Here is the graph of the recursive function $g_n = 9n \times \frac{1}{2}$ for $n > 1$ and $g_1 = 12$. The values of $g_n$ grow linearly as $n$ increases, following the defined pattern.

Let me know if you would like further details or adjustments.

Here are some related questions you might consider:

1. How does the function behave for larger values of $n$?
2. What would happen if you changed the initial condition, $g_1$?
3. How does halving the term $9n$ affect the function's growth rate?
4. Can this function be modified for different ranges of $n$?
5. What if we applied a different operation instead of multiplying by $\frac{1}{2}$?

Tip: Recursive functions are powerful tools for defining sequences and processes step by step. Always ensure the base case (like $g_1$) is clear to prevent undefined behavior!