has a sideways curve, a circle or an ellipse.
- An x-value in a table appears with two different y-values.
- A vertical segment appears as part of the graph.
The method
- Imagine, or draw with a ruler, a vertical line at some x-value on the graph.
- Slide the line across the whole graph, from the far left to the far right.
- If any position of the line touches the graph in two or more places, answer: not a function.
- If no position does, answer: it is a function.
Vertical Line Test: solved example
Problem
The graph of \(x = y^2 – 4\) is a parabola lying on its side, opening to the right. Does it represent y as a function of x?
Solution
Draw the vertical line \(x = 0\). Then \(0 = y^2 – 4\), so \(y^2 = 4\) and \(y = 2\) or \(y = -2\).
The line meets the graph at \((0, 2)\) and \((0, -2)\): one input, two outputs. The graph fails the test, so it is not a function.
Try it
Question 1
Does the line \(y = -2x + 3\) pass the test?
Show the answer
Yes. Every vertical line meets a slanted line once, so it is a function.
Question 2
Is the graph of the vertical line \(x = 4\) a function?
Show the answer
No. The line \(x = 4\) itself meets the graph at every point, so one input has infinitely many outputs.
Vertical Line Test vs. Function
A function is the relationship, where every input gets one output. The vertical line test is just a visual way to check that on a graph. A table or an equation has no graph to slice, so for those you look for repeated x-values with different y-values instead.
Keep going
Lessons
Worksheets and quizzes
- Functions and Function Notation – Worksheet
- Functions and Function Notation – Quiz
- Understanding Functions – Worksheet
- Understanding Functions – Quiz
Related words
Whole dictionary: A–Z · Algebra 1 · Functions and Graphs