Transformations
Section: Position and Transformation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Three Types of Transformation
- Translation: sliding a shape without turning or flipping it - described by a column vector.
- Reflection: flipping a shape over a mirror line, creating a mirror image.
- Rotation: turning a shape about a fixed centre by a given angle and direction.
Translation, reflection, and rotation all produce a congruent image
Translations
A translation is described by a column vector (a, b): a is the horizontal movement (positive = right, negative = left), and b is the vertical movement (positive = up, negative = down).
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Question: Translate point A(2, 3) by the vector (4, −1). Find the image A′.
- Step 1: A′ = (2 + 4, 3 + (−1))
- Answer: A′ = (6, 2)
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Question: Translate point B(−3, 5) by the vector (−2, 3). Find the image B′.
- Step 1: B′ = (−3 + (−2), 5 + 3)
- Answer: B′ = (−5, 8)
Reflections
- Reflection in the x-axis: (x, y) → (x, −y)
- Reflection in the y-axis: (x, y) → (−x, y)
- Reflection in the line y = x: (x, y) → (y, x)
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Question: Reflect point P(3, 5) in the x-axis.
- Step 1: Keep x the same, flip the sign of y: P′ = (3, −5)
- Answer: P′ = (3, −5)
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Question: Reflect point R(2, 7) in the line y = x.
- Step 1: Swap the x and y coordinates: R′ = (7, 2)
- Answer: R′ = (7, 2)
Rotations
For a rotation about the origin: 90° clockwise: (x, y) → (y, −x). 90° anticlockwise: (x, y) → (−y, x). 180°: (x, y) → (−x, −y).
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Question: Rotate point A(3, 2) by 90° clockwise about the origin.
- Step 1: (x, y) → (y, −x): A′ = (2, −3)
- Answer: A′ = (2, −3)
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Question: Rotate point B(4, 1) by 180° about the origin.
- Step 1: (x, y) → (−x, −y): B′ = (−4, −1)
- Answer: B′ = (−4, −1)
Identifying a Transformation from an Object and Image
To fully describe a transformation, always state the type plus its specific details: the vector (translation), the mirror line (reflection), or the centre, angle, and direction (rotation).
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Question: A shape has vertices (1, 1), (3, 1), (3, 4). Its image has vertices (1, −1), (3, −1), (3, −4). Describe the transformation.
- Step 1: Compare each point: (1,1)→(1,−1), (3,1)→(3,−1), (3,4)→(3,−4) - x stays the same, y changes sign each time
- Answer: Reflection in the x-axis
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Question: A shape has vertices (2, 3), (4, 3), (4, 5). Its image has vertices (5, 5), (7, 5), (7, 7). Describe the transformation.
- Step 1: Compare each point: every vertex moves by (+3, +2)
- Answer: Translation by vector (3, 2)
Combined Transformations
When two transformations are applied one after another, work through them in the order given, one step at a time.
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Question: Point A(2, 1) is first reflected in the y-axis, then translated by vector (3, −2). Find the final image.
- Step 1 (reflect in y-axis): (2, 1) → (−2, 1)
- Step 2 (translate by (3, −2)): (−2 + 3, 1 + (−2))
- Answer: (1, −1)
Congruence After Transformation
Translations, reflections, and rotations are all rigid transformations - they preserve every side length and angle, so the image is always congruent to the object (identical size and shape), just possibly in a different position or orientation. This is different from enlargement, which changes size.
Real-World Applications
Transformations appear throughout design and technology:
- Computer graphics and animation: sprites moved, flipped, or spun on screen.
- Art and design: tessellating and symmetric patterns.
- Manufacturing: robotic arm movements combining rotations and translations.
- Architecture: symmetric building designs using reflections.
- Dance and choreography: movement patterns described using transformations.
Exam Tips
Common Mistakes
MistakeConfusing a translation vector's x and y components
Fixin a vector (a,b), a is always the horizontal movement and b is always the vertical movement
MistakeSwapping the rules for reflecting in the x-axis and y-axis
Fixreflecting in the x-axis flips the y sign; reflecting in the y-axis flips the x sign
MistakeDescribing a rotation without stating its centre, angle, and direction
Fixalways fully describe a transformation - type plus its specific details
MistakeAssuming a translation, reflection, or rotation changes a shape's size
Fixthese always produce a congruent (same size and shape) image
MistakeApplying combined transformations in the wrong order
Fixapply each transformation in the order given in the question
For Exams
- Learn the three rigid transformations: translation (slide), reflection (flip), rotation (turn).
- Always state full details when describing a transformation, not just its type.
- Remember: the image is always congruent to the object.
- For combined transformations, work through them one step at a time, in order.
- Check your coordinates carefully, especially signs.
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