The equation given is:

\[
y = -2f(x - 3) - 2
\]

This equation represents a transformation of the function \( f(x) \). Let's break down the transformations step-by-step:

1. **Horizontal Shift**: The term \( (x - 3) \) inside the function suggests a horizontal shift. The graph of \( f(x) \) is shifted **3 units to the right**.
   
2. **Vertical Stretch and Reflection**: The multiplication by -2 outside the function means two things:
   - The graph is **stretched vertically** by a factor of 2.
   - The negative sign indicates a **reflection across the x-axis**.
   
3. **Vertical Shift**: The term \(-2\) outside the function suggests a **shift 2 units downward**.

### Summary of transformations:
- The graph is shifted **3 units to the right**.
- It is **reflected** across the x-axis and **stretched vertically** by a factor of 2.
- The graph is shifted **2 units downward**.

Would you like more details or have any questions on this?

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Here are five questions to expand your understanding:
1. What happens to the graph if the coefficient of \( f(x) \) was positive instead of negative?
2. How would the graph change if \( x - 3 \) was replaced with \( x + 3 \)?
3. How does the transformation affect the function’s domain and range?
4. What would the transformations look like if \( f(x) \) was a quadratic function, say \( f(x) = x^2 \)?
5. Can you describe the effect of each transformation using an example of a specific function?

**Tip**: Always apply transformations in this order: horizontal shifts, reflections, stretches, and vertical shifts.

Explore the transformations of the function y = -2f(x - 3) - 2. This example covers horizontal shifts, vertical stretches, reflections across the x-axis, and vertical shifts, offering a step-by-step breakdown suitable for Grades 9-11.

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