https://file.aiquickdraw.com/mathbot/file/17294736439709u1donln.jpg

Let's break down how to solve each part of the problem:

### Part (a): \( y = 2f(x - 3) \)
1. **Horizontal Shift**: The transformation \( f(x - 3) \) means a shift to the **right** by 3 units. So, we move the graph of \( f(x) \) three units to the right.
2. **Vertical Stretch**: The multiplication by 2, \( 2f(x - 3) \), means the graph will stretch **vertically** by a factor of 2. Every y-coordinate on the graph will be multiplied by 2.

To summarize:
- Move the graph of \( f(x) \) right by 3 units.
- Multiply all y-values by 2 to stretch the graph vertically.

### Part (b): \( y = h(-x) - 1 \)
1. **Horizontal Reflection**: The transformation \( h(-x) \) reflects the graph across the **y-axis**. So, the graph will flip horizontally.
2. **Vertical Shift**: The subtraction of 1, \( h(-x) - 1 \), shifts the graph **down** by 1 unit. Every point on the graph will move down 1 unit.

To summarize:
- Reflect the graph of \( h(x) \) across the y-axis.
- Move the entire graph down by 1 unit.

Would you like further details or examples for this? Here are five related questions you might want to consider:

1. How do different transformations (like reflections and stretches) affect the domain and range of a function?
2. Can vertical and horizontal transformations change the overall shape of the graph?
3. What happens if we have a combination of multiple transformations?
4. How would the transformations be different if we had \( f(2x) \) instead of \( 2f(x) \)?
5. Can these transformations affect symmetry properties of a function?

**Tip**: Always apply horizontal transformations (like shifts and reflections) before vertical ones, when multiple transformations are present!

The graph of y = f(x) is shown. Draw the graph of y = 2f(x - 3) and the graph of y = h(x) is shown. Draw the graph of y = h(-x) - 1.

Learn how to transform the graph of functions by shifting, stretching, reflecting, and translating. In this problem, we cover transformations of y = f(x) and y = h(x), applying concepts such as horizontal shifts and vertical stretches. Suitable for students in Grades 9-12.

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