A quadratic function y, equals, f, of, x*y*=*f*(*x*) is plotted on a graph and the vertex of the resulting parabola is left parenthesis, minus, 3, comma, minus, 6, right parenthesis(−3,−6). What is the vertex of the function defined as g, of, x, equals, minus, f, of, x, minus, 3*g*(*x*)=−*f*(*x*)−3?

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.

A quadratic function y = f(x) is plotted on a graph and the vertex of the resulting parabola is (-3, -6). What is the vertex of the function defined as g(x) = -f(x) - 3?

Learn how to find the vertex of a transformed quadratic function g(x) = -f(x) - 3. This problem involves key concepts like reflections, vertex transformations, and vertical shifts. Perfect for students in Grades 9-12 exploring advanced quadratic functions.

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