Intro to Transformations
A transformation is a function that moves, flips, or turns every point of a figure to create a new figure. Master the vocabulary of pre-image and image before diving into the specific types.
Introduction
Up to now, every figure you've studied has been sitting still. Transformations let us move geometry — slide it, flip it, turn it — and still talk precisely about the result. This lesson introduces the language every other lesson in Unit 4 depends on.
Past Knowledge
Coordinate plane (1.2). Points, segments, and angle measure (Units 1–3).
Today's Goal
Define transformation, pre-image, image, and prime notation. Classify transformations as rigid or non-rigid.
Future Success
Translations (4.1.2), reflections (4.1.4), and rotations (4.1.6) all use today's notation.
Key Concepts
Core Vocabulary
| Term | Definition | Example |
|---|---|---|
| Transformation | A function that maps each point of a figure to a new location | |
| Pre-image | The original figure (input) | |
| Image | The new figure after the transformation (output) | |
| Prime notation | A tick mark (′) on a letter to label the image point | (read “A prime”) |
Rigid vs. Non-Rigid Transformations
Rigid (isometry): preserves distances and angle measures — the shape and size stay the same. Translations, reflections, and rotations are rigid.
Non-rigid: changes the size of the figure. Dilations are non-rigid (Unit 7).
Mapping Notation
We can describe a transformation as a mapping rule:
This reads: “the point maps to the point .”
Worked Examples
Naming Image Points
is transformed. Name the image triangle and its vertices.
Add a prime to every vertex label:
The image is .
Identifying the Transformation Type
A figure is moved so that every point slides the same distance in the same direction. What type of transformation is this? Is it rigid or non-rigid?
“Every point slides the same distance in the same direction” is the definition of a translation.
Translations preserve distance and angle measure, so this is a rigid transformation (isometry).
Translation — rigid (isometry).
Using a Mapping Rule
A transformation maps . Find the image of , , and .
Common Pitfalls
Confusing Pre-image and Image
The pre-image is the original (no prime). The image is the result (with prime). Always label carefully — and are different points!
Thinking “Rigid” Means “Doesn't Move”
Rigid transformations do move figures — the word “rigid” means the shape and size are preserved, not that nothing moves.
Forgetting Vertex Order
When you write , the correspondence is , , . Mixing up the order means you're mislabelling which point went where.
Real-Life Applications
Computer Graphics & Animation
Every frame of a video game or animated movie applies hundreds of transformations per second — translating characters, rotating cameras, and reflecting light. The mapping-rule notation you learned today is exactly how game engines store these operations.
Textile & Wallpaper Design
Patterns on fabric and wallpaper are created by repeatedly applying translations, reflections, and rotations to a single motif. Understanding which transformation was used helps designers build seamless, repeating patterns.
Practice Quiz
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