Transformations on the Coordinate Plane

What are Transformations?

Transformations move or change a figure on the coordinate plane.

Four types:

• Translation: Slide
• Reflection: Flip
• Rotation: Turn
• Dilation: Resize (covered separately)

Image: The new figure after transformation Pre-image: The original figure

Translations (Slides)

Translation moves every point the same distance in the same direction.

No change in: Size, shape, or orientation

Notation: (x, y) → (x + a, y + b)

• a = horizontal change (+ right, − left)
• b = vertical change (+ up, − down)

Example 1: Translate a Point

Point A(2, 3) Translation: 4 units right, 3 units up

Rule: (x, y) → (x + 4, y + 3)

Calculate:

• A(2, 3) → A'(2 + 4, 3 + 3) = A'(6, 6)

Answer: New point is A'(6, 6)

Example 2: Translate Left and Down

Point B(5, 7) Translation: 3 units left, 2 units down

Rule: (x, y) → (x − 3, y − 2)

Calculate:

• B(5, 7) → B'(5 − 3, 7 − 2) = B'(2, 5)

Answer: B'(2, 5)

Example 3: Translate a Triangle

Triangle vertices: A(1, 2), B(3, 2), C(2, 4) Translation: Right 2, up 1

Apply to each vertex:

• A(1, 2) → A'(3, 3)
• B(3, 2) → B'(5, 3)
• C(2, 4) → C'(4, 5)

Result: Triangle A'B'C' with vertices (3,3), (5,3), (4,5)

Reflections (Flips)

Reflection flips a figure over a line.

Common lines of reflection:

• x-axis: (x, y) → (x, −y)
• y-axis: (x, y) → (−x, y)
• y = x: (x, y) → (y, x)

Distance from line stays the same!

Example 1: Reflect Over x-axis

Point D(3, 5)

Rule: (x, y) → (x, −y)

Calculate:

• D(3, 5) → D'(3, −5)

Answer: D'(3, −5)

Note: x stays same, y becomes opposite

Example 2: Reflect Over y-axis

Point E(−2, 4)

Rule: (x, y) → (−x, y)

Calculate:

• E(−2, 4) → E'(2, 4)

Answer: E'(2, 4)

Note: y stays same, x becomes opposite

Example 3: Reflect Over y = x

Point F(3, 7)

Rule: (x, y) → (y, x)

Calculate:

• F(3, 7) → F'(7, 3)

Answer: F'(7, 3)

Note: Swap x and y coordinates

Example 4: Reflect a Shape

Square vertices: (1, 1), (3, 1), (3, 3), (1, 3) Reflect over x-axis

Apply rule (x, y) → (x, −y):

• (1, 1) → (1, −1)
• (3, 1) → (3, −1)
• (3, 3) → (3, −3)
• (1, 3) → (1, −3)

Result: Square reflected below x-axis

Rotations (Turns)

Rotation turns a figure around a point (usually the origin).

Common rotations around origin:

• 90° counterclockwise: (x, y) → (−y, x)
• 180°: (x, y) → (−x, −y)
• 270° counterclockwise: (x, y) → (y, −x)

Note: 90° clockwise = 270° counterclockwise

Example 1: Rotate 90° Counterclockwise

Point G(4, 2)

Rule: (x, y) → (−y, x)

Calculate:

• G(4, 2) → G'(−2, 4)

Answer: G'(−2, 4)

Example 2: Rotate 180°

Point H(3, −5)

Rule: (x, y) → (−x, −y)

Calculate:

• H(3, −5) → H'(−3, 5)

Answer: H'(−3, 5)

Note: Both coordinates become opposite

Example 3: Rotate 270° Counterclockwise

Point J(−3, 2)

Rule: (x, y) → (y, −x)

Calculate:

• J(−3, 2) → J'(2, 3)

Answer: J'(2, 3)

Example 4: Rotate a Triangle

Triangle: A(2, 1), B(4, 1), C(3, 3) Rotate 90° counterclockwise

Apply rule (x, y) → (−y, x):

• A(2, 1) → A'(−1, 2)
• B(4, 1) → B'(−1, 4)
• C(3, 3) → C'(−3, 3)

Composition of Transformations

Composition: Performing multiple transformations in sequence

Order matters! Different orders give different results.

Example: Translation then Reflection

Point P(2, 3)

Step 1: Translate right 3

• P(2, 3) → (5, 3)

Step 2: Reflect over x-axis

• (5, 3) → (5, −3)

Final image: (5, −3)

Identifying Transformations

Given pre-image and image, identify the transformation!

Example: What Transformation?

Pre-image: A(2, 3) Image: A'(2, −3)

Observation: x same, y opposite

Answer: Reflection over x-axis

Example: Multiple Possibilities

Pre-image: B(3, 4) Image: B'(7, 8)

Change: +4 in x, +4 in y

Answer: Translation 4 right, 4 up

Properties Preserved

Translations, Reflections, Rotations preserve:

• Size (lengths stay same)
• Shape (angles stay same)
• Area (same area)

These are called rigid motions or isometries

NOT preserved (sometimes):

• Position
• Orientation (reflections reverse orientation)

Real-World Applications

Video games: Moving characters (translation)

Architecture: Symmetric designs (reflection)

Ferris wheel: Rotating seats (rotation)

Tile patterns: Repeated transformations

Animation: Creating movement sequences

Kaleidoscopes: Multiple reflections

Practice