How to Solve: Transformations (Translation, Rotation, Reflection, Enlargement)

How to Solve: Transformations (Translation, Rotation, Reflection, Enlargement)

For GCSE & A-Level Maths (Edexcel/AQA/OCR)

Introduction

"Mastering transformations lets you ace 6–8 marks on your GCSE paper—enough to boost your grade by one level. Architects, game designers, and even surgeons use these exact rules to move, flip, and resize objects. Today, you’ll learn the step-by-step method to solve any transformation question in under 2 minutes."

What You Need To Know First

1. Coordinates (x, y): You must plot points on a grid.
2. Vectors: Understand column vectors (e.g., (3, -2) means 3 right, 2 down).
3. Angles & Directions: Clockwise vs. anticlockwise, 90° vs. 180° turns.

Key Vocabulary

Formulas To Know

1. Translation

• Vector: (a, b)
• a = horizontal shift (right if +, left if –).
• b = vertical shift (up if +, down if –).
• MEMORISE THIS: Add the vector to every point of the shape.

2. Rotation

• Centre (h, k): The fixed point the shape turns around.
• Angle θ: Degrees turned (90°, 180°, 270°).
• Direction: Clockwise (CW) or anticlockwise (ACW).
• MEMORISE THIS:
  • 90° ACW (or 270° CW): (x, y) → (-y, x)
  • 180°: (x, y) → (-x, -y)
  • 90° CW (or 270° ACW): (x, y) → (y, -x)

3. Reflection

• Mirror Line: Equation of the line (e.g., y = x, x = 2).
• MEMORISE THIS:
  • Reflect over x-axis: (x, y) → (x, -y)
  • Reflect over y-axis: (x, y) → (-x, y)
  • Reflect over y = x: (x, y) → (y, x)
  • Reflect over y = -x: (x, y) → (-y, -x)

4. Enlargement

• Scale Factor (k): How much bigger/smaller the shape gets.
• k > 1: Shape gets bigger.
• 0 < k < 1: Shape gets smaller.
• k negative: Shape flips (enlarges on the opposite side).
• Centre (h, k): Fixed point the shape grows/shrinks from.
• MEMORISE THIS:
  • New point = (h + k(x – h), k + k(y – k))
  • Or: Distance from centre × scale factor in the same direction.

Step-by-Step Method

Step 1: Identify the Transformation

• Read the question carefully.
• Underline keywords: translate, rotate, reflect, enlarge.
• Note any given details (e.g., vector, angle, mirror line, scale factor).

Step 2: Find Key Points

• List the coordinates of the object’s vertices (corners).
• Example: Triangle ABC with points A(1,2), B(3,4), C(2,1).

Step 3: Apply the Transformation

• Translation: Add the vector to each point.
• Rotation: Use the rotation rules (memorised formulas).
• Reflection: Use the reflection rules (memorised formulas).
• Enlargement: Multiply distances from the centre by the scale factor.

Step 4: Plot the Image

• Write the new coordinates (A’, B’, C’).
• Plot them on the grid.
• Join the points to draw the transformed shape.

Step 5: Check for Invariance (If Needed)

• For rotation/enlargement: Is the centre invariant?
• For reflection: Are any points on the mirror line invariant?

Step 6: Verify

• Count squares for translations.
• Use tracing paper for rotations/reflections (if allowed).
• Check scale factor for enlargements (e.g., sides double if k=2).

Worked Examples

Example 1 – Basic: Translation

Question: Translate triangle ABC with vertices A(1,2), B(3,4), C(2,1) by the vector (4, -1).

Step 1: Identify transformation → Translation by (4, -1). Step 2: List points: A(1,2), B(3,4), C(2,1). Step 3: Add vector to each point: - A’ = (1+4, 2-1) = (5,1) - B’ = (3+4, 4-1) = (7,3) - C’ = (2+4, 1-1) = (6,0) Step 4: Plot A’(5,1), B’(7,3), C’(6,0) and join. What we did and why: Added the vector to each point to slide the shape 4 right and 1 down.

Example 2 – Medium: Rotation

Question: Rotate point P(3,4) 90° clockwise about the origin (0,0).

Step 1: Identify transformation → 90° CW rotation about (0,0). Step 2: Use the formula: (x, y) → (y, -x). Step 3: Apply to P(3,4): - P’ = (4, -3) Step 4: Plot P’(4,-3). What we did and why: Used the memorised rule for 90° CW rotation to swap x and y, then negate the new y.

Example 3 – Exam-Style: Reflection + Enlargement

Question: Shape A is reflected over the line y = x, then enlarged by scale factor 2 from centre (0,0). Find the final image of point (2,1).

Step 1: First transformation → Reflection over y = x. - Rule: (x, y) → (y, x). - (2,1) → (1,2). Step 2: Second transformation → Enlargement, scale factor 2, centre (0,0). - Rule: Multiply coordinates by scale factor. - (1,2) → (2,4). Step 3: Final image: (2,4). What we did and why: Applied transformations in order—first reflection, then enlargement—using the rules for each.

Common Mistakes

Exam Traps

1-Minute Recap (Night Before Exam)

"You’ve got this! Here’s the cheat sheet: 1. Translation: Add the vector to every point. 2. Rotation: Memorise 90°/180° rules—swap and negate. 3. Reflection: Use the mirror line rules (x-axis flips y, y=x swaps x and y). 4. Enlargement: Multiply distances from the centre by the scale factor. For combined transformations, do them one at a time. Double-check directions (CW vs. ACW) and centres. Plot points carefully—every mark counts! Now go ace that exam!