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?

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).