https://file.aiquickdraw.com/mathbot/file/1726987242789rmsnhxhs.jpg

To analyze the problem, we have the graph of \( f(x) \) shown on the right and two transformations defined:

1. \( g(x) = -2f(x + 2) - 3 \)
2. \( h(x) = f(-x) \)

### Part (a): \( g(x) = -2f(x + 2) - 3 \)
This transformation involves a series of steps:

- **Horizontal shift**: The term \( f(x + 2) \) indicates a shift of the graph 2 units to the left.
- **Vertical stretch and reflection**: The coefficient \( -2 \) means the graph is vertically stretched by a factor of 2 and reflected over the x-axis.
- **Vertical shift**: The term \( -3 \) means the graph is shifted down 3 units.

### Part (b): \( h(x) = f(-x) \)
This transformation reflects the graph of \( f(x) \) across the y-axis.

### Steps to graph the transformations:
- For part (a), take the points on the original graph, shift them left by 2 units, reflect over the x-axis, stretch vertically by 2, and shift down by 3.
- For part (b), simply reflect all points across the y-axis.

Would you like a more detailed step-by-step guide on how to graph these transformations, or any specific questions about interpreting them?

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Here are 5 follow-up questions:
1. How does reflecting a graph over the x-axis affect its shape?
2. What effect does shifting a graph left or right have on its equation?
3. How do we interpret vertical stretching and compressing in transformations?
4. How does adding or subtracting constants outside a function affect its vertical position?
5. What happens when a function is reflected across the y-axis?

**Tip:** Always apply transformations to the graph in the correct order (horizontal shifts, stretches/reflections, and then vertical shifts).

The graph of f(x) is shown, and two transformations are given: g(x) = -2f(x + 2) - 3 and h(x) = f(-x).

Analyze the graph of f(x) and apply transformations to find g(x) = -2f(x + 2) - 3 and h(x) = f(-x). This problem explores reflections, vertical shifts, and stretches. Suitable for high school math students in Grades 9-12.

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