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.