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.