Shape Shifts Reference Sheet
Grade 8 Geometry Transformations
Arrow Notation: Translations
Reference Sheet
\( (x,y) \to (x+a,\, y+b) \)
A translation slides every point of a shape the exact same distance and direction. The image remains congruent (\(\triangle ABC \cong \triangle A'B'C'\)).
Anatomy of Translation Notation
"Maps original point to image point"
\( (x, y) \)
Pre-Image
Original vertex coordinate
\( \to \)
Slides / Maps To
Transformation rule applied
\( x + a \)
Horizontal Shift
\( y + b \)
Vertical Shift
Visual Example: Slide 5 Right, 3 Down
\((x,y) \to (x+5,\, y-3)\)
x y -4 -2 2 4 4 2 -2 -4 +5 right -3 down A(-4, 1) B(-1, 4) C(-1, 1) A'(1, -2) B'(4, 1) C'(4, -2)
Pre-Image (\(\triangle ABC\))
Image (\(\triangle A'B'C'\))
→ ↓ Shift Vector
The Direction Compass
→ Right \(h\) units
\(x + h\)
← Left \(h\) units
\(x - h\)
↑ Up \(k\) units
\(y + k\)
↓ Down \(k\) units
\(y - k\)
Word-to-Rule Translator
3 left, 7 up \( (x-3,\, y+7) \)
6 right, 2 down \( (x+6,\, y-2) \)
4 right only \( (x+4,\, y) \)
5 down only \( (x,\, y-5) \)
3-Step Translation Method
- Write: List pre-image vertices \((x,y)\).
- Substitute: Plug into rule \((x+a,\, y+b)\).
- Graph: Plot new image points \((x',\, y')\).
Vertex Computation Table
Rule applied: \((x,y) \to (x+5,\, y-3)\)
| Pre-Image Vertex | Horizontal Math (\(x + 5\)) | Vertical Math (\(y - 3\)) | Image Vertex (Prime) |
|---|
| \(A(-4,\, 1)\) | \(-4 + 5 = \mathbf{1}\) | \(1 - 3 = \mathbf{-2}\) | \(A'(1,\, -2)\) |
| \(B(-1,\, 4)\) | \(-1 + 5 = \mathbf{4}\) | \(4 - 3 = \mathbf{1}\) | \(B'(4,\, 1)\) |
| \(C(-1,\, 1)\) | \(-1 + 5 = \mathbf{4}\) | \(1 - 3 = \mathbf{-2}\) | \(C'(4,\, -2)\) |
Golden Translation Rules
- • Congruent Shapes: Side lengths and angle measures stay equal.
- • Orientation Preserved: The shape slides; it never rotates or reflects.
- • Prime Marks: Always label transformed points with primes (\(A', B', C'\)).
Watch Out! Common Pitfalls
- • Axis Mix-Up: \(x\) controls left/right only; \(y\) controls up/down only.
- • Sign Swaps: "Left" and "Down" mean subtract (\(-\)).
- • Negative Inputs: Watch signs: \(-4 + 5 = +1\), not \(-9\).
