Overview
This lesson explains the vertical line test, a method for determining if a graph represents a function by checking if vertical lines intersect the graph at more than one point.
Functions and Relations
- A function is a relation where each input (x) has exactly one output (y).
- Not all relations are functions; the key is the uniqueness of outputs for each input.
The Vertical Line Test
- The vertical line test states a graph is a function if no vertical line intersects it more than once.
- If a vertical line hits the graph more than once, some input has multiple outputs, so it is not a function.
Applying the Vertical Line Test: Examples
- If all vertical lines intersect the graph at most once, the graph passes and represents a function.
- If any vertical line intersects the graph more than once, the relation is not a function.
- A hole in the graph with a point elsewhere for the same x-value is still a function if only one intersection exists at each x.
- Discrete points: If no vertical line goes through more than one plotted point, it passes the test and is a function.
- In cases of two closed points with the same x-value, the graph fails the test and is not a function.
- For vertical asymptotes or breaks where a vertical line does not intersect the graph at all, this does not violate the test.
Key Terms & Definitions
- Function — A relation in which each input (x) has exactly one output (y).
- Relation — A set of ordered pairs (x, y).
- Vertical Line Test — A method to determine if a graph is a function; a graph is a function if no vertical line intersects it more than once.
- Hole — An open point in the graph where the function is undefined.
- Asymptote — A line that the graph approaches but does not cross.
Action Items / Next Steps
- Practice applying the vertical line test to various graphs to determine if they represent functions.