Visual examples of the vertical line test to determine if a graph represents a function.

Problem Description:

The image shows six graphs, and the task is to apply the Vertical Line Test to determine which of these graphs represent functions. The Vertical Line Test states that a graph represents a function if and only if every vertical line intersects the graph at most once.

Solution:

We will analyze each graph one by one using the Vertical Line Test.

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#### Graph 1: \( y = x \)
- This is a straight line passing through the origin with a slope of 1.
- If we draw any vertical line, it will intersect this graph at exactly one point.
- Conclusion: This graph passes the Vertical Line Test and represents a function.

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#### Graph 2: \( x^2 + y^2 = 9 \)
- This is the equation of a circle centered at the origin with radius 3.
- If we draw a vertical line through any point on the x-axis within the interval \([-3, 3]\), it will intersect the circle at two points (one above and one below the x-axis).
- Conclusion: This graph fails the Vertical Line Test and does not represent a function.

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#### Graph 3: \( y = x^2 \)
- This is a parabola opening upwards with its vertex at the origin.
- If we draw any vertical line, it will intersect this graph at exactly one point.
- Conclusion: This graph passes the Vertical Line Test and represents a function.

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#### Graph 4: \( y^2 = x \)
- This is a parabola opening to the right with its vertex at the origin.
- If we draw a vertical line through any point on the x-axis where \( x \geq 0 \), it will intersect the graph at two points (one above and one below the x-axis).
- Conclusion: This graph fails the Vertical Line Test and does not represent a function.

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#### Graph 5: \( y = \sin x \)
- This is the sine wave, which oscillates between -1 and 1.
- If we draw any vertical line, it will intersect this graph at exactly one point.
- Conclusion: This graph passes the Vertical Line Test and represents a function.

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#### Graph 6: \( x = a \)
- This is a vertical line parallel to the y-axis.
- By definition, a vertical line cannot pass the Vertical Line Test because it intersects itself at infinitely many points.
- Conclusion: This graph fails the Vertical Line Test and does not represent a function.

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Final Answer:

The graphs that represent functions are:
1. \( y = x \)
2. \( y = x^2 \)
3. \( y = \sin x \)

The graphs that do not represent functions are:
1. \( x^2 + y^2 = 9 \)
2. \( y^2 = x \)
3. \( x = a \)

\[
\boxed{y = x, \, y = x^2, \, y = \sin x}
\]