To find the coordinates of the image of the point [tex]$(-4, 1)$[/tex] after rotating it 180 degrees counterclockwise about the origin [tex]$(0,0)$[/tex], let's go through the necessary steps:
### Step-by-step Solution:
1. Understand the Rotation:
Rotating a point [tex]$(x, y)$[/tex] by 180 degrees counterclockwise about the origin is equivalent to changing its coordinates to [tex]$(-x, -y)$[/tex]. In other words, both the [tex]$x$[/tex] and [tex]$y$[/tex] coordinates of the original point will be negated.
2. Original Point:
The coordinates of the original point are [tex]$(-4, 1)$[/tex].
3. Negate the Coordinates:
- The original [tex]$x$[/tex]-coordinate is [tex]$-4$[/tex], so negating it gives:
[tex]\[
-(-4) = 4
\][/tex]
- The original [tex]$y$[/tex]-coordinate is [tex]$1$[/tex], so negating it gives:
[tex]\[
-(1) = -1
\][/tex]
4. Resulting Coordinates:
After applying the rotation, the new coordinates of the point are [tex]$(4, -1)$[/tex].
### Conclusion:
The coordinates of the image of the point [tex]$(-4, 1)$[/tex] after rotating it 180 degrees counterclockwise about the origin [tex]$(0, 0)$[/tex] are [tex]$(4, -1)$[/tex].
Thus, the correct option is:
[tex]\[ \boxed{(4, -1)} \][/tex]
