how will the following transformation rule move a preimage? (x,y) -> (-x,-y)

Answer 1

The transformation rule \((x, y) \to (-x, -y)\) represents a reflection through the origin on a Cartesian coordinate plane. Here's how it works:

1. Negation of Coordinates: The x-coordinate and y-coordinate of the point \((x, y)\) are both negated. This means:

   • The x-coordinate changes sign: if \(x\) is positive, it becomes negative, and if \(x\) is negative, it becomes positive. If \(x = 0\), it remains \(0\).
   • The y-coordinate changes sign similarly: if \(y\) is positive, it becomes negative, and if \(y\) is negative, it becomes positive. If \(y = 0\), it remains \(0\).

2. Geometric Interpretation: This transformation effectively rotates the point \((x, y)\) by 180 degrees around the origin \((0, 0)\). For example:

   • The point \((1, 2)\) would move to \((-1, -2)\).
   • The point \((-3, 4)\) would move to \((3, -4)\).

3. Overall Shift: The overall effect is that every point in the plane moves to the opposite side of the origin, maintaining the same distance from the origin.

In summary, the transformation \((x, y) \to (-x, -y)\) moves each point in the coordinate plane to its diagonally opposite location relative to the origin.