The question asks us to use the vertical line test to determine which of the graphs represent functions.

Vertical Line Test:

The vertical line test states that if any vertical line intersects a graph at more than one point, then the graph does not represent a function. If every vertical line intersects the graph at most one point, then the graph does represent a function.

Analyzing Each Graph:

1. Top-left graph:

   • This is a parabolic curve opening upward, which resembles a quadratic function.
   • Any vertical line drawn through this graph would only intersect it at one point, so it passes the vertical line test.
   • Conclusion: This graph represents a function.

2. Top-right graph:

   • This is a curve with multiple bends, possibly a higher-order polynomial.
   • Any vertical line will intersect this graph at only one point throughout the graph's domain, so it also passes the vertical line test.
   • Conclusion: This graph represents a function.

3. Bottom-left graph:

   • This is a straight diagonal line, which resembles a linear function.
   • Any vertical line would intersect this line at only one point, so it passes the vertical line test.
   • Conclusion: This graph represents a function.

4. Bottom-right graph:

   • This graph has a curve that appears to double back horizontally, making it fail the vertical line test at certain points (e.g., vertical lines near the origin will intersect the graph at more than one point).
   • Conclusion: This graph does not represent a function.

Summary:

• Top-left, top-right, and bottom-left graphs all represent functions.
• Bottom-right graph does not represent a function.

Would you like more details or have any questions?

Here are five questions that build on this explanation:

1. How does the vertical line test relate to the definition of a function?
2. What are some examples of real-world scenarios where a relation is not a function?
3. How does the horizontal line test differ from the vertical line test, and what does it check?
4. Can a graph fail the vertical line test but still represent a valid mathematical relation?
5. What types of functions commonly have graphs that pass the vertical line test?

Tip: For a relation to be a function, each input (x-value) must correspond to exactly one output (y-value).