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.