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Let's tackle the two parts of this question step by step:

### Part (a) 
We need to find the coordinates of point A for the graph of \( y = -f(x - 2) \).

- Point A on the original graph \( y = f(x) \) has coordinates \( (2, -1) \).
- The transformation \( y = -f(x - 2) \) involves two changes:
  1. **Horizontal shift**: The \( x - 2 \) means a shift to the right by 2 units.
  2. **Reflection in the x-axis**: The negative sign in front of \( f(x) \) reflects the graph vertically.

So, applying these transformations:
- Shift \( A(2, -1) \) 2 units to the right: the new x-coordinate becomes \( 2 + 2 = 4 \).
- Reflect the y-coordinate: the original y-coordinate is \( -1 \), and reflecting it gives \( 1 \).

Thus, the new coordinates of point A are \( (4, 1) \).

### Part (b) 
We need to find the coordinates of point B for the graph of \( y = 2f(3x) + 1 \).

- Point B on the original graph \( y = f(x) \) has coordinates \( (6, 4) \).
- The transformation \( y = 2f(3x) + 1 \) involves three changes:
  1. **Horizontal compression**: The factor of \( 3x \) compresses the graph horizontally by a factor of \( \frac{1}{3} \).
  2. **Vertical stretch**: The factor of \( 2 \) stretches the graph vertically by 2.
  3. **Vertical shift**: The \( +1 \) shifts the graph 1 unit upward.

Let's apply these transformations step by step:
- Compress the x-coordinate by a factor of 3: the new x-coordinate becomes \( \frac{6}{3} = 2 \).
- Stretch the y-coordinate by a factor of 2: the original y-coordinate is \( 4 \), so after stretching, it becomes \( 2 \times 4 = 8 \).
- Shift the y-coordinate up by 1: the new y-coordinate becomes \( 8 + 1 = 9 \).

Thus, the new coordinates of point B are \( (2, 9) \).

### Summary:
- (a) The new coordinates of A are \( (4, 1) \).
- (b) The new coordinates of B are \( (2, 9) \).

Would you like more details on these transformations, or do you have any further questions?

Here are 5 related questions you can explore:
1. What would be the new coordinates of point B for the graph of \( y = -f(x + 1) \)?
2. How does a horizontal stretch affect the coordinates of a point?
3. What happens to the graph if we only reflect \( f(x) \) in the y-axis?
4. Can you find the equation of a transformation that moves A to \( (0, 2) \)?
5. What would be the effect of the transformation \( y = f(2x + 3) - 5 \) on point A?

**Tip:** When applying transformations, always handle horizontal changes before vertical ones for clarity.

Part (a): Write down the coordinates of A for the graph of y = -f(x - 2). Part (b): Write down the coordinates of B for the graph of y = 2f(3x) + 1.

Learn how to find the new coordinates of points A and B after applying transformations to a graph. This problem involves transformations such as vertical reflections, horizontal shifts, and stretches/compressions. Suitable for high school students in Grades 9-12.

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