IGCSE MATH: TRANSFORMATIONS

IGCSE Mathematics (0580) – Transformations Complete Guide

Master IGCSE transformations using grid methods, tracing paper techniques, and coordinate analysis.

1. Reflection – Using Perpendicular Distances

Method: Find perpendicular distance from each point to the line. Image is same distance on opposite side.

Example: Reflection in x = 1

Triangle: (4,2), (6,1), (5,4)

(4,2) → 3 right → 3 left → (-2,2)
(6,1) → 5 right → 5 left → (-4,1)
(5,4) → 4 right → 4 left → (-3,4)

Tip: Use tracing paper! Trace, fold along mirror line, trace image.

2. Rotation – Center Point Methods

Method:

Find distances from center to point
Apply rotation rules
Add back to center

90° clockwise: (x,y) → (y,-x)

90° anticlockwise: (x,y) → (-y,x)

180°: (x,y) → (-x,-y)

Example: 90° ACW about (6,0)

Point (8,3):

Distances: 2 right, 3 up
Apply: (-3,2)
Add: 6+(-3)=3, 0+2=2 → (3,2)

3. Translation – Counting Squares

Example: Vector (-4,5)

Triangle: (2,1), (4,1), (3,3)

(2,1) → (-2,6)
(4,1) → (0,6)
(3,3) → (-1,8)

4. Enlargement – Ray Method

Center C(cₓ,cᵧ), point P(pₓ,pᵧ), SF k:

Image = (cₓ + k(pₓ-cₓ), cᵧ + k(pᵧ-cᵧ))

Example: SF ½, center (-5,-3)

Point (-3,1):

Differences: 2, 4
×½: (1,2)
Add: (-4,-1)

Negative SF → image on opposite side of center

5. Area Relationships

New area = Original area × (scale factor)²

Original: 20 cm², SF: 1.2

New area = 20 × 1.44 = 28.8 cm²

6. Identifying Transformations

Step	Check	Conclusion
1	Orientation	Same: translation/reflection Different: rotation/enlargement
2	Size	Same: translation/rotation/reflection Different: enlargement
3	Coordinates	Find pattern

7. Essential Formulae

Transformation	Formula
Reflection x=a	(x,y)→(2a-x,y)
Reflection y=b	(x,y)→(x,2b-y)
90° CW (origin)	(x,y)→(y,-x)
Translation (a,b)	(x,y)→(x+a,y+b)
Area after enlargement	Original × k²

8. Practice Questions

Question 1:

Triangle P: (-1,2), (1,1), (0,4)

(a) Reflect in y=x

(b) Rotate 90° CW about (2,-1)

Question 2:

Rectangle area: 32 cm²

Enlarged SF 1.5, then SF ⅔

Find final area.

Exam Tip: Use sharp pencil, count carefully, draw light lines, check all points.

9. Common Mistakes

Mistake	Correction
Wrong reflection	Check perpendicular distances
Wrong rotation	Clockwise = to the right
Wrong center	Measure from given center
Area error	Square the scale factor

Enlargement of Shapes on a Grid - Complete Guide

Enlargement is transforming a shape to make it larger or smaller from a fixed centre point, using a scale factor.

Key Terms

Centre of Enlargement: The fixed starting point for the enlargement
Scale Factor: How much to enlarge or reduce the shape
Ray Method: Drawing lines from centre through each vertex

How to Perform an Enlargement

Method: Using Rays from the Centre

Mark the centre of enlargement on the grid
For each vertex of the shape, draw a line from the centre through that vertex
Multiply the distance from centre to vertex by the scale factor
Plot the new vertex at this calculated distance
Connect all new vertices to form the enlarged shape

Types of Scale Factors

Positive Scale Factors

Scale Factor > 1: Shape gets larger (e.g., scale factor 2 = double size)
Scale Factor = 1: Shape stays the same size
Scale Factor between 0 and 1: Shape gets smaller (e.g., scale factor ½ = half size)

Example: If a point is 3 squares from centre and scale factor is 2, new point will be 6 squares from centre along the same line.

Negative Scale Factors

Shape appears on the opposite side of the centre
Scale factor -1: Same size but inverted through the centre
Scale factor -2: Double size and on opposite side of centre
Scale factor -½: Half size and on opposite side of centre

Example: If scale factor is -2, a point 3 squares right of centre moves to 6 squares LEFT of centre.

Step-by-Step Process

For Positive Scale Factors:

Count horizontal and vertical squares from centre to each vertex
Multiply both distances by the scale factor
Plot new vertex using these multiplied distances

Example: Vertex at (2,3) from centre, scale factor 2 → new vertex at (4,6) from centre

For Negative Scale Factors:

Count horizontal and vertical squares from centre to each vertex
Multiply both distances by the scale factor (ignore the negative sign for calculation)
Plot new vertex in the opposite direction from centre

Example: Vertex at (2,3) from centre, scale factor -2 → new vertex at (-4,-6) from centre

Finding Centre and Scale Factor from Two Shapes

To find the Centre of Enlargement:

Draw straight lines through matching vertices of both shapes
Where all these lines meet is the centre of enlargement

To find the Scale Factor:

Pick a vertex from original shape and its matching vertex from enlarged shape
Measure distance from centre to original vertex
Measure distance from centre to enlarged vertex
Divide: Scale Factor = (Distance to enlarged vertex) ÷ (Distance to original vertex)

Example: If original vertex is 2 squares from centre and enlarged vertex is 6 squares from centre, scale factor = 6 ÷ 2 = 3

Quick Reference

Scale Factor 2: Double distance from centre
Scale Factor ½: Half distance from centre
Scale Factor -1: Same distance but opposite side
Scale Factor -2: Double distance and opposite side

Remember: Negative scale factors create images on the opposite side of the centre point!

Rotation Rules - General Formulas

Here are the general rules for rotating points about the origin and any point (a,b) on a coordinate grid.

Rotation about the Origin (0,0)

Angle & Direction	Rule	Original Point (x,y)	New Point
90° Anticlockwise	(x,y) → (-y,x)	(3,2)	(-2,3)
90° Clockwise	(x,y) → (y,-x)	(3,2)	(2,-3)
180° (Either direction)	(x,y) → (-x,-y)	(3,2)	(-3,-2)
270° Anticlockwise	(x,y) → (y,-x)	(3,2)	(2,-3)
270° Clockwise	(x,y) → (-y,x)	(3,2)	(-2,3)

Rotation about Any Point (a,b)

Method: Use a 3-step process:

Translate the shape so point (a,b) becomes the origin
Rotate using the origin rules above
Translate back to original position

Angle & Direction	Rule	Original Point (x,y)	Centre (a,b)	New Point
90° Anticlockwise	(x,y) → (-(y-b)+a, (x-a)+b)	(5,4)	(2,1)	(-1,4)
90° Clockwise	(x,y) → ((y-b)+a, -(x-a)+b)	(5,4)	(2,1)	(5,-2)
180° (Either direction)	(x,y) → (2a-x, 2b-y)	(5,4)	(2,1)	(-1,-2)
270° Anticlockwise	(x,y) → ((y-b)+a, -(x-a)+b)	(5,4)	(2,1)	(5,-2)
270° Clockwise	(x,y) → (-(y-b)+a, (x-a)+b)	(5,4)	(2,1)	(-1,4)

Simplified Method for Any Centre (a,b)

Step 1: Subtract (a,b) from your point: (x-a, y-b)
Step 2: Apply the origin rotation rules to (x-a, y-b)
Step 3: Add (a,b) back to the result

Example: Rotate (5,4) 90° anticlockwise about (2,1)

(5-2, 4-1) = (3,3)
Rotate (3,3): (-3,3)
(-3+2, 3+1) = (-1,4)

Quick Memory Tips

90° Anticlockwise: Swap coordinates and negate the new x
90° Clockwise: Swap coordinates and negate the new y
180°: Negate both coordinates
For any centre: "Shift, rotate, shift back"

Note: 270° anticlockwise = 90° clockwise, and 270° clockwise = 90° anticlockwise

Reflecting Points Through Lines - Study Notes

Reflecting Points Through Lines: Complete Guide

When You Don't Need the Formula

Special Cases (Easy Reflections)

1. Reflecting over the x-axis (y = 0)

Rule: (a, b) → (a, -b)
Just flip the sign of the y-coordinate
Example: (3, 5) → (3, -5)

2. Reflecting over the y-axis (x = 0)

Rule: (a, b) → (-a, b)
Just flip the sign of the x-coordinate
Example: (3, 5) → (-3, 5)

3. Reflecting over y = x

Rule: (a, b) → (b, a)
Swap the coordinates
Example: (3, 5) → (5, 3)

4. Reflecting over y = -x

Rule: (a, b) → (-b, -a)
Swap and negate both coordinates
Example: (3, 5) → (-5, -3)

General Method: Reflecting (a, b) over y = mx + c

The Step-by-Step Process

Step 1: Check if the point is on the line

Substitute: Does b = ma + c?
If YES: The point reflects to itself!
If NO: Continue to Step 2

Step 2: Find the perpendicular slope

Line slope: m
Perpendicular slope: -1/m
(Remember: perpendicular slopes multiply to give -1)

Step 3: Write the perpendicular line through (a, b)

Use point-slope form: y - b = (-1/m)(x - a)
Simplify to: y = (-1/m)x + (b + a/m)

Step 4: Find where the perpendicular meets y = mx + c

Set the equations equal: mx + c = (-1/m)x + (b + a/m)
Solve for x, then find y
This gives you point P (the foot of the perpendicular)

Step 5: Use the midpoint formula

P is the midpoint between (a, b) and its reflection (a', b')
Set up: P = ((a + a')/2, (b + b')/2)
Solve for a' and b':
- a' = 2P_x - a
- b' = 2P_y - b

The Quick Formula (Advanced)

If you want a direct formula, the reflection of (a, b) over y = mx + c is:

a' = a - 2m(ma - b + c)/(m² + 1)

b' = b + 2(ma - b + c)/(m² + 1)

How to use this formula:

Calculate: d = ma - b + c (this measures distance from point to line)
Calculate: k = m² + 1
Then: a' = a - 2md/k and b' = b + 2d/k

Worked Example

Reflect (2, 3) over y = 2x + 1

Step 1: Check if on line

3 = 2(2) + 1 = 5? NO

Step 2: Perpendicular slope

m = 2, so perpendicular slope = -1/2

Step 3: Perpendicular line through (2, 3)

y - 3 = (-1/2)(x - 2)
y = -1/2x + 4

Step 4: Find intersection P

2x + 1 = -1/2x + 4
2.5x = 3
x = 1.2, y = 3.4
P = (1.2, 3.4)

Step 5: Find reflection

a' = 2(1.2) - 2 = 0.4
b' = 2(3.4) - 3 = 3.8
Answer: (0.4, 3.8) or (2/5, 19/5)

Common Mistakes to Avoid

❌ Forgetting to use the perpendicular slope (not the same slope!)
❌ Not checking if the point is already on the line
❌ Making arithmetic errors when solving simultaneous equations
❌ Forgetting that the intersection point P is the midpoint, not the reflection

Quick Check

To verify your answer, check that:

The original point and reflection are equidistant from the line
The line connecting them is perpendicular to y = mx + c
The midpoint of the original and reflection lies on y = mx + c

Practice Problems

Try these:

Reflect (0, 0) over y = x + 2
Reflect (4, 1) over y = -x + 3
Reflect (-1, 2) over y = 3x - 1
Reflect (5, 5) over y = x (use the special case!)

Answers:

(4, -4)
(2, 5)
(1.4, -2.2) or (7/5, -11/5)
(5, 5) - already on the line!

Translating Points on a Grid - Study Notes

Translating Points on a Grid: Complete Guide

What is Translation?

Translation means sliding or moving a point (or shape) from one position to another on a grid. Every point moves the same distance in the same direction.

Key Idea: Translation does NOT change the size, shape, or orientation of an object. It only changes its position!

The Translation Rule

General Formula:

If you translate point (x, y) by vector (a, b):

New point = (x + a, y + b)

What does the vector mean?

a = horizontal movement (left/right)
b = vertical movement (up/down)

Sign conventions:

Positive a → move RIGHT
Negative a → move LEFT
Positive b → move UP
Negative b → move DOWN

Step-by-Step Method

To translate a point (x, y) by vector (a, b):

Step 1: Identify the original point coordinates (x, y)

Step 2: Identify the translation vector (a, b)

a = horizontal shift
b = vertical shift

Step 3: Apply the formula

New x-coordinate = x + a
New y-coordinate = y + b

Step 4: Write the new point as (x + a, y + b)

Common Translation Examples

Translation Vector	Movement Description	Example: (3, 2) becomes...
(5, 0)	5 units RIGHT	(8, 2)
(-3, 0)	3 units LEFT	(0, 2)
(0, 4)	4 units UP	(3, 6)
(0, -2)	2 units DOWN	(3, 0)
(2, 3)	2 RIGHT, 3 UP	(5, 5)
(-1, -4)	1 LEFT, 4 DOWN	(2, -2)

Worked Examples

Example 1: Simple Translation

Question: Translate point (2, 5) by vector (3, -2)

Solution:

Original point: (2, 5)
Translation vector: (3, -2)
New x = 2 + 3 = 5
New y = 5 + (-2) = 3
Answer: (5, 3)

Check: We moved 3 units right and 2 units down ✓

Example 2: Negative Coordinates

Question: Translate point (-3, 1) by vector (4, -5)

Solution:

Original point: (-3, 1)
Translation vector: (4, -5)
New x = -3 + 4 = 1
New y = 1 + (-5) = -4
Answer: (1, -4)

Example 3: Finding the Translation Vector

Question: Point A(2, 3) is translated to B(7, 1). What is the translation vector?

Solution:

Start: (2, 3), End: (7, 1)
Horizontal change: 7 - 2 = 5
Vertical change: 1 - 3 = -2
Answer: Translation vector is (5, -2)

Formula to find translation vector:
Vector = (x₂ - x₁, y₂ - y₁)

Translating Shapes

When translating a shape, translate each vertex using the same vector.

Example: Translate Triangle

Question: Triangle ABC has vertices A(1, 2), B(3, 5), C(4, 1). Translate by vector (2, -3).

Solution:

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

Answer: New triangle A'B'C' has vertices (3, -1), (5, 2), (6, -2)

Special Cases

1. No Movement

Vector (0, 0) means no translation
Point stays in the same place
Example: (5, 3) + (0, 0) = (5, 3)

2. Reverse Translation

To undo a translation, use the opposite vector
If you translate by (a, b), reverse it with (-a, -b)
Example: (3, 4) + (2, 5) = (5, 9)
Reverse: (5, 9) + (-2, -5) = (3, 4) ✓

Common Mistakes to Avoid

❌ Forgetting the negative signs in the vector
❌ Mixing up x and y coordinates
❌ Subtracting instead of adding (remember: always ADD the vector)
❌ Translating different vertices by different amounts
❌ Confusing translation with reflection or rotation

Tips for Success

✓ Always write your starting point clearly
✓ Draw arrows on your grid to show the movement
✓ Check your signs: positive right/up, negative left/down
✓ Count squares on the grid to verify your answer
✓ For shapes, translate ALL vertices the same way

Quick Reference Guide

Translation Formula:
(x, y) + vector (a, b) = (x + a, y + b)

Finding Translation Vector:
From (x₁, y₁) to (x₂, y₂) = (x₂ - x₁, y₂ - y₁)

Reverse Translation:
Opposite of (a, b) is (-a, -b)

Practice Problems

Try these:

Basic:

Translate (4, 3) by vector (2, 5)
Translate (-2, 6) by vector (3, -4)
Translate (0, 0) by vector (-3, 2)

Finding Vectors:

Find the vector that translates (1, 4) to (5, 7)
Find the vector that translates (3, -2) to (0, 3)

Shapes:

Translate square with vertices (1,1), (1,3), (3,3), (3,1) by vector (2, -1)

Answers:

(6, 8)
(1, 2)
(-3, 2)
(4, 3)
(-3, 5)
New vertices: (3, 0), (3, 2), (5, 2), (5, 0)

Translation vs Other Transformations

Transformation	What it does	Changes size/shape?
Translation	Slides/moves position	NO
Reflection	Flips across a line	NO
Rotation	Turns around a point	NO
Enlargement	Makes bigger/smaller	YES