To solve this problem, we need to determine the image of point $A$ located at $(-2, 1)$ after a $270^\circ$ clockwise rotation about the point $(3, -3)$.

Here’s the step-by-step approach:

1. Translate the Points: Shift the entire setup so that the center of rotation, $(3, -3)$, moves to the origin. This is done by translating both points by $(-3, +3)$:

   • New coordinates for $A$ after translation: $(-2 - 3, 1 + 3) = (-5, 4)$

2. Rotate $270^\circ$ Clockwise: A $270^\circ$ clockwise rotation around the origin is equivalent to a $90^\circ$ counterclockwise rotation. The transformation rule for a $90^\circ$ counterclockwise rotation is: $(x, y) \rightarrow (-y, x)$ Applying this to $(-5, 4)$: $(-5, 4) \rightarrow (-4, -5)$

3. Translate Back: Now, move the rotated point back to the original position by translating it by $(+3, -3)$: $(-4 + 3, -5 - 3) = (-1, -8)$

Thus, the image of point $A$ after a $270^\circ$ clockwise rotation about $(3, -3)$ is $(-1, -8)$.

Would you like more details on any step, or do you have questions?

Here are 5 related questions you might explore:

1. How would the result change with a $90^\circ$ clockwise rotation instead?
2. What is the effect of rotating a point $180^\circ$ about a given point?
3. How can rotation transformations be applied to a line segment or shape instead of a single point?
4. How does translating affect the coordinates before and after rotation?
5. Can rotation transformations be applied to three-dimensional coordinates?

Tip: When working with rotation, translation can simplify the process by making the rotation center the origin. This makes applying rotation rules straightforward.