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Perform translations and reflections
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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.
Transformations move or change shapes in predictable ways.
Four main types:
This topic focuses on translations and reflections.
A translation slides every point of a shape the same distance in the same direction.
Think of it as:
Translate point A(3, 5) by the rule (x, y) โ (x + 4, y - 2).
Apply the translation rule:
(x, y) โ (x + 4, y - 2)
A(3, 5) โ A'(3 + 4, 5 - 2)
A'(7, 3)
Answer: A'(7, 3)
Reflect point B(4, -2) over the x-axis.
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Properties:
Notation: (x, y) โ (x + a, y + b)
Where:
Example translations:
(x, y) โ (x + 3, y + 5)
(x, y) โ (x - 4, y + 2)
(x, y) โ (x + 1, y - 6)
(x, y) โ (x - 2, y - 3)
Example 1: Translate point A(2, 3) by the rule (x, y) โ (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) โ (x - 3, y + 2)
A(1, 2) โ A'(1 - 3, 2 + 2) = A'(-2, 4) B(3, 5) โ B'(3 - 3, 5 + 2) = B'(0, 7) C(4, 1) โ 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) โ (x - 4, y + 5)
A reflection flips a shape over a line (called the line of reflection).
Think of it as:
Properties:
1. Reflection over the x-axis
Rule: (x, y) โ (x, -y)
The x-coordinate stays the same, y-coordinate changes sign.
Example: A(3, 5) โ A'(3, -5) B(-2, 4) โ B'(-2, -4) C(1, -3) โ C'(1, 3)
Pattern: Points above the x-axis flip below, and vice versa.
2. Reflection over the y-axis
Rule: (x, y) โ (-x, y)
The y-coordinate stays the same, x-coordinate changes sign.
Example: A(4, 2) โ A'(-4, 2) B(-3, 5) โ B'(3, 5) C(6, -1) โ C'(-6, -1)
Pattern: Points right of y-axis flip left, and vice versa.
3. Reflection over the line y = x
Rule: (x, y) โ (y, x)
Swap the coordinates!
Example: A(2, 5) โ A'(5, 2) B(3, 1) โ B'(1, 3) C(-4, 2) โ C'(2, -4)
Pattern: The diagonal line y = x is the mirror.
4. Reflection over the line y = -x
Rule: (x, y) โ (-y, -x)
Swap coordinates AND change both signs.
Example: A(3, 2) โ A'(-2, -3) B(1, 4) โ B'(-4, -1) C(-2, 5) โ C'(-5, 2)
| Line of Reflection | Rule | What Changes |
|---|---|---|
| x-axis | (x, y) โ (x, -y) | y changes sign |
| y-axis | (x, y) โ (-x, y) | x changes sign |
| y = x | (x, y) โ (y, x) | Coordinates swap |
| y = -x | (x, y) โ (-y, -x) | Swap and both change sign |
Example 1: Reflect point P(4, -3) over the x-axis.
Solution: Rule: (x, y) โ (x, -y) P(4, -3) โ 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) โ (-x, y)
A(2, 1) โ A'(-2, 1) B(5, 4) โ B'(-5, 4) C(3, 6) โ 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) โ (y, x) Q(-3, 5) โ Q'(5, -3)
Answer: Q'(5, -3)
You can perform multiple transformations in sequence!
Example: Point A(2, 3) is translated by (x, y) โ (x + 1, y - 2), then reflected over the x-axis. Find the final position.
Solution:
Step 1: Translation A(2, 3) โ (2 + 1, 3 - 2) = (3, 1)
Step 2: Reflection over x-axis (3, 1) โ (3, -1)
Answer: Final position is (3, -1)
Both translations and reflections preserve:
What changes:
Translations:
Computer graphics: Moving sprites in video games
Animation: Sliding objects across screen
Maps: Shifting a view on GPS
Manufacturing: Moving parts on assembly line
Reflections:
Mirror images: Photography, design
Letter symmetry: Some letters reflect over vertical line
Architecture: Symmetric building designs
Nature: Butterfly wings, faces (approximate symmetry)
Logos: Many company logos use reflections
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 โ -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) โ (x + 3, y - 2)
โ Mistake 1: Confusing which coordinate changes in reflections
โ Mistake 2: Adding when you should subtract in translations
โ Mistake 3: Forgetting to swap in y = x reflection
โ Mistake 4: Applying transformation to only one point
โ Mistake 5: Sign errors with negative coordinates
For Translations:
For Reflections:
Translations: (x, y) โ (x + a, y + b)
Reflections:
Properties Preserved:
What Changes:
Translations slide shapes without changing orientation:
Reflections flip shapes over a line:
Both are rigid transformations:
Understanding these transformations builds foundation for advanced geometry and real-world applications!
Reflection over x-axis rule: (x, y) โ (x, -y)
B(4, -2) โ B'(4, 2)
The x-coordinate stays the same, y-coordinate changes sign.
Answer: B'(4, 2)
Reflect point C(-3, 5) over the y-axis.
Reflection over y-axis rule: (x, y) โ (-x, y)
C(-3, 5) โ C'(3, 5)
The y-coordinate stays the same, x-coordinate changes sign.
Answer: C'(3, 5)
Point D(2, 6) is translated 5 units left and 3 units down. Then it is reflected over the x-axis. Find the final position.
Step 1: Translation (5 left, 3 down means -5, -3) (x, y) โ (x - 5, y - 3) D(2, 6) โ D'(2 - 5, 6 - 3) = D'(-3, 3)
Step 2: Reflect over x-axis (x, y) โ (x, -y) D'(-3, 3) โ D''(-3, -3)
Answer: D''(-3, -3)
Triangle ABC has vertices A(1, 2), B(4, 2), C(3, 5). Reflect it over the line y = x. Find the new coordinates.
Reflection over y = x rule: (x, y) โ (y, x) [swap coordinates]
A(1, 2) โ A'(2, 1) B(4, 2) โ B'(2, 4) C(3, 5) โ C'(5, 3)
Answer: A'(2, 1), B'(2, 4), C'(5, 3)