How do you rotate 270 degrees counterclockwise about the origin

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The Correct Answer and Explanation is:

To rotate a point 270 degrees counterclockwise about the origin, there is a specific rule for transforming the coordinates of the point. The rule for rotating a point ((x, y)) 270 degrees counterclockwise is:

[
(x, y) \to (y, -x)
]

Step-by-Step Explanation

1. Understanding Rotations: Rotation of a figure refers to turning the shape around a fixed point—in this case, the origin ((0, 0)). Rotating counterclockwise means the shape turns to the left, while rotating clockwise means it turns to the right.
2. Rotation by 270 Degrees: A full circle is 360 degrees, and rotating 270 degrees counterclockwise is equivalent to rotating 90 degrees clockwise. This is important because the rule for 270 degrees counterclockwise is derived from the understanding of how the coordinates shift with smaller rotations.
3. Coordinate Transformation: For a 270-degree counterclockwise rotation, you apply the rule ((x, y) \to (y, -x)). What this means is:

• The x-coordinate becomes the new y-coordinate.
• The y-coordinate becomes the negative of the original x-coordinate. Example:
• If the original point is ((2, 3)), after a 270-degree counterclockwise rotation, the new coordinates will be:
  [
  (2, 3) \to (3, -2)
  ]
• Another example, for the point ((-1, 4)):
  [
  (-1, 4) \to (4, 1)
  ]

1. Rotation by 270 Degrees Clockwise: If you were to rotate 270 degrees clockwise instead, the rule would be:
   [
   (x, y) \to (-y, x)
   ]
   This rule reflects the rotation in the opposite direction (clockwise).
2. General Tips:

• To visualize a 270-degree counterclockwise rotation, imagine the point on the coordinate plane and think of how it moves: it moves to the left by three-quarters of a turn.
• The 270-degree rotation counterclockwise rule ((x, y) \to (y, -x)) applies to all points in the coordinate plane, whether they are in quadrants I, II, III, or IV.

In summary, a 270-degree counterclockwise rotation about the origin transforms the point ((x, y)) into the coordinates ((y, -x)). This transformation involves switching the x- and y-values, while also changing the sign of the original x-coordinate.