Key takeaway: The vertical line test says a graph shows a function exactly when no vertical line crosses it at more than one point.
Grade 9 – Algebra 1F-IF.A.1Functions and Graphs
Definition of Vertical Line Test
The vertical line test is a quick visual check for functions. Imagine sweeping a vertical line, like a pencil held upright, from the far left of a graph to the far right.
If at every position the line meets the graph at most once, each input has only one output, so the graph is a function. If even one vertical line meets the graph twice or more, some input has two different outputs, and the graph is not a function. A vertical line is itself an equation \(x = c\), which never passes the test because it touches the line at every point.
Formula and notation
A vertical line has the form
\[x = c\]
where \(c\) is a constant. For a set of points, apply the same idea without drawing: if the same \(x\)-value appears in two pairs with different \(y\)-values, such as \((2, 7)\) and \((2, -1)\), the relation fails.
Key facts
- Passing the test means one output for each input, which is the definition of a function.
- Circles, sideways parabolas and vertical lines fail the test.
- Lines that are not vertical, and parabolas that open up or down, pass it.
- The test checks outputs per input; it does not check whether the function is one-to-one.
Vertical Line Test example
Does the relation \(\{(1, 4), (2, 7), (2, -1), (5, 3)\}\) represent a function? Apply the vertical line test idea to its points.
-
Plot the points; a vertical line through \(x = 2\) passes through both \((2, 7)\) and \((2, -1)\).
-
So the input 2 has two outputs, 7 and \(-1\).
-
The other inputs, 1 and 5, each have only one output.
The relation is not a function, because the input 2 has two outputs.
Common mistake
Watch out
Rhea sees that a horizontal line crosses a parabola twice and says the parabola \(y = x^{2}\) is not a function.
Why it is wrong
She used a horizontal line. A horizontal line asks whether an output repeats, which is allowed. The vertical line test is the one that decides functions: every vertical line crosses \(y = x^{2}\) just once, so it is a function.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 1 vocabulary · Functions and Graphs