You can determine if the graph of a relation represents a function by using the Vertical Line Test.
Vertical Line Test Explained
The Vertical Line Test is a visual method to quickly assess if a graphed relation is a function. Here's how it works:
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Draw a Vertical Line: Imagine drawing a vertical line anywhere on the graph. This can be done mentally or physically with a ruler.
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Check for Intersections: Observe how many times the vertical line intersects the graph.
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Determine if it's a Function:
- If the vertical line intersects the graph at only one point (or not at all) for every possible vertical line, then the graph represents a function. This means for every x-value, there is only one corresponding y-value.
- If any vertical line intersects the graph at more than one point, then the graph does not represent a function. This means at least one x-value has multiple corresponding y-values, violating the definition of a function.
Example
Let's consider two examples:
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Example 1: A Parabola opening sideways. If you graph
x = y^2, you'll see a parabola opening to the right. A vertical line drawn atx = 4would intersect the parabola at two points (e.g., y = 2 and y = -2). Therefore,x = y^2is not a function. -
Example 2: A Parabola opening upwards. If you graph
y = x^2, you'll see a parabola opening upwards. No matter where you draw a vertical line, it will only ever intersect the graph at one point. Therefore,y = x^2is a function.
Why does this work?
A function, by definition, assigns only one output (y-value) for each input (x-value). The Vertical Line Test is a visual representation of this definition. If a vertical line intersects the graph more than once, it means that for that particular x-value, there are multiple corresponding y-values, violating the single output requirement of a function.
In summary, use the Vertical Line Test: If any vertical line intersects the graph more than once, it is not a function. If no vertical line intersects the graph more than once, it is a function.
