Here is the graph of the function $g(x) = -2\sqrt{9 - x^2} + 6$, which matches the given equation from the image. The shape of the graph is a downward-facing semi-circle, as expected for a function based on the square root of $9 - x^2$, shifted upward by 6 units and scaled by a factor of -2.

Would you like further details on how this function was derived, or any other clarification?

Here are 5 related questions to expand your understanding:

1. How does changing the coefficient of the square root (e.g., from -2 to a different value) affect the graph?
2. What effect does modifying the constant inside the square root (e.g., changing 9 to another value) have on the graph?
3. How would shifting the graph vertically by adjusting the constant outside the square root change the graph?
4. How does the domain of $g(x)$ affect its graph, and why is it limited to the range $[-3, 3]$?
5. How would the graph change if we replaced the square root with a different root (e.g., cube root)?

Tip: For square root functions involving expressions like $9 - x^2$, always consider the domain restrictions (here, $-3 \leq x \leq 3$) since the square root of a negative number is undefined in the real number system.