To determine which graph represents $Y$ as a function of $X$, we need to recall the definition of a function. For a graph to represent $Y$ as a function of $X$, it must pass the Vertical Line Test. This means that a vertical line drawn anywhere on the graph should intersect it at most once.

Let's analyze each option:

1. First Graph: This appears to have multiple vertical lines that intersect the graph in more than one place, so it likely fails the vertical line test.
2. Second Graph: This graph seems to have a curve that also would fail the vertical line test due to multiple intersections with some vertical lines.
3. Third Graph: This graph passes the vertical line test, as any vertical line would intersect it only once, indicating that it represents $Y$ as a function of $X$.
4. Fourth Graph: This is a circle, and any vertical line through the center would intersect it twice, so it does not represent $Y$ as a function of $X$.

Based on this analysis, the third graph is the correct answer, as it represents $Y$ as a function of $X$.

Would you like more details or have any questions?

Here are some related questions for further understanding:

1. What is the Vertical Line Test, and why does it determine if a relation is a function?
2. How can we identify functions just by looking at their equations instead of graphs?
3. What are some examples of graphs that always represent functions of $X$?
4. How can transformations (like shifts and reflections) affect whether a graph represents a function?
5. Can a circle ever represent a function if we modify it?

Tip: Remember that passing the Vertical Line Test is crucial to identifying whether a graph represents a function in terms of $Y$ as a function of $X$.