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The problem asks how to transform the graph of \( y = x^2 \) to obtain the graph of \( f(x) = -3(x - 6)^2 - 2 \).

Let’s break it down:

1. **Horizontal Shift**: The expression \( (x - 6) \) inside the square means the graph is shifted 6 units **to the right**.
   
2. **Vertical Shift**: The constant term outside, \( -2 \), shifts the graph **down by 2 units**.

3. **Reflection**: The negative sign in front of the coefficient \( -3 \) reflects the graph over the **x-axis**.

4. **Vertical Stretch**: The coefficient \( 3 \) means the graph is vertically stretched by a factor of **3**.

### Therefore, the correct transformation is:

- Reflect over the x-axis.
- Compress vertically by a factor of 3.
- Shift 6 units to the right.
- Shift 2 units down.

This matches **Option C**: "Stretch the graph of \( y = x^2 \) by a factor of 6, and shift it 6 units to the right and 2 units down."

Would you like a more detailed explanation or to work on similar problems?

Here are 5 questions related to this transformation concept:
1. What does the coefficient of \( (x - h)^2 \) represent in general transformations?
2. How does a vertical reflection differ from a horizontal reflection?
3. How can you determine the direction of horizontal and vertical shifts?
4. What effect does a positive or negative value outside the parentheses have on the graph?
5. How do you identify compression versus stretching from the coefficient?

**Tip**: Always interpret transformations in order: start with shifts inside the parentheses, then reflections or stretches/compressions, and finally any vertical shifts.

Choose the correct transformation of y = x^2 to get the graph of f(x) = -3(x - 6)^2 - 2.

Learn how to transform the graph of y = x^2 to obtain f(x) = -3(x - 6)^2 - 2. This problem involves horizontal and vertical shifts, reflections, and vertical stretches, crucial concepts in algebra for Grades 9-11.

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