4.5.2 • Using the Vertical Line Test
Activity
A table for the relation is below.
| 1 | 2 | 3 | 1 | 4 | 5 | |
| -1 | 5 | -2 | 1 | 3 | -2 |
1. Is this relation a function? Explain your answer.
Compare your answer:
is not a function because and .
2. Use the Desmos graphing tool or technology outside the course. Graph the points in the relation.
Interact with the graph. Graph the points in the relation on a coordinate grid.
Compare your answer:
3. What do you notice about the points and ?
Compare your answer:
They are on the same vertical line.
VERTICAL LINE TEST
A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.
If any vertical line intersects the graph in more than one point, the graph does not represent a function.
If the graph does represent a function, we say it “passes the vertical line test.”
4. For the function graphed below, determine if the graph passes the vertical line test and then explain how this identifies the graph as a function or not.
Compare your answer:
Yes, this passes the vertical line test (see the sample blue lines below) for all points on the line so the graph is a function.
5. For the function graphed below, determine if the graph passes the vertical line test and then explain how this identifies the graph as a function or not.
Compare your answer:
No, this graph does not pass the vertical line test, so it is not a function. For example, the vertical line crosses the graph more than once.
Notice that the graph of a non-vertical line will always be a function. This is called a linear function.
Video: Interpreting Function Notation
Watch the following video to learn more about function notation.
Self Check
Additional Resources
Determine If a Graph Is a Function
A relation is a function if every input has exactly one output value. So the relation defined by the equation is a function. Notice that a line will always be a function, so it is called a linear function.
If we look at the graph, each vertical dashed line intersects the line at only one point. This makes sense because in a function, for every -value there is only one -value.
If the vertical line hit the graph twice, the -value would be mapped to two -values, so the graph would not represent a function.
This leads us to the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.
Look at the graphs below:
Graph a passes the vertical line test because if an imaginary vertical line were drawn anywhere on the graph, it would only touch one time.
Graph a is a function.
Graph b does not pass the vertical line test because when a vertical line is drawn down the graph, there is at least one place where the vertical line touches twice.
Graph b is not a function.
Try it
Determine if a Graph Is a Function
Determine if each graph is a function. Use the vertical line test to explain.
Compare your answser:
Here is how to determine if a graph is a function:
Graph a does not pass the vertical line test because if a vertical line were drawn down the graph, there would be at least one place where the vertical line would touch more than once. For example, a vertical line drawn at would cross the graph more than once.
Graph a is not a function.
Graph b does pass the vertical line test because a vertical line drawn anywhere on the graph will only touch one time.
Graph b is a function.