Questions and Answers
Which transformation maintains the shape and size of a figure?
What effect does a rotation have on an object on a coordinate plane?
Which of the following points is the reflection of point (3, 4) over the y-axis?
In a translation, what happens to a figure on a coordinate plane?
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What is the main characteristic of similar figures on a coordinate plane?
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If a graph is rotated 180 degrees around the origin, which of the following transformations occurs?
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What is the new coordinate of the point (4, 5) after a 90-degree counterclockwise rotation about the origin?
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Which type of transformation does not change the graph's shape or size?
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What are the new coordinates of the point (5, -2) after a 90° clockwise rotation?
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Which transformation occurs when applying a translation of vector (-3, 4) to the point (6, -2)?
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What will be the coordinates of the point (4, -1) after a 180° rotation about the origin?
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After reflecting the point (-3, 5) over the x-axis, what will be its new coordinates?
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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.
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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)).
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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.
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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.
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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.
