To determine the image of the point $(-5, -4)$ after a rotation of $180^\circ$ counterclockwise about the origin, we can use the rotation rule:

When a point $(x, y)$ is rotated $180^\circ$ counterclockwise about the origin, its new coordinates are $(-x, -y)$.

Solution:

Given the point $(-5, -4)$: $\text{New coordinates} = (-(-5), -(-4)) = (5, 4)$

Final Answer:

The image of the point $(-5, -4)$ after a rotation of $180^\circ$ counterclockwise about the origin is $(5, 4)$.

Would you like a visual representation or further explanation?
Here are 5 related questions you might find interesting:

1. What are the coordinates of a point after a $90^\circ$ counterclockwise rotation?
2. How do you find the image of a point after reflection over the y-axis?
3. How do rotation matrices work in transformations?
4. What happens to the distance of a point from the origin after rotation?
5. Can a $270^\circ$ counterclockwise rotation be equivalent to a $90^\circ$ clockwise rotation?

Tip: Always check the rotation direction (clockwise or counterclockwise) as it affects the result.