Identify the point that applies a rotation function mapping for P located at (1,4) in a 270-degree counterclockwise rotation about the origin.
The point that applies a rotation function mapping for P located at (1,4) in a 270-degree counterclockwise rotation about the origin is (-4, 1).
To apply a rotation function mapping for point P located at (1,4) in a 270-degree counterclockwise rotation about the origin, we can use the rotation formula:
(x', y') = (x * cosθ - y * sinθ, x * sinθ + y * cosθ)
where (x', y') are the coordinates of the new rotated point, (x, y) are the coordinates of the original point, and θ is the angle of rotation in radians.
In this case, the angle of rotation is 270 degrees, which is equivalent to 3π/2 radians.
Let's substitute the values into the formula:
(x', y') = (1 * cos(3π/2) - 4 * sin(3π/2), 1 * sin(3π/2) + 4 * cos(3π/2))
= (1 * 0 - 4 * (-1), 1 * (-1) + 4 * 0)
= (0 + 4, -1 + 0)
= (4, -1)
Therefore, the point that applies a rotation function mapping for P located at (1,4) in a 270-degree counterclockwise rotation about the origin is (4, -1).
To find the point after a rotation function mapping, we can use the following steps:
1. Identify the coordinates of the point to be rotated. In this case, the point is P located at (1,4).
2. Determine the angle of rotation. In this case, the rotation is 270 degrees counterclockwise.
3. Recall that, for a counterclockwise rotation about the origin, we can use the rotation formula:
- x' = x * cos(theta) - y * sin(theta)
- y' = x * sin(theta) + y * cos(theta)
4. Substitute the values into the rotation formula:
For x': x' = 1 * cos(270) - 4 * sin(270)
For y': y' = 1 * sin(270) + 4 * cos(270)
5. Simplify the equations:
For x': x' = 1 * 0 - 4 * (-1) = 0 + 4 = 4
For y': y' = 1 * (-1) + 4 * 0 = -1 + 0 = -1
So, the point after a 270-degree counterclockwise rotation about the origin is (4, -1).