Final answer:
The transformations (x, -y), (-x, y), and (-x, -y) reflect a point vertically downward, horizontally to the left, and diagonally across the coordinate system, respectively. Calculating change (A) involves subtracting the initial value from the final value. The distance of a point to the origin remains the same under rotations, represented by x² + y².
Step-by-step explanation:
When applying transformations to a point (x, y) in the coordinate system:
- The transformation (x, -y) reflects the point across the x-axis, effectively switching the point from vertically upward to vertically downward in the coordinate system, or vice versa.
- The transformation (-x, y) reflects the point across the y-axis, moving it from the positive x direction to the negative x direction of the coordinate system, or vice versa.
- The combination of both, (-x, -y), results in a point reflection about the origin, sending the original point diagonally across the coordinate system to the opposite quadrant.
When substituting the x and t values of chosen points into an equation, it is crucial to use the final value minus the initial value to calculate the change (A). This will give us the amount of change between the two points in terms of position or time.
To show that the distance of point P to the origin is invariant under rotations of the coordinate system, you would use the relation x² + y². This sum represents the distance squared from the origin to the point P, which does not change regardless of the angle of rotation.