Transformations in Geometry

Study Notes

Transformations in Geometry

Rigid Transformation: This transformation maintains both the shape and size of a figure. Examples include translations, rotations, and reflections.

Rotation Effects

Effect of Rotation: When an object is rotated on a coordinate plane, its position changes around a fixed point (typically the origin), but its shape and size remain unchanged.

Reflection Example

Reflection of (3, 4) over the Y-Axis: The reflection of the point (3, 4) over the y-axis is (-3, 4), as the x-coordinate changes sign while the y-coordinate remains the same.

Translation Characteristics

Translation: In a translation, every point of a figure moves the same distance in the same direction on a coordinate plane. The size and shape of the figure remain unchanged.

Similar Figures on a Coordinate Plane

Characteristics of Similar Figures: Similar figures have the same shape but may differ in size. Their corresponding angles are equal, and their sides are in proportion.

Translations

A translation alters the position of a graph without modifying its shape or orientation.
Horizontal Translation:
- Moving right involves replacing (f(x)) with (f(x - h)) where (h) is positive.
- Moving left replaces (f(x)) with (f(x + h)).
Vertical Translation:
- An upward shift is represented as (f(x) \to f(x) + k) with (k) being positive.
- A downward shift involves (f(x) \to f(x) - k).
Combined Translation: Achieved through (f(x) \to f(x - h) + k), resulting in a right shift by (h) and an upward shift by (k).

Reflections

A reflection alters the orientation of a graph by flipping it across a designated line or axis.
Reflection over the x-axis: This transformation is represented by (f(x) \to -f(x)).
Reflection over the y-axis: Achieved through (f(x) \to f(-x)).
Reflection over the line (y = x): Denoted by (f(x) \to f^{-1}(x)), indicating the inverse function.
Reflection over the line (y = -x): Represented by (f(x) \to -f^{-1}(x)).

Rotations

A rotation involves turning the graph around a fixed point (commonly the origin) at a specific angle.
Rotation about the origin (0,0):
- 90 degrees counterclockwise: The transformation changes coordinates from ((x, y)) to ((-y, x)).
- 180 degrees: Coordinates transition from ((x, y)) to ((-x, -y)).
- 90 degrees clockwise: Transformation shifts coordinates from ((x, y)) to ((y, -x)).
Rotation about a point (h, k): Necessary to first translate the graph to center around the origin, perform the rotation, and then translate back to the original position.
General transformations adjust the coordinates based on the angle of rotation and the specified point of rotation, aiding in visualizing function behavior changes in the coordinate plane.

Rotation Rules

A 90° counterclockwise rotation transforms a point (x, y) to (-y, x).
A 90° clockwise rotation changes (x, y) to (y, -x).
A 180° rotation reverses the coordinates, resulting in (-x, -y).
A 270° counterclockwise rotation is equivalent to a 90° clockwise rotation, and updates (x, y) to (y, -x).
A 270° clockwise rotation corresponds to a 90° counterclockwise rotation and updates (x, y) to (-y, x).

Reflection Rules

Reflecting a point over the x-axis changes it from (x, y) to (x, -y).
Reflecting over the y-axis transforms the coordinates to (-x, y).
A reflection over the line y = x swaps the coordinates, resulting in (y, x).
Reflecting over the line y = -x reverses both coordinates, changing (x, y) to (-y, -x).
Reflecting a point over a horizontal line y = k modifies it to (x, 2k - y).
Reflecting over a vertical line x = h changes it to (2h - x, y).

Translation Rules

A translation by vector (a, b) moves a point by adding to its coordinates, changing (x, y) to (x + a, y + b).
Horizontal translation involves adjusting the x-coordinate; (x + a, y) indicates movement right (a > 0) or left (a < 0).
Vertical translation adjusts the y-coordinate; (x, y + b) indicates movement up (b > 0) or down (b < 0).
Combined translation applies both horizontal and vertical changes simultaneously, adjusting both coordinates at once.

Description

This quiz explores the concepts of translations, rotations, and reflections in geometry, specifically focusing on their properties and effects on figures on a coordinate plane. You'll also encounter questions about similar figures and their characteristics. Test your understanding of how these transformations maintain shape and size.