To sketch $y = -f(2 - x) - 3$ based on the given graph of $y = f(x)$, let's break down the transformations step by step:

1. Start with $f(x)$: You already have the graph of $y = f(x)$.

2. Horizontal reflection and shift $f(2 - x)$:

   • $f(2 - x)$ involves two transformations:
     • Reflection across the vertical axis (because of the $-x$).
     • Horizontal shift to the right by 2 units (since $f(2 - x)$ shifts the graph horizontally).

3. Vertical reflection $-f(2 - x)$:

   • The negative sign in front of $f(2 - x)$ reflects the graph vertically (across the x-axis).

4. Vertical shift downward by 3 units $-f(2 - x) - 3$:

   • Finally, subtracting 3 shifts the entire graph down by 3 units.

Summary of transformations:

• Reflect the graph horizontally across the y-axis.
• Shift it 2 units to the right.
• Reflect it vertically across the x-axis.
• Shift the entire graph downward by 3 units.

If you'd like, I can describe further details about each step, or you can upload a photo of the original graph for a more specific sketch.

Would you like more details, or have any questions about the transformations?

Five related questions:

1. How do horizontal shifts affect the graph of a function?
2. What does reflecting a function across the y-axis mean geometrically?
3. How do you identify vertical shifts from a function's equation?
4. What is the effect of multiplying a function by a negative sign?
5. How would you graph $y = f(x) + c$ for any constant $c$?

Tip:

Always apply transformations in the correct order: start with reflections and shifts inside the function before handling external transformations like vertical shifts and reflections.