Translations and Reflections

Transformations change the position or orientation of shapes on a coordinate plane! Translations slide shapes, while reflections flip them. Understanding these transformations is essential for geometry, computer graphics, and real-world applications.

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What Are Transformations?

Transformations move or change shapes in predictable ways.

Four main types:

1. Translation - Slide
2. Reflection - Flip
3. Rotation - Turn
4. Dilation - Resize

This topic focuses on translations and reflections.

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Translations (Slides)

A translation slides every point of a shape the same distance in the same direction.

Think of it as:

• Moving a shape without rotating or flipping it
• Sliding on a sheet of paper
• Same shape, same size, new location

Properties:

• Shape and size stay the same
• Orientation stays the same (doesn't flip or turn)
• All points move the same distance and direction

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Translation Rules

Notation: (x, y) → (x + a, y + b)

Where:

• a = horizontal shift (positive = right, negative = left)
• b = vertical shift (positive = up, negative = down)

Example translations:

(x, y) → (x + 3, y + 5)

• Move 3 units right and 5 units up

(x, y) → (x - 4, y + 2)

• Move 4 units left and 2 units up

(x, y) → (x + 1, y - 6)

• Move 1 unit right and 6 units down

(x, y) → (x - 2, y - 3)

• Move 2 units left and 3 units down

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Performing Translations

Example 1: Translate point A(2, 3) by the rule (x, y) → (x + 4, y - 1)

Solution: Original: (2, 3) Apply rule: (2 + 4, 3 - 1) = (6, 2)

Answer: A' = (6, 2)

The prime symbol (') means "after transformation"

Example 2: Translate triangle ABC where A(1, 2), B(3, 5), C(4, 1) by moving 3 units left and 2 units up.

Solution: Rule: (x, y) → (x - 3, y + 2)

A(1, 2) → A'(1 - 3, 2 + 2) = A'(-2, 4) B(3, 5) → B'(3 - 3, 5 + 2) = B'(0, 7) C(4, 1) → C'(4 - 3, 1 + 2) = C'(1, 3)

Answer: A'(-2, 4), B'(0, 7), C'(1, 3)

Example 3: A point moves from (5, -2) to (1, 3). Describe the translation.

Solution: Original: (5, -2) New: (1, 3)

Horizontal change: 1 - 5 = -4 (moved 4 left) Vertical change: 3 - (-2) = 5 (moved 5 up)

Answer: Translation rule is (x, y) → (x - 4, y + 5)

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Reflections (Flips)

A reflection flips a shape over a line (called the line of reflection).

Think of it as:

• Looking in a mirror
• Folding paper along a line
• Creating a mirror image

Properties:

• Shape and size stay the same
• Orientation reverses (flips)
• Distance from line of reflection stays the same

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Common Lines of Reflection

1. Reflection over the x-axis

Rule: (x, y) → (x, -y)

The x-coordinate stays the same, y-coordinate changes sign.

Example: A(3, 5) → A'(3, -5) B(-2, 4) → B'(-2, -4) C(1, -3) → C'(1, 3)

Pattern: Points above the x-axis flip below, and vice versa.

2. Reflection over the y-axis

Rule: (x, y) → (-x, y)

The y-coordinate stays the same, x-coordinate changes sign.

Example: A(4, 2) → A'(-4, 2) B(-3, 5) → B'(3, 5) C(6, -1) → C'(-6, -1)

Pattern: Points right of y-axis flip left, and vice versa.

3. Reflection over the line y = x

Rule: (x, y) → (y, x)

Swap the coordinates!

Example: A(2, 5) → A'(5, 2) B(3, 1) → B'(1, 3) C(-4, 2) → C'(2, -4)

Pattern: The diagonal line y = x is the mirror.

4. Reflection over the line y = -x

Rule: (x, y) → (-y, -x)

Swap coordinates AND change both signs.

Example: A(3, 2) → A'(-2, -3) B(1, 4) → B'(-4, -1) C(-2, 5) → C'(-5, 2)

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Comparing Reflections

| Line of Reflection | Rule | What Changes | |-------------------|------|--------------| | x-axis | (x, y) → (x, -y) | y changes sign | | y-axis | (x, y) → (-x, y) | x changes sign | | y = x | (x, y) → (y, x) | Coordinates swap | | y = -x | (x, y) → (-y, -x) | Swap and both change sign |

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Performing Reflections

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

Solution: Rule: (x, y) → (x, -y) P(4, -3) → P'(4, 3)

Answer: P'(4, 3)

Example 2: Reflect triangle ABC where A(2, 1), B(5, 4), C(3, 6) over the y-axis.

Solution: Rule: (x, y) → (-x, y)

A(2, 1) → A'(-2, 1) B(5, 4) → B'(-5, 4) C(3, 6) → C'(-3, 6)

Answer: A'(-2, 1), B'(-5, 4), C'(-3, 6)

Example 3: Reflect point Q(-3, 5) over the line y = x.

Solution: Rule: (x, y) → (y, x) Q(-3, 5) → Q'(5, -3)

Answer: Q'(5, -3)

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Combining Transformations

You can perform multiple transformations in sequence!

Example: Point A(2, 3) is translated by (x, y) → (x + 1, y - 2), then reflected over the x-axis. Find the final position.

Solution:

Step 1: Translation A(2, 3) → (2 + 1, 3 - 2) = (3, 1)

Step 2: Reflection over x-axis (3, 1) → (3, -1)

Answer: Final position is (3, -1)

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Properties Preserved

Both translations and reflections preserve:

• Size - distances don't change
• Shape - angles and proportions stay the same
• Congruence - original and image are congruent

What changes:

• Position - both move the shape
• Orientation - reflections flip the shape (translations don't)

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Real-World Applications

Translations:

Computer graphics: Moving sprites in video games

• Character moves right: (x, y) → (x + speed, y)

Animation: Sliding objects across screen

Maps: Shifting a view on GPS

Manufacturing: Moving parts on assembly line

Reflections:

Mirror images: Photography, design

• Symmetric designs use reflections

Letter symmetry: Some letters reflect over vertical line

• A, H, M, T, U, V, W, Y

Architecture: Symmetric building designs

Nature: Butterfly wings, faces (approximate symmetry)

Logos: Many company logos use reflections

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Identifying Transformations

Given original and image, determine the transformation:

Example: Point A(3, 2) becomes A'(3, -2). What transformation?

Solution: x stayed the same (3 = 3) y changed sign (2 → -2)

Answer: Reflection over the x-axis

Example: Point B(4, 5) becomes B'(7, 3). What transformation?

Solution: x changed by +3 (4 + 3 = 7) y changed by -2 (5 - 2 = 3)

Answer: Translation (x, y) → (x + 3, y - 2)

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Common Mistakes to Avoid

❌ Mistake 1: Confusing which coordinate changes in reflections

• Wrong: Reflect over x-axis: (x, y) → (-x, y)
• Right: Reflect over x-axis: (x, y) → (x, -y)

❌ Mistake 2: Adding when you should subtract in translations

• Wrong: Move 3 left: (x, y) → (x + 3, y)
• Right: Move 3 left: (x, y) → (x - 3, y)

❌ Mistake 3: Forgetting to swap in y = x reflection

• Wrong: (3, 5) over y = x stays (3, 5)
• Right: (3, 5) over y = x becomes (5, 3)

❌ Mistake 4: Applying transformation to only one point

• Wrong: Translate only one vertex of a triangle
• Right: Apply to ALL points of the shape

❌ Mistake 5: Sign errors with negative coordinates

• Be careful: -(-3) = +3!

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Step-by-Step Strategy

For Translations:

1. Identify the rule (x + a, y + b)
2. Add a to every x-coordinate
3. Add b to every y-coordinate
4. Plot new points

For Reflections:

1. Identify the line of reflection
2. Apply the correct rule:
   • x-axis: flip y
   • y-axis: flip x
   • y = x: swap x and y
   • y = -x: swap and flip both
3. Plot new points

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Quick Reference

Translations: (x, y) → (x + a, y + b)

• +a = right, -a = left
• +b = up, -b = down

Reflections:

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

Properties Preserved:

• Size, shape, congruence

What Changes:

• Position (both)
• Orientation (reflections only)

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Summary

Translations slide shapes without changing orientation:

• All points move same distance and direction
• Rule: (x, y) → (x + a, y + b)
• Preserves size, shape, and orientation

Reflections flip shapes over a line:

• Creates mirror image
• Four common lines: x-axis, y-axis, y = x, y = -x
• Preserves size and shape, reverses orientation

Both are rigid transformations:

• Shapes stay congruent
• Only position/orientation changes
• Essential for geometry, graphics, and design

Understanding these transformations builds foundation for advanced geometry and real-world applications!