The problem involves transforming a quadratic function and determining how the vertex changes.

Given:

• The function $f(x)$ is quadratic, and its vertex is at $(-3, -6)$.
• The new function is defined as $g(x) = -f(x) - 3$.

Step-by-step solution:

1. Original function: Since $f(x)$ is quadratic and its vertex is $(-3, -6)$, we can assume the general form of the quadratic function is: $f(x) = a(x + 3)^2 - 6$ Here, $a$ is the leading coefficient of the quadratic function, and the vertex form of a quadratic is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. In this case, $h = -3$ and $k = -6$.

2. Transformation: Now, let's analyze the new function $g(x) = -f(x) - 3$.

   • First, we reflect $f(x)$ across the x-axis by multiplying by -1: $-f(x) = -[a(x + 3)^2 - 6] = -a(x + 3)^2 + 6$
   • Then, we subtract 3 from this result: $g(x) = -a(x + 3)^2 + 6 - 3 = -a(x + 3)^2 + 3$

3. New vertex: The new function $g(x) = -a(x + 3)^2 + 3$ is still in the vertex form, and the vertex of this function is at $(-3, 3)$.

   • The horizontal shift is the same as for $f(x)$, so the x-coordinate of the vertex remains $-3$.
   • The vertical shift changes because of the transformations. The original function had a vertex at $(-3, -6)$. Reflecting it across the x-axis moved the vertex to $(-3, 6)$, and then subtracting 3 moved the vertex down to $(-3, 3)$.

Final Answer:

The vertex of $g(x) = -f(x) - 3$ is $\boxed{(-3, 3)}$.

Would you like more details or have any questions about this transformation?

Here are 5 related questions to expand your understanding:

1. How does changing the leading coefficient $a$ affect the vertex of a quadratic function?
2. What is the effect of reflecting a quadratic function across the x-axis?
3. How does adding or subtracting constants outside the function affect its graph?
4. How can you find the vertex of a quadratic function given its standard form?
5. What happens to the symmetry of the parabola when you reflect it?

Tip: The vertex form of a quadratic function is a powerful tool for quickly identifying key transformations like shifts and reflections.