Final answer:
To determine the coordinates of the point (-4, 1) after a 180° counterclockwise rotation around the center (-3, 0), a vector is computed and then inverted to find the image position relative to the center. The image point is located at (-2, -1).
Step-by-step explanation:
The point (-4, 1) is rotated 180° counterclockwise around the center (-3, 0). To determine the coordinates of the rotated point, you can consider the geometric nature of a 180° rotation. This specific transformation is equivalent to a half-turn, which means the point will move along a straight line away from the center of rotation, keeping the same distance, but in the opposite direction.
First, we find the vector from the center of rotation to the point by subtracting the center's coordinates from the point's coordinates:
- x-coordinates: (-4) - (-3) = -1
- y-coordinates: 1 - 0 = 1
Then, we invert this vector to get the new position relative to the center:
- x-coordinates: -(-1) = 1
- y-coordinates: -(1) = -1
Finally, we can add these vector components to the center's coordinates to get the rotated point's coordinates:
- x-coordinates: (-3) + 1 = -2
- y-coordinates: 0 - 1 = -1
The coordinates of the image after a 180° counterclockwise rotation around (-3, 0) are (-2, -1).