Shape and Space: Transformations and Coordinate Geometry
The coordinate plane has four quadrants, and every point is named by an ordered pair (x, y): x first (horizontal), y second (vertical). Signs identify the quadrant: (+, +) is Quadrant I, (−, +) is II, (−, −) is III, and (+, −) is IV. Points lying on an axis belong to no quadrant.
Transformations change a figure's position but not its size or shape, so the image is congruent to the original. A translation slides every point the same distance in the same direction. A reflection flips the figure over a line, producing a mirror image. A rotation turns the figure about a fixed point, usually the origin, by 90°, 180°, or 270°.
Coordinate rules worth memorizing:
- Reflection over the x-axis: (x, y) → (x, −y)
- Reflection over the y-axis: (x, y) → (−x, y)
- Rotation 180° about the origin: (x, y) → (−x, −y)
- Rotation 90° clockwise about the origin: (x, y) → (y, −x)
Symmetry questions ask how many lines divide a figure into matching mirror halves.
Traps: writing (y, x) instead of (x, y); reflecting over the wrong axis; assuming a translation or rotation reverses orientation the way a reflection does; and rotating about a vertex when the question specifies the origin.
Worked example: start with (−3, 5). Reflect it over the x-axis: the x-coordinate stays and the y-coordinate changes sign, giving (−3, −5). Rotate that result 180° about the origin: both signs flip, giving (3, 5). A quick sketch confirms both moves.