Reflections Across Axes
A reflection flips a figure over a line of reflection, creating a mirror image. This lesson focuses on the two most common lines: the x-axis and the y-axis.
Introduction
Imagine folding a piece of graph paper along the x-axis or y-axis — whichever side your figure is on, the image appears on the opposite side at the same distance. That “fold” is a reflection.
Past Knowledge
Transformations vocabulary (4.1.1). Coordinate plane & plotting (1.2).
Today's Goal
Reflect points and figures across the x-axis and y-axis using coordinate rules.
Future Success
Reflections across other lines (4.1.5), symmetry (4.2.1), and congruence via isometries (4.2.4).
Key Concepts
Reflection Rules
| Line of Reflection | Rule | What Changes |
|---|---|---|
| x-axis | The -coordinate flips sign | |
| y-axis | The -coordinate flips sign |
Memory Trick
Reflecting across the x-axis changes y (the “other” coordinate). Reflecting across the y-axis changes x (the “other” coordinate). The coordinate that matches the axis name stays the same.
Key Properties of Reflections
- Rigid — distances and angles are preserved.
- Orientation is reversed — a clockwise labeling becomes counterclockwise (or vice versa). This is the key difference from translations.
- Each point and its image are equidistant from the line of reflection.
- The segment connecting a point to its image is perpendicular to the line of reflection.
Worked Examples
Reflecting a Point Across the x-axis
Reflect across the x-axis.
Rule:
— same , opposite .
Reflecting a Triangle Across the y-axis
Reflect with , , across the y-axis.
Rule:
Identifying the Line of Reflection
and . Identify the line of reflection.
Check: -coordinates stay the same; -coordinates change sign.
The rule is .
The line of reflection is the x-axis.
Common Pitfalls
Negating the Wrong Coordinate
Across the x-axis → change y. Across the y-axis → change x. Students frequently negate the coordinate that matches the axis name instead of the other one.
Double-Negating a Negative Coordinate
Reflecting across the x-axis gives , not . The negative of is . Watch your signs!
Thinking Reflections = Translations
A reflection reverses orientation (the figure appears “mirrored”), while a translation preserves it. If your image doesn't look like a mirror image, double-check which transformation you applied.
Real-Life Applications
Mirrors & Optics
A flat mirror produces a reflection across the mirror's plane. The image appears the same distance behind the mirror as the object is in front of it — exactly how coordinate reflections work.
Water Reflections in Photography
A landscape reflected in a still lake creates a natural x-axis reflection. Photographers use this effect intentionally — the water line acts as the axis of reflection.
Practice Quiz
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