The problem involves determining the image of point $Q$ after it is rotated by $-150^\circ$ around a center of rotation that is clearly marked on the diagram. The negative angle indicates a clockwise rotation.

Steps to Solve:

1. Identify the center of rotation: It's the black dot labeled "Center" in the middle of the diagram.

2. Locate point $Q$: $Q$ is marked as a blue dot.

3. Visualize a $-150^\circ$ (clockwise) rotation: Starting from $Q$, we need to rotate $Q$ in the clockwise direction around the center by $150^\circ$.

   Based on a rough visual estimate of $150^\circ$, the image of $Q$ should land near one of the labeled points (A, B, C, or D).

Analysis:

Looking at the relative positions, the point that seems to match a $150^\circ$ clockwise rotation is point C, as it aligns with the direction of a large clockwise angle from $Q$.

Answer:

The image of point $Q$ after a $-150^\circ$ rotation appears to be point C.

Would you like further clarification on rotation transformations? Here are 5 related questions for further practice:

1. How do you determine the angle of rotation in a counterclockwise direction?
2. What is the difference between clockwise and counterclockwise rotations?
3. How can you rotate a point using trigonometric functions?
4. What is the image of a point under a $90^\circ$ counterclockwise rotation?
5. How can you find the coordinates of a rotated point without a graph?

Tip: For precise rotations, try sketching smaller intermediate angles (like $90^\circ$) to help visualize the full rotation more clearly.