Overview
Concise notes on geometric transformations: translations, reflections, and rotations on the coordinate plane. Includes rules, coordinate formulas, and worked examples.
Translations
- Translation moves a figure without changing size or orientation.
- Directions: left/right affect x; up/down affect y.
- Right by a units: add a to x; left by a units: subtract a from x.
- Up by b units: add b to y; down by b units: subtract b from y.
- Combined moves are applied to x and y independently.
Translation Coordinate Rules
| Original | Move | New Point |
|---|
| (x, y) | Right a | (x + a, y) |
| (x, y) | Left a | (x − a, y) |
| (x, y) | Up b | (x, y + b) |
| (x, y) | Down b | (x, y − b) |
| (x, y) | Right a, Down b | (x + a, y − b) |
| (x, y) | Left a, Down b | (x − a, y − b) |
Examples
- Triangle A(−4,1), B(−2,3), C(−1,1) → right 5, down 4:
- A′(1,−3), B′(3,−1), C′(4,−3)
- Quadrilateral A(1,1), B(2,4), C(5,4), D(6,1) → left 7, down 6:
- A′(−6,−5), B′(−5,−2), C′(−2,−2), D′(−1,−5)
Reflections
- Reflection flips a figure across a line (x-axis, y-axis) or the origin.
- Over x-axis: x unchanged; y changes sign.
- Over y-axis: y unchanged; x changes sign.
- About origin: both coordinates change sign; equivalent to x-axis then y-axis reflection (order irrelevant).
Reflection Coordinate Rules
| Reflection | Rule | Effect on Quadrant |
|---|
| Over x-axis | (x, y) → (x, −y) | I ↔ IV, II ↔ III |
| Over y-axis | (x, y) → (−x, y) | I ↔ II, IV ↔ III |
| About origin | (x, y) → (−x, −y) | I ↔ III, II ↔ IV |
Examples
- Reflect over x-axis: A(−6,1), B(−4,4), C(−1,1) → A′(−6,−1), B′(−4,−4), C′(−1,−1).
- Reflect over y-axis: A(2,1), B(3,3), C(5,4), D(6,1) → A′(−2,1), B′(−3,3), C′(−5,4), D′(−6,1).
- Reflect about origin: A(−4,−1), B(−1,1), C(−1,−5) → A′(4,1), B′(1,−1), C′(1,5).
Rotations
- Rotations are about the origin unless stated; preserve size and shape.
- 90° clockwise (CW): swap coordinates; new y negated.
- 90° counterclockwise (CCW): swap coordinates; new x negated.
- 180° (CW or CCW): both coordinates negated; same as origin reflection.
Rotation Coordinate Rules
| Rotation | Rule | Quadrant Movement (from I) |
|---|
| 90° CW | (x, y) → (y, −x) | I → IV |
| 90° CCW | (x, y) → (−y, x) | I → II |
| 180° | (x, y) → (−x, −y) | I → III |
Examples
- 90° CW: A(2,1), B(5,4), C(5,1) → A′(1,−2), B′(4,−5), C′(1,−5).
- 90° CCW: A(2,1), B(5,1), C(7,5) → A′(−1,2), B′(−1,5), C′(−5,7).
- 180°: A(2,1), B(5,1), C(3,−3), D(1,3) → A′(−2,−1), B′(−5,−1), C′(−3,3), D′(−1,−3).
Quadrants Reference
- Quadrant I: x > 0, y > 0
- Quadrant II: x < 0, y > 0
- Quadrant III: x < 0, y < 0
- Quadrant IV: x > 0, y < 0
Key Terms & Definitions
- Translation: Shifts a figure horizontally/vertically without rotation or reflection.
- Reflection: Flip across a line (x-axis, y-axis) or about the origin; creates a mirror image.
- Rotation: Turn a figure about the origin by a specified angle and direction.
- Line of symmetry: Line acting as a mirror in a reflection (e.g., x-axis or y-axis).
Action Items / Next Steps
- Practice applying each transformation rule to multiple points and plot results.
- Memorize coordinate rules for reflections and rotations.
- Verify quadrant changes after transformations to check for sign errors.