Final answer:
Point P(-4, 6) undergoes transformation by changing signs of coordinates, moving downwards, rightward, and diagonally opposite without changing its distance from the origin.
Step-by-step explanation:
The student is looking at transformations of a point P in the Cartesian coordinate system. Understanding these transformations helps clarify how position and distance change under various operations.
Transformations of Point P(-4, 6)
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- a. If we change the sign of the y-coordinate while keeping the x-coordinate the same, the new coordinates of P would be (-4, -6). This transformation moves the point vertically downwards in the coordinate system.
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- b. If we change the sign of the x-coordinate but keep the y-coordinate the same, point P becomes (4, 6). This transformation shifts the point horizontally to the right side of the coordinate system.
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- c. When we change the signs of both coordinates, point P ends up at (4, -6). This moves the point diagonally to the quadrant opposite to its initial position.
Invariance of Distance
It's important to know that the distance of point P from the origin is given by \(\sqrt{x^2 + y^2}\). Under rotations of the coordinate system, this distance does not change because the new coordinates (x', y') will satisfy \(x'^2 + y'^2 = x^2 + y^2\). This is because rotation is a distance-preserving transformation or an isometry.
For example, the distance of the original point P from the origin is the square root of (\(-4\))^2 + 6^2, which will remain the same after rotating the coordinate system.