To find the coordinates of $A''$, the final position of point $A$ after the transformations, we need to follow these steps:

1. Rotate $A$ by $90^\circ$ counterclockwise about the origin.

   Given point $A(-3, -3)$, when we rotate any point $(x, y)$ by $90^\circ$ counterclockwise about the origin, the new coordinates $(x', y')$ become: $(x', y') = (-y, x)$

   Applying this to point $A(-3, -3)$: $A' = (-(-3), -3) = (3, -3)$

2. Translate $A'$ by the vector $\langle -6, -3 \rangle$.

   To translate a point by a vector, we add the vector components to the coordinates of the point. For $A'(3, -3)$ and the translation vector $\langle -6, -3 \rangle$: $A'' = (3 + (-6), -3 + (-3)) = (-3, -6)$

Final Answer:

The coordinates of $A''$ are $(-3, -6)$.

Would you like more details on any of the steps?

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Here are five related questions that build on this solution:

1. What are the coordinates of $B''$ after applying the same transformations?
2. What are the coordinates of $C''$ after the same transformations?
3. How would the process differ if the rotation were $180^\circ$ instead of $90^\circ$?
4. How can we determine the overall effect of a rotation followed by a translation on any point in the plane?
5. What transformation matrix could represent the combined rotation and translation?

Tip: When rotating points around the origin, use the general rotation formulas for clarity and avoid errors with negative coordinates.