Transformations | FractionRush

Welcome to Transformations!

A transformation moves or changes a shape on a coordinate grid. The original shape is called the object; the result is the image. In IGCSE 0580 you need to both perform transformations (find image coordinates) and describe them fully (state all the required information). There are four types: Translation, Reflection, Rotation and Enlargement.

4 types: Translation · Reflection · Rotation · Enlargement

Translation

Column vector, image coords

Reflection

x-axis, y-axis, y=x, y=−x, x=a, y=b

Rotation

Centre, angle, direction

Enlargement

Scale factor, centre

Combined & Inverse

Two-step transformations

Invariant Points

Points that don't move

1. Translation

A translation slides every point of a shape by the same amount. It is described using a column vector. The shape does not rotate or change size.

Column vector: ^a/_b — moves a right (negative = left) and b up (negative = down)

To find the image: Add the column vector to every coordinate.
If vector = ³/₋₂ and object point is (1, 4), then image point = (1+3, 4+(−2)) = (4, 2)

To fully describe a translation, you must state:
1. The word "Translation"
2. The column vector (e.g. ³/₋₂)

Object and image are congruent (same shape and size). No invariant points — every point moves by the same vector.

2. Reflection

A reflection flips a shape in a mirror line. Every point and its image are equidistant from the mirror line, on opposite sides, and the line joining them is perpendicular to the mirror.

Object and image are mirror images — congruent, but orientation is reversed

Mirror line	Rule for (x, y) → image
x-axis (y = 0)	(x, y) → (x, −y)
y-axis (x = 0)	(x, y) → (−x, y)
y = x	(x, y) → (y, x)
y = −x	(x, y) → (−y, −x)
x = a (vertical line)	(x, y) → (2a − x, y)
y = b (horizontal line)	(x, y) → (x, 2b − y)

Example — reflect (3, 1) in y = x:
Swap x and y: image = (1, 3)

Example — reflect (5, 2) in x = 3:
x-image = 2(3) − 5 = 6 − 5 = 1, y stays: image = (1, 2)

To fully describe a reflection, you must state:
1. The word "Reflection"
2. The equation of the mirror line (e.g. y = x)

Invariant points: any point that lies ON the mirror line maps to itself.

3. Rotation

A rotation turns a shape through an angle about a fixed point called the centre of rotation. Distances from the centre are preserved.

Rotation about the origin — key rules:

Rotation	Rule for (x, y) → image
90° clockwise (= 270° anticlockwise)	(x, y) → (y, −x)
90° anticlockwise (= 270° clockwise)	(x, y) → (−y, x)
180° (either direction)	(x, y) → (−x, −y)

Rotating about a point other than the origin — method:
1. Subtract the centre from the point: (x − cx, y − cy)
2. Apply the rotation rule
3. Add the centre back

Example: Rotate (5, 2) by 90° anticlockwise about centre (2, 1)
Step 1: (5−2, 2−1) = (3, 1)
Step 2: 90° ACW: (−y, x) → (−1, 3)
Step 3: Add centre: (−1+2, 3+1) = (1, 4)

To fully describe a rotation, you must state:
1. The word "Rotation"
2. The angle (e.g. 90°, 180°)
3. The direction (clockwise or anticlockwise) — not needed for 180°
4. The centre of rotation as a coordinate (e.g. (0, 0))

A 180° rotation is the same in both directions. The only invariant point is the centre of rotation itself.

4. Enlargement

An enlargement scales a shape from a fixed point called the centre of enlargement by a scale factor (SF). Image distances from the centre = SF × object distances.

Image vector from centre = SF × object vector from centre

Scale factor	Effect
SF > 1	Enlargement — image larger than object, same side of centre
0 < SF < 1	Reduction — image smaller than object, same side of centre
SF < 0 (negative)	Image on opposite side of centre; \|SF\| > 1 enlarges, \|SF\| < 1 reduces

Method — enlarge point P with centre C and scale factor k:
Image = C + k × (P − C)

Example: Enlarge (4, 1) with centre (2, −1), SF = 3
Vector from centre: (4−2, 1−(−1)) = (2, 2)
Multiply by SF: 3 × (2, 2) = (6, 6)
Add centre: (2+6, −1+6) = (8, 5)

Negative SF example: Enlarge (3, 2) with centre (1, 1), SF = −2
Vector from centre: (3−1, 2−1) = (2, 1)
Multiply by −2: (−4, −2)
Add centre: (1+(−4), 1+(−2)) = (−3, −1) — image is on the opposite side of the centre.

To fully describe an enlargement, you must state:
1. The word "Enlargement"
2. The scale factor (including sign if negative)
3. The centre of enlargement as a coordinate

Finding SF from object and image: SF = image length / object length. Check with a side length ratio or compare distances from the centre.
Finding the centre: Draw lines through corresponding vertices — they all meet at the centre.

5. Combined and Inverse Transformations

A combined transformation applies two (or more) transformations in sequence. Apply them one at a time in order — the image of the first becomes the object for the second.

Example: Shape A is translated by ²/₁ to give B, then B is reflected in the x-axis to give C.
If A has vertex (1, 3):
After translation: (1+2, 3+1) = (3, 4)
After reflection in x-axis: (3, −4)

Inverse transformation: The transformation that maps the image back to the object.
• Inverse of translation ^a/_b is translation ^−a/_−b
• Inverse of a reflection is the same reflection
• Inverse of rotation 90° CW is rotation 90° ACW (or 270° CW)
• Inverse of enlargement SF k, centre C is enlargement SF 1/k, same centre C

When a question says "describe the single transformation that maps A onto C", look at the overall effect — sometimes two reflections combine to give a rotation, or two translations combine to give a single translation.

6. Invariant Points

An invariant point is a point that maps to itself under a transformation — it does not move.

Invariant points by transformation type:
• Translation: None (every point moves, unless vector is zero)
• Reflection: All points on the mirror line
• Rotation: Only the centre of rotation
• Enlargement SF ≠ 1: Only the centre of enlargement
• Enlargement SF = 1: Every point (identity transformation)

To find invariant points algebraically: set image = object and solve. For example, reflection in y = x maps (x, y) → (y, x). Invariant when (x, y) = (y, x), i.e. x = y — so all points on the line y = x are invariant.

Example 1 — Perform: Translate a triangle

Object: Triangle with vertices A(1, 2), B(3, 2), C(3, 5). Vector = ⁻³/₁.

Add the vector to each vertex:
A′ = (1+(−3), 2+1) = (−2, 3)
B′ = (3+(−3), 2+1) = (0, 3)
C′ = (3+(−3), 5+1) = (0, 6)

Check: All points moved 3 left and 1 up. Shape unchanged in size and orientation.

Example 2 — Describe: Identify a reflection

Object: A(2, 1), B(5, 1), C(5, 4). Image: A′(2, −1), B′(5, −1), C′(5, −4).

Look at what changed: x-coordinates stayed the same; y-coordinates changed sign. This is reflection in the x-axis.

Full description: Reflection in the line y = 0 (the x-axis).

Example 3 — Perform: Rotate 90° anticlockwise about origin

Object: P(4, 1), Q(4, 3), R(1, 1). Rotate 90° anticlockwise about (0, 0).

Rule 90° ACW: (x, y) → (−y, x)
P′ = (−1, 4)
Q′ = (−3, 4)
R′ = (−1, 1)

Full description of this transformation: Rotation, 90° anticlockwise, centre (0, 0).

Example 4 — Perform: Enlarge with positive scale factor

Object: A(3, 1), B(5, 1), C(5, 4). Centre of enlargement (1, 1), SF = 2.

Method: Image = Centre + SF × (Point − Centre)
A′ = (1,1) + 2×(3−1, 1−1) = (1,1) + 2×(2,0) = (1+4, 1+0) = (5, 1)
B′ = (1,1) + 2×(5−1, 1−1) = (1,1) + 2×(4,0) = (9, 1)
C′ = (1,1) + 2×(5−1, 4−1) = (1,1) + 2×(4,3) = (1+8, 1+6) = (9, 7)

Check: Each side of the image is twice the length of the object. Centre (1,1) is invariant.

Example 5 — Describe: Find centre and SF of an enlargement

Object: A(2, 1), B(4, 1), C(4, 3). Image: A′(−1, −2), B′(5, −2), C′(5, 4).

Find SF: AB = 2, A′B′ = 6. SF = 6/2 = 3. Since image is larger and on the same side, SF = 3.

Find centre: Draw a line through A(2,1) and A′(−1,−2): slope = (−2−1)/(−1−2) = −3/−3 = 1, line: y = x − 1. Draw line through B(4,1) and B′(5,−2): slope = (−2−1)/(5−4) = −3. Line: y − 1 = −3(x − 4) → y = −3x + 13. Intersection: x − 1 = −3x + 13 → 4x = 14 → x = 3.5, y = 2.5. Centre = (3.5, 2.5)... wait, let us re-check with integer centre approach. Ratio check: Centre C satisfies C + 3(A − C) = A′. 3A − 2C = A′ → 2C = 3A − A′ = (6,3) − (−1,−2) = (7,5) → C = (3.5, 2.5). SF = 3, centre (3.5, 2.5).

Full description: Enlargement, scale factor 3, centre (3.5, 2.5).

Example 6 — Combined: Two transformations in sequence

Object triangle T has vertices (1, 0), (3, 0), (1, 2). First, rotate 180° about origin to get T′. Then reflect T′ in y = x to get T″.

Step 1 — Rotation 180°: (x,y)→(−x,−y)
(1,0)→(−1,0) (3,0)→(−3,0) (1,2)→(−1,−2)

Step 2 — Reflect in y = x: (x,y)→(y,x)
(−1,0)→(0,−1) (−3,0)→(0,−3) (−1,−2)→(−2,−1)

T″ vertices: (0,−1), (0,−3), (−2,−1). The single equivalent transformation mapping T to T″ is a rotation of 90° clockwise about the origin.

Transformation Visualiser

Pick a transformation and see it applied to a triangle. The pink shape is the object; the blue shape is the image.

Type:

Select a transformation and click Apply.

Exercise 1 — Translation

Find the image coordinates after each translation. Enter x then y for the single vertex marked.

1. Translate (2, 3) by vector ⁴/₁. Give the image x-coordinate.

2. Translate (2, 3) by vector ⁴/₁. Give the image y-coordinate.

3. Translate (−1, 5) by vector ³/₋₂. Give the image x-coordinate.

4. Translate (−1, 5) by vector ³/₋₂. Give the image y-coordinate.

5. Translate (0, −3) by vector ⁻²/₄. Give the image x-coordinate.

6. Translate (0, −3) by vector ⁻²/₄. Give the image y-coordinate.

7. Triangle vertices: A(1,1), B(4,1), C(1,4). Translate by ²/₋₃. Give x-coord of A′.

8. Same triangle. Give y-coord of A′.

9. A point P′ = (5, −1) is the image of P under translation ³/₂. Give the x-coord of P.

10. Same. Give the y-coord of P.

Exercise 2 — Reflection

Give the image coordinate (x or y as asked) after reflecting the given point.

1. Reflect (3, 5) in the x-axis. Give the image y-coordinate.

2. Reflect (−2, 4) in the y-axis. Give the image x-coordinate.

3. Reflect (3, 7) in the line y = x. Give the image x-coordinate.

4. Reflect (3, 7) in the line y = x. Give the image y-coordinate.

5. Reflect (4, 2) in the line y = −x. Give the image x-coordinate.

6. Reflect (4, 2) in the line y = −x. Give the image y-coordinate.

7. Reflect (5, 3) in the line x = 2. Give the image x-coordinate.

8. Reflect (1, 6) in the line y = 4. Give the image y-coordinate.

9. Reflect (−3, 2) in the x-axis. Give the image y-coordinate.

10. Reflect (6, 1) in the line x = 1. Give the image x-coordinate.

Exercise 3 — Rotation

All rotations are about the origin unless stated. Give the requested image coordinate.

1. Rotate (3, 1) by 90° clockwise about origin. Give image x-coordinate.

2. Rotate (3, 1) by 90° clockwise about origin. Give image y-coordinate.

3. Rotate (2, 5) by 90° anticlockwise about origin. Give image x-coordinate.

4. Rotate (2, 5) by 90° anticlockwise about origin. Give image y-coordinate.

5. Rotate (−3, 4) by 180° about origin. Give image x-coordinate.

6. Rotate (−3, 4) by 180° about origin. Give image y-coordinate.

7. Rotate (4, 1) by 90° anticlockwise about centre (1, 1). Give image x-coordinate.

8. Rotate (4, 1) by 90° anticlockwise about centre (1, 1). Give image y-coordinate.

9. Rotate (5, 2) by 180° about centre (2, 2). Give image x-coordinate.

10. Rotate (5, 2) by 180° about centre (2, 2). Give image y-coordinate.

Exercise 4 — Enlargement

Find image coordinates under each enlargement. Give the coordinate asked.

1. Enlarge (4, 2) from origin, SF = 3. Give image x-coordinate.

2. Enlarge (4, 2) from origin, SF = 3. Give image y-coordinate.

3. Enlarge (6, 4) from origin, SF = ½. Give image x-coordinate.

4. Enlarge (6, 4) from origin, SF = ½. Give image y-coordinate.

5. Enlarge (5, 1) from centre (1, 1), SF = 2. Give image x-coordinate.

6. Enlarge (5, 1) from centre (1, 1), SF = 2. Give image y-coordinate.

7. Enlarge (3, 5) from centre (1, 1), SF = −1. Give image x-coordinate.

8. Enlarge (3, 5) from centre (1, 1), SF = −1. Give image y-coordinate.

9. Enlarge (2, 4) from centre (0, 0), SF = −2. Give image x-coordinate.

10. Enlarge (2, 4) from centre (0, 0), SF = −2. Give image y-coordinate.

Exercise 5 — Describing Transformations

Each question gives object and image coordinates. Enter the numerical value required to fully describe the transformation.

1. A(2,1)→A′(2,−1), B(5,3)→B′(5,−3). This is a reflection. The mirror line is y = ? Enter the value.

2. A(1,3)→A′(3,1), B(2,5)→B′(5,2). Reflection. Mirror line is y = mx. Enter m.

3. A(4,2)→A′(−2,4), B(3,1)→B′(−1,3). This is a rotation about origin. Enter the angle in degrees (positive = ACW).

4. A(1,2)→A′(−1,−2), B(3,4)→B′(−3,−4). Rotation about origin. Enter the angle in degrees.

5. A(2,1)→A′(6,3), B(4,1)→B′(12,3). Enlargement from origin. Enter the scale factor.

6. A(0,2)→A′(0,−4), B(3,2)→B′(−6,−4). Enlargement. Centre is origin. Enter the scale factor.

7. A(3,1)→A′(1,3): this is reflection in y = x? Enter 1 for yes, 0 for no.

8. Translate P(5,−1) back to Q(2,3). Give the x-component of the column vector (Q − P).

9. Translate P(5,−1) back to Q(2,3). Give the y-component of the column vector.

10. Object AB has length 2. Image A′B′ has length 6. What is the scale factor of enlargement?

Practice — 20 Questions

Mixed practice covering all transformation skills. Give the numerical answer requested.

1. Translate (3, −2) by ⁵/₃. Give x-coordinate of image.

2. Translate (3, −2) by ⁵/₃. Give y-coordinate of image.

3. Reflect (7, −3) in x-axis. Give y-coordinate of image.

4. Reflect (−4, 6) in y-axis. Give x-coordinate of image.

5. Reflect (2, 9) in y = x. Give x-coordinate of image.

6. Reflect (5, 3) in y = −x. Give x-coordinate of image.

7. Reflect (7, 2) in line x = 3. Give image x-coordinate.

8. Reflect (4, 1) in line y = 5. Give image y-coordinate.

9. Rotate (2, 6) by 90° CW about origin. Give image x-coordinate.

10. Rotate (2, 6) by 90° CW about origin. Give image y-coordinate.

11. Rotate (−3, 4) by 90° ACW about origin. Give image x-coordinate.

12. Rotate (1, 5) by 180° about origin. Give image y-coordinate.

13. Enlarge (3, 4) from origin, SF = 4. Give image x-coordinate.

14. Enlarge (5, 10) from origin, SF = ½. Give image y-coordinate.

15. Enlarge (7, 3) from centre (1, 3), SF = 2. Give image x-coordinate.

16. Rotate (6, 2) by 90° CW about centre (2, 2). Give image x-coordinate.

17. Rotate (6, 2) by 90° CW about centre (2, 2). Give image y-coordinate.

18. A′(3, 0) is the image of A under translation ⁻²/₄. Give x-coord of A.

19. Object side = 4, image side = 1. Scale factor = ?

20. Enlarge (4, 6) from centre (2, 2), SF = −1. Give image x-coordinate.

Challenge — 8 Questions

Harder questions: combined transformations, negative enlargements, and finding centres. Give the numerical answer.

1. Triangle T has vertex A(1,2). Rotate A 90° ACW about origin to get A′, then translate A′ by ⁻¹/₃. Give x-coordinate of final image.

2. Same as Q1. Give y-coordinate of final image.

3. Point P(4, 3). Reflect in y-axis, then reflect the result in x-axis. Give x-coordinate of final image.

4. Same as Q3. Give y-coordinate of final image.

5. Enlarge (5, 3) from centre (3, 1), SF = −2. Give image x-coordinate.

6. Enlarge (5, 3) from centre (3, 1), SF = −2. Give image y-coordinate.

7. Object A(1,0)→Image A′(3,4). Object B(3,0)→Image B′(7,4). The transformation is an enlargement. SF = image AB / object AB. AB = 2, A′B′ = 4. Give the scale factor.

8. Triangle vertices: (2,1),(4,1),(2,4). Rotate 90° CW about origin, then enlarge from origin by SF = 2. The vertex originally at (2,1) ends up at (x, y). Give x-coordinate.