Here's the sketch of the graph for $f(x) = \frac{3x^2 + 1}{x^2 - 9}$. Key points from the analysis are:

• Domain: All real $x$, except $x = 3$ and $x = -3$.
• Vertical Asymptotes: Located at $x = 3$ and $x = -3$.
• Horizontal Asymptote: $y = 3$.
• Intercepts: The y-intercept is at $(0, -\frac{1}{9})$; no x-intercepts.

Would you like further details on any part of this analysis, or have any specific questions?

Here are five related questions for deeper exploration:

1. How would the graph change if the numerator were a linear function?
2. How would different values for the coefficients affect the horizontal asymptote?
3. What happens to the graph near the asymptotes?
4. How does the degree of the polynomial in the numerator affect the shape of the graph?
5. Could there be any transformations applied to this function to alter its intercepts?

Tip: When finding vertical asymptotes, always check where the denominator is zero, as it indicates undefined points in the function.