1.1 Notation and Vocabulary — Transformations

Standards: G-CO.A.2 · G-CO.A.4

Given transformation notation, you will label the transformation name, parameters, input, preimage, and image, then write each object with the correct brackets.

Definitions and rules

Adding a negative moves the other way

6 + (−2) = 4 up −2 is down 2

Function notation: a name, an input, a rule

f(x) = x + 4 f(10) = 10 + 4 = 14

Whatever sits in the parentheses is the input.

The same notation, with a point as the input

T⟨a, b⟩(x, y) = (x + a, y + b)

Four jobs, not three

T	Name. Which kind of transformation.
⟨a, b⟩	Parameters. Which particular one. Set once, then fixed.
(x, y)	Input. The point going in. Changes for every vertex.
(x + a, y + b)	Rule. What to do with the input.

Same shape as y = mx + b, where m and b pick which line and x moves along it.

A point is written (x, y) — horizontal first

(3, 2) means right 3, then up 2

The words used for every transformation, not just one.
Term	What it means
Transformation	Any rule that takes a figure and produces a new figure.
Preimage	The original figure. Plain letters: A, B.
Image	The result. Prime marks: A′, read “A prime”.
Rigid motion	A transformation that preserves size and shape, so the image is congruent.
Similar	Same shape, possibly different size. Angles preserved, lengths scaled.

Worked examples

Example 1 — a complete transformation statement

Interpret T⟨−3, 2⟩(P) = P′ when P = (4, −1). Name the transformation, vector, input, preimage, and image, then compute P′.

Suggested work

Givens

T⟨−3, 2⟩(P) = P′ P = (4, −1)

Tools

T⟨a, b⟩(x, y) = (x + a, y + b)

Primitives

transformation name: T vector: ⟨−3, 2⟩ input and preimage: P image label: P′

Computation

P(4, −1) → ((4) + (−3), (−1) + (2)) = P′(1, 1)

Notice: Angle brackets hold the vector, round parentheses hold a point, and the prime mark names the image.

Why the symbols have separate jobs

The transformation needs an instruction and an input. Angle brackets package the displacement; round parentheses package the point being moved.

The prime mark distinguishes the output from the original label without changing which point corresponds to which.

Example 2 — three objects built from the same numbers

Explain the roles of ⟨5, −2⟩, T⟨5, −2⟩, and (5, −2).

Suggested work

Givens

⟨5, −2⟩ T⟨5, −2⟩ (5, −2)

Tools

angle brackets = vector T with a vector = translation round parentheses = point

Primitives

The numbers match; the brackets and leading symbol change the object.

Computation

⟨5, −2⟩ is a vector: right 5 and down 2. T⟨5, −2⟩ is the translation using that vector. (5, −2) is the point with x = 5 and y = −2.

Notice: Matching numbers do not make the three expressions interchangeable.

Why the brackets matter

Geometry uses the container to identify the data type. A vector is an instruction, a transformation applies the instruction, and a point is an input or output.

Example 3 — a point and its image

Point A(−2, 4) maps to A′(3, 4). Identify the preimage and image, then state the horizontal and vertical location of A.

Suggested work

Givens

A(−2, 4) → A′(3, 4)

Tools

(x, y) lists horizontal location, then vertical location plain label = preimage prime label = image

Primitives

x = −2 y = 4

Computation

Preimage: A(−2, 4) Image: A′(3, 4) A is 2 units left and 4 units up from the origin.

Why horizontal comes first

The ordered pair fixes one convention for every point: x locates left or right, then y locates down or up.

Common traps

Notation to keep straight

⟨a, b⟩ with angle brackets is the vector. T⟨a, b⟩ is the transformation. (a, b) with round parentheses is a point. Three different things, three different brackets, used consistently throughout.

Practice

In R90(B) = B′, which label names the preimage and which label names the image?
Answer — tap to show
B is the preimage, and B′ is the image.
Suggested work — tap to show
Givens
R90(B) = B′
Tools
plain label = preimage prime label = image
Primitives
input: B output: B′
Computation
B goes into the transformation, so B is the preimage. B′ comes out, so B′ is the image.
Write the vector 4 units left and 3 units up, then write the point with the same horizontal and vertical values.
Answer — tap to show
Vector: ⟨−4, 3⟩. Point: (−4, 3).
For T⟨2, −5⟩(C) = C′, label the transformation name, parameters, input, and output.
Answer — tap to show
Name: T. Parameters: ⟨2, −5⟩. Input: C. Output: C′.
Suggested work — tap to show
Givens
T⟨2, −5⟩(C) = C′
Tools
name + parameters + input = output
Primitives
T | ⟨2, −5⟩ | C | C′
Computation
T names the transformation. ⟨2, −5⟩ fixes its displacement. C is the input. C′ is the output.
Read T⟨−6, 1⟩(D) = D′ in words.
Answer — tap to show
Translate point D 6 units left and 1 unit up; the result is D prime.
Suggested work — tap to show
Givens
T⟨−6, 1⟩(D) = D′
Tools
negative horizontal component = left positive vertical component = up
Primitives
a = −6 b = 1
Computation
D moves left 6 and up 1, producing D′.
A classmate says, T⟨2, −3⟩ is a point because it contains two coordinates. Diagnose the error.
Answer — tap to show
The classmate ignored the leading T and the angle brackets. T⟨2, −3⟩ is a translation. The vector is ⟨2, −3⟩, while the point with those values is (2, −3).
Suggested work — tap to show
Givens
claim: T⟨2, −3⟩ is a point
Tools
T⟨a, b⟩ = translation ⟨a, b⟩ = vector (a, b) = point
Primitives
leading symbol: T brackets: angle brackets
Computation
The expression names a translation and its vector. A point would use round parentheses and no leading T.