Identify the point that applies a rotation function mapping for P located at (4,1) in a 270-degree counterclockwise rotation about the origin.
To find the new coordinates of point P after a 270-degree counterclockwise rotation about the origin, we can use the rotation formula:
(x', y') = (x*cos(θ) - y*sin(θ), x*sin(θ) + y*cos(θ))
where (x', y') are the new coordinates after the rotation, (x, y) are the original coordinates, and θ is the angle of rotation.
In this case, the original coordinates are (4, 1) and the angle of rotation is 270 degrees.
Using the rotation formula, we can calculate the new coordinates:
(x', y') = (4*cos(270) - 1*sin(270), 4*sin(270) + 1*cos(270))
Since cos(270) = 0 and sin(270) = -1, we can simplify the equation:
(x', y') = (4*0 - 1*(-1), 4*(-1) + 1*0)
(x', y') = (0 + 1, -4 + 0)
(x', y') = (1, -4)
Therefore, the point P located at (4, 1) will move to the point (1, -4) after a 270-degree counterclockwise rotation about the origin.
To apply a 270-degree counterclockwise rotation about the origin, we can use the following rotation matrix:
```
[ cos(theta) -sin(theta) ]
[ sin(theta) cos(theta) ]
```
where `theta` is the angle of rotation in radians.
Since we want to rotate P(4, 1) by 270 degrees counterclockwise, we can let `theta` be `(270 * pi) / 180`, which is equal to `3*pi/2`.
Applying the rotation matrix to point P(4, 1):
```
[ cos(3*pi/2) -sin(3*pi/2) ] [ 0 -1 ] [ 1 ]
[ sin(3*pi/2) cos(3*pi/2) ] * [ 1 0 ] * [ 4 ]
[ 0 1 ] [ 1 ]
```
Calculating:
```
[ 0 -1 ] [ 1 ] [ -1 ]
[ 1 0 ] * [ 4 ] = [ 4 ]
[ 0 1 ] [ 1 ] [ 1 ]
```
Therefore, the point that applies the rotation function mapping for P(4, 1) in a 270-degree counterclockwise rotation about the origin is (-1, 4).
To find the point after a 270-degree counterclockwise rotation about the origin, we can use the rotation matrix for a 270-degree rotation.
The rotation matrix for a counterclockwise rotation of θ degrees about the origin is:
[cos(θ) -sin(θ)]
[sin(θ) cos(θ)]
In this case, since we want a 270-degree counterclockwise rotation, θ is equal to 270 degrees.
Plugging θ = 270 into the rotation matrix, we get:
[cos(270) -sin(270)]
[sin(270) cos(270)]
Evaluating the trigonometric functions for 270 degrees, we get:
[0 -1]
[1 0]
Now we can apply this matrix to the coordinates of P, which are (4, 1):
[0 -1] [4] = [0*(4) + (-1)*(1)] = [-1]
[1 0] [1] [1*(4) + 0*(1)] [4]
Therefore, the point after a 270-degree counterclockwise rotation about the origin is (-1, 4).