Question

Write a rule to describe each transformation

Answer 1

I’m missing which transformations you mean. Do you mean geometric (plane) transformations, function/graph transformations, matrix linear transformations, string/data transformations, chemical reactions, or something else?

Below are two helpful things I can do—pick one or tell me which domain you want:

1) If you want a general template for writing a rule for any transformation:
- Name: short identifier (e.g., Translation by (a,b)).
- Operation: concise mathematical description of what is done to an input.
- Rule (mapping): explicit input→output formula.
- Domain/conditions: inputs for which the rule applies.
- Inverse (if any): formula for undoing the transformation.
- Example(s): one or two specific inputs and outputs.

2) Quick examples for common domains

A. Plane (2D) geometric transformations (mapping of point (x,y)):
- Translation by (a,b)
- Rule: (x,y) → (x + a, y + b)
- Inverse: (x,y) → (x − a, y − b)
- Rotation about origin by angle θ (counterclockwise)
- Rule: (x,y) → (x cosθ − y sinθ, x sinθ + y cosθ)
- Inverse: rotation by −θ (use −θ in formula)
- Reflection across x-axis
- Rule: (x,y) → (x, −y)
- Inverse: same rule (self-inverse)
- Reflection across line y = x
- Rule: (x,y) → (y, x)
- Dilation (scaling) with factor k about origin
- Rule: (x,y) → (k x, k y)
- Inverse: scale by 1/k (if k ≠ 0)

B. Function/graph transformations (given y = f(x)):
- Vertical shift by c
- Rule: y = f(x) → y = f(x) + c
- Effect: whole graph moves up c (down if c < 0)
- Horizontal shift by h
- Rule: y = f(x) → y = f(x − h)
- Effect: graph moves right by h (left if h < 0)
- Vertical stretch/compression by factor a
- Rule: y = f(x) → y = a·f(x)
- Effect: multiply y-values by a (a > 1 stretch; 0 < a < 1 compress)
- Horizontal stretch/compression by factor b
- Rule: y = f(x) → y = f(x / b)
- Effect: x-values multiplied by b (b > 1 stretches horizontally)
- Reflection in x-axis
- Rule: y = f(x) → y = −f(x)

C. Linear transformations via matrices (in R^n):
- For matrix A (n×n), linear map T_A: R^n → R^n
- Rule: v → A v
- Composition: T_B∘T_A corresponds to matrix product B A
- Inverse: A^−1 exists if det(A) ≠ 0; rule v → A^−1 v

D. String transformation (example rules):
- Lowercase to uppercase
- Rule: map each character c → uppercase(c)
- Remove vowels
- Rule: for each character, drop if in {a,e,i,o,u} (optionally case-insensitive)

If you tell me which domain and which specific transformations you need rules for, I’ll write precise rules in the format you prefer.