The vertical line test is a quick way to decide whether a graph represents a function. A function assigns each input exactly one output, so one x-value cannot lead to two different y-values. On a coordinate graph, vertical lines help reveal whether any input is paired with more than one output.
This matters because functions are used to model predictable relationships in algebra, science, economics, and engineering.
To use the test, imagine sliding a vertical line across the entire graph. If the line ever touches the graph at more than one point at the same time, the relation is not a function. If every vertical line touches the graph at zero or one point, the relation is a function.
This matches the mapping idea: each input from the domain must point to only one output in the range.
Key Facts
- A relation is a function if each input x has exactly one output y.
- Vertical line test: if any vertical line intersects a graph more than once, the graph is not a function.
- If every vertical line intersects the graph at most once, the graph represents a function.
- In y = f(x), one x-value can produce only one f(x)-value.
- The graph of y = x^2 passes the vertical line test because each x has one y.
- The circle x^2 + y^2 = 9 fails the vertical line test because many x-values have two y-values.
Vocabulary
- Function
- A relation in which every input is paired with exactly one output.
- Relation
- A set of ordered pairs that shows how inputs and outputs are connected.
- Domain
- The set of all possible input values, usually the x-values.
- Range
- The set of all possible output values, usually the y-values.
- Vertical Line Test
- A graph test that checks whether any vertical line crosses a relation more than once.
Common Mistakes to Avoid
- Using horizontal lines instead of vertical lines. Horizontal lines test whether a function is one-to-one, not whether it is a function.
- Thinking a graph must cross every vertical line to be a function. A function can have a limited domain, so some vertical lines may not touch the graph at all.
- Counting two points with different x-values as a failure. The vertical line test only fails when the same x-value has more than one y-value.
- Assuming all curved graphs are not functions. Curves like parabolas and exponential graphs can pass the vertical line test.
Practice Questions
- 1 Determine whether the relation {(1, 3), (2, 5), (3, 5), (4, 7)} is a function. Explain using inputs.
- 2 Determine whether the relation {(0, 2), (1, 4), (1, 6), (3, 8)} is a function. Identify the x-value that decides the answer.
- 3 A graph shows a sideways parabola opening to the right. Explain why it fails the vertical line test and connect your explanation to the idea of one input having more than one output.