Vertical-Line Test
Apply the Vertical-Line Test in Grade 9 algebra to determine if a graph represents a function: if any vertical line intersects the graph more than once, the relation is not a function.
Key Concepts
Property A graph on the coordinate plane represents a function if any vertical line intersects the graph in exactly one point.
Examples A straight line like $y = x + 2$ passes the test, as a vertical line only ever hits it once. A circle fails the test because a vertical line can slice through it at two points simultaneously. A U shaped parabola like $y = x^2$ passes the test because each vertical line crosses it only once.
Explanation Imagine sliding a vertical ruler across a graph from left to right. If your ruler only ever touches the graphed line at a single point at any time, congratulations, you've got a function! But if the ruler hits the graph in two or more spots at once, it fails the test. It's a super quick, visual trick.
Common Questions
What is the Vertical-Line Test?
The Vertical-Line Test checks if a graph represents a function. Draw or imagine vertical lines across the graph. If every vertical line intersects the graph in at most one point, the graph is a function. Two intersections anywhere means it is not a function.
Why does a circle fail the Vertical-Line Test?
A vertical line passing through the interior of a circle intersects it at two points. This means a single x-value maps to two different y-values, violating the definition of a function where each input has exactly one output.
What is the connection between the Vertical-Line Test and the function definition?
A function requires each x-value (input) to have exactly one y-value (output). The Vertical-Line Test is a visual check of this: a vertical line at any x-coordinate represents all possible y-values at that x, so two intersections signal two outputs for one input.