Which graph represents a function?

Question

On a coordinate plane, a straight vertical line crosses the y-axis 0 times and the x-axis 1 time.

On a coordinate plane, a curved line crosses the y-axis 2 times and the x-axis 1 time.

On a coordinate plane, 2 curved lines are shown. Both lines touch the y-axis 1 time and the x-axis 1 time.

On a coordinate plane, a straight horizontal line crosses the y-axis 1 time and the x-axis 0 times.

Answer 1

To determine which graph represents a function, we can use the **Vertical Line Test**. The Vertical Line Test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.

Let's analyze each option based on this test:

1. **Vertical Line Crossing the y-axis 0 times and the x-axis 1 time:**
- This is a vertical line.
- Vertical lines fail the Vertical Line Test because they intersect any vertical line at infinitely many points (the line itself).
- Therefore, this is **not** a function.

2. **Curved Line Crossing the y-axis 2 times and the x-axis 1 time:**
- A curve that crosses the y-axis more than once implies there are multiple y-values for the same x-value.
- It fails the Vertical Line Test because at least one vertical line will intersect the curve at two points.
- Therefore, this is **not** a function.

3. **Two Curved Lines Touching the y-axis 1 time and the x-axis 1 time:**
- If these two curved lines are distinct and do not touch or overlap each other, but they each touch the y-axis once and the x-axis once, we need to check if a vertical line intersects them at more than one point.
- If any vertical line intersects both curves at any point, it fails the Vertical Line Test.
- Given they are two separate curved lines, they likely intersect at more than one point when tested vertically.
- Therefore, this is **not** a function.

4. **Horizontal Line Crossing the y-axis 1 time and the x-axis 0 times:**
- This is a horizontal line.
- Horizontal lines pass the Vertical Line Test because any vertical line will intersect a horizontal line at exactly one point.
- Therefore, this **is** a function.

Based on this analysis, the graph that represents a function is: **the straight horizontal line that crosses the y-axis 1 time and the x-axis 0 times**.