Here is a graph of y=f(x). On the same graph, sketch y=-f(2-x)-3

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.

Here is a graph of y=f(x). On the same graph, sketch y=-f(2-x)-3.

Learn how to sketch y = -f(2 - x) - 3 based on the graph of y = f(x). This detailed guide walks through horizontal reflections, shifts, and vertical transformations, making it ideal for high school students studying graph transformations in algebra.

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