We are tasked with determining the coordinates of point $Q'$ after rectangle $PQRS$ is rotated $90^\circ$ counterclockwise about the origin.

Step-by-Step Solution:

1. Coordinates of Point $Q$:
   From the graph, $Q$ is located at $(5, 3)$.

2. Rule for $90^\circ$ Counterclockwise Rotation:
   When a point $(x, y)$ is rotated $90^\circ$ counterclockwise about the origin, its new coordinates become $(-y, x)$.

3. Apply the Rule:
   For $Q(5, 3)$, the rotation transforms it into: $Q'(x', y') = (-y, x) = (-3, 5)$

4. Result:
   The coordinates of $Q'$ are $(-3, 5)$.

Thus, the correct answer is $Q'(-3, 5)$.

Would you like further details or explanations? Here are related questions you might find helpful:

1. What are the new coordinates of other points $P, R, S$ after the same rotation?
2. How does the transformation rule differ for $90^\circ$ clockwise rotation?
3. How can this process be visualized on the coordinate plane?
4. What happens if the rectangle is rotated $180^\circ$ instead?
5. Can we find the rotation matrix for $90^\circ$ counterclockwise transformations?

Tip: Always verify the rotation rule by substituting a simple point like $(1, 0)$ or $(0, 1)$ to ensure you apply it correctly!