You can determine if a graph represents a function by using the vertical line test.
The Vertical Line Test Explained
The vertical line test is a visual method to check if a relation, depicted as a graph, is a function. Here's how it works:
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Imagine a vertical line: Visualize a vertical line moving across the entire graph from left to right (or you can physically draw a vertical line on the graph).
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Check for intersections: Observe how many times the vertical line intersects the graph at any given point.
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Determine if it's a function:
- If the vertical line intersects the graph at only one point at any given vertical line position, then the graph represents a function.
- If the vertical line intersects the graph at more than one point at any given vertical line position, then the graph does not represent a function.
Why the Vertical Line Test Works
The vertical line test is based on the definition of a function: for every input (x-value), there must be only one output (y-value). If a vertical line intersects the graph at two or more points, it means that for a single x-value, there are multiple corresponding y-values, violating the definition of a function.
Examples
- Function: A straight, non-vertical line, a parabola opening upwards or downwards, or an exponential curve. A vertical line drawn anywhere on these graphs will only intersect at one point.
- Not a Function: A circle, a parabola opening sideways, or a vertical line. A vertical line can be drawn on these graphs that intersect at more than one point.
Summary
The vertical line test is a simple and effective way to visually determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.
