Transformations - Seventh Grade

Translations, Reflections & Rotations

1. Understanding Transformations

What is a Transformation?

A transformation is a change in

the position, size, or orientation of a shape

• Original shape = PRE-IMAGE

• Transformed shape = IMAGE

• Image is marked with prime notation (A'B'C')

Three Types of Transformations (7th Grade)

1. Translation: SLIDE the shape (move without turning)

2. Reflection: FLIP the shape (create mirror image)

3. Rotation: TURN the shape (spin around a point)

Key Point: All three are RIGID transformations

Shape and size stay the SAME (congruent)

Only position or orientation changes!

2. Translations (Slide)

What is a Translation?

A translation SLIDES every point

of a shape the SAME distance

in the SAME direction

Think: Moving a piece on a checkerboard

Translation Rule

(x, y) → (x + h, y + k)

Where:

h = horizontal shift (left/right)

k = vertical shift (up/down)

Direction Rules:

• Move RIGHT: h is positive (add)

• Move LEFT: h is negative (subtract)

• Move UP: k is positive (add)

• Move DOWN: k is negative (subtract)

Examples

Example 1: Translate point A(3, 5) right 4 units and up 2 units.

h = +4 (right), k = +2 (up)

(x, y) → (x + 4, y + 2)

A(3, 5) → A'(3 + 4, 5 + 2)

A(3, 5) → A'(7, 7)

Answer: A'(7, 7)

Example 2: Translate point B(6, -2) left 5 units and down 3 units.

h = -5 (left), k = -3 (down)

(x, y) → (x - 5, y - 3)

B(6, -2) → B'(6 - 5, -2 - 3)

B(6, -2) → B'(1, -5)

Answer: B'(1, -5)

3. Reflections (Flip)

What is a Reflection?

A reflection FLIPS a shape

over a line of reflection

Creates a MIRROR IMAGE

Each point is the same distance from the line on opposite sides

Reflection Rules

Reflection Over X-Axis

(x, y) → (x, -y)

X stays the same, Y changes sign

Reflection Over Y-Axis

(x, y) → (-x, y)

Y stays the same, X changes sign

Reflection Over Line y = x

(x, y) → (y, x)

X and Y swap places

Reflection Through Origin

(x, y) → (-x, -y)

Both coordinates change sign (same as 180° rotation)

Examples

Example 1: Reflect point A(4, 3) over the x-axis.

Rule: (x, y) → (x, -y)

A(4, 3) → A'(4, -3)

Answer: A'(4, -3)

Example 2: Reflect point B(-2, 5) over the y-axis.

Rule: (x, y) → (-x, y)

B(-2, 5) → B'(2, 5)

Answer: B'(2, 5)

Example 3: Reflect point C(3, -4) over the line y = x.

Rule: (x, y) → (y, x)

C(3, -4) → C'(-4, 3)

Answer: C'(-4, 3)

4. Rotations (Turn)

What is a Rotation?

A rotation TURNS a shape

around a fixed point (center of rotation)

• Usually rotated around the ORIGIN (0, 0)

• Measured in degrees

• Can be clockwise or counterclockwise

Rotation Rules (Counterclockwise about Origin)

Rotation 90° Counterclockwise

(x, y) → (-y, x)

Swap coordinates, make old y negative

Rotation 180° (Either Direction)

(x, y) → (-x, -y)

Both coordinates change sign

Rotation 270° Counterclockwise

(x, y) → (y, -x)

Swap coordinates, make old x negative

(Same as 90° clockwise)

Clockwise vs Counterclockwise

Important:

• 90° clockwise = 270° counterclockwise: (x, y) → (y, -x)

• 270° clockwise = 90° counterclockwise: (x, y) → (-y, x)

Examples

Example 1: Rotate point A(3, 2) 90° counterclockwise about the origin.

Rule: (x, y) → (-y, x)

A(3, 2) → A'(-2, 3)

Answer: A'(-2, 3)

Example 2: Rotate point B(-4, 5) 180° about the origin.

Rule: (x, y) → (-x, -y)

B(-4, 5) → B'(4, -5)

Answer: B'(4, -5)

Example 3: Rotate point C(2, -3) 270° counterclockwise about the origin.

Rule: (x, y) → (y, -x)

C(2, -3) → C'(-3, -2)

Answer: C'(-3, -2)

5. Identifying Transformations

How to Identify Which Transformation Occurred

Translation (Slide)

✓ Shape moves but doesn't turn or flip

✓ Orientation stays the same

✓ All points move the same distance in the same direction

Reflection (Flip)

✓ Shape is flipped over a line

✓ Orientation reverses (mirror image)

✓ Image and pre-image are equidistant from the line of reflection

Rotation (Turn)

✓ Shape turns around a point

✓ Orientation changes (different angle)

✓ All points are same distance from center of rotation

Quick Check Method

Step 1: Compare orientation

Same orientation → Translation or Rotation

Reversed orientation → Reflection

Step 2: Check coordinates

Apply transformation rules to verify

Quick Reference: All Transformation Rules

Transformation	Rule
Translation	(x, y) → (x + h, y + k)
Reflection over x-axis	(x, y) → (x, -y)
Reflection over y-axis	(x, y) → (-x, y)
Reflection over y = x	(x, y) → (y, x)
Reflection through origin	(x, y) → (-x, -y)
Rotation 90° CCW	(x, y) → (-y, x)
Rotation 180°	(x, y) → (-x, -y)
Rotation 270° CCW	(x, y) → (y, -x)