Transformations

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Notes

Translations

A **translation** moves a shape without changing its size or orientation; object and image are **congruent**.
Movement is described by a **column vector** $\begin{pmatrix} x \\ y \end{pmatrix}$ : $x$ = horizontal shift (right positive, left negative), $y$ = vertical shift (up positive, down negative).
To translate a shape, move each vertex by the vector and join the new vertices.
To describe a translation, state it is a translation and give the vector (e.g., $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$ ).
To reverse a translation, use the same vector with both signs changed.

Reflections

A **reflection** flips a shape across a **mirror line** (line of reflection); object and image are congruent.
Each point and its image are equidistant from the mirror line, measured perpendicularly.
Points on the mirror line are **invariant** (do not move).
Common mirror lines: vertical $x = k$ , horizontal $y = k$ , diagonal $y = x$ or $y = -x$ .
To describe a reflection, state it is a reflection and give the equation of the mirror line.
To reverse a reflection, apply the same reflection again.

Rotations

A **rotation** turns a shape about a fixed **centre of rotation**; object and image are congruent.
You must specify the centre, angle (90°, 180°, 270°) and direction (clockwise or anticlockwise). For 180° direction is not needed.
Use tracing paper: draw the shape, place pencil on centre, rotate by the given angle, then draw the image.
To find the centre of rotation for 90° or 270°, use trial and error with tracing paper. For 180°, draw lines connecting corresponding vertices; they intersect at the centre.
To reverse a rotation, rotate by the same angle in the opposite direction about the same centre.

Enlargements

An **enlargement** changes the size of a shape by a **scale factor** (SF) about a **centre of enlargement** (CoE).
If $SF > 1$ , image is larger; if $0 < SF < 1$ , image is smaller (fractional enlargement).
For a positive SF, measure horizontal and vertical distances from CoE to a vertex, multiply by SF, and count from CoE to find the image vertex.
For a **negative SF**, the image is rotated 180° and lies on the opposite side of the CoE; distances are multiplied by |SF|.
To describe an enlargement, state it is an enlargement, give the SF and the coordinates of the CoE.
To reverse an enlargement, use the reciprocal SF with the same CoE (sign stays for negative SF).
**Area scale factor** $= (linear$ SF)²; multiply original area by this to find image area.

Combined Transformations

When two or more transformations are applied sequentially, the final image can often be described by a **single transformation**.
To find the single transformation, consider the net effect on position, orientation and size.
Common combinations: rotation + translation can sometimes be a single rotation about a different centre; two reflections can be a rotation or translation.

Translation by vector (3, -1)

Reflection in line x = -1

Rotation 90° clockwise about (0,2)

Enlargement with scale factor 2, centre (2,1)

Practice questions

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1.What is the name of the transformation that moves a shape without changing its size or orientation?
Easy
- ATranslation
- BReflection
- CRotation
- DEnlargement
2.A shape is translated by the vector $\begin{pmatrix} -3 \\ 5 \end{pmatrix}$ . What does this mean?
Easy
- A3 left and 5 up
- B3 right and 5 down
- C3 left and 5 down
- D3 right and 5 up
3.What is the equation of the mirror line for a reflection that maps a point (2, 3) to (-2, 3)?
Easy
- A $x = 0$
- B $y = 0$
- C $x = 2$
- D $y = 3$
4.A rotation turns a shape. Which of the following is NOT needed to fully describe a rotation?
Easy
- AScale factor
- BCentre of rotation
- CAngle of rotation
- DDirection of rotation
5.An enlargement has scale factor 0.5. What does this do to the size of the shape?
Easy
- AMakes it smaller
- BMakes it larger
- CKeeps it the same size
- DFlips it over
6.When describing a translation, what must you include?
Easy
- AA column vector
- BA mirror line
- CA centre of rotation
- DA scale factor
7.Triangle A has vertices (1, 2), (3, 2), (1, 5). It is translated by vector $\begin{pmatrix} -2 \\ 4 \end{pmatrix}$ . What are the coordinates of the image of vertex (1, 2)?
Medium
- A(-1, 6)
- B(3, -2)
- C(-1, -2)
- D(3, 6)
8.A shape is reflected in the line $y = 2$ . Which of the following points is invariant?
Medium
- A(3, 2)
- B(2, 3)
- C(0, 0)
- D(1, 1)

Summary

## Transformations (IGCSE Maths Extended) ### Key Concepts - **Object**: original shape; **Image**: transformed shape (vertices labelled A', B', etc.) - Four types: **translation**, **reflection**, **rotation**, **enlargement** ### Translations - Moves shape without changing size/orientation; object and image are **congruent**. - Described by a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$: - $x$: horizontal (+ right, – left) - $y$: vertical (+ up, – down) - **To describe**: state "translation" + vector. - **Reverse**: change signs of both components. ### Reflections - Flips shape across a **mirror line** (line of reflection). - Perpendicular distances from mirror line are equal; points on line are **invariant**. - Mirror lines: vertical $x = k$, horizontal $y = k$, diagonal $y = x$ or $y = -x$. - **To describe**: state "reflection" + equation of mirror line. - **Reverse**: same reflection again. ### Rotations - Turns shape about a **centre of rotation** by a given angle and direction. - Angles: 90°, 180°, 270°; direction: clockwise or anti-clockwise (omit for 180°). - Use tracing paper: draw arrow up, rotate, check orientation. - **To describe**: state "rotation", centre, angle, direction. - **Reverse**: opposite direction, same centre and angle. ### Enlargements - Changes size by **scale factor** (sf): - $|sf| > 1$: larger - $0 < |sf| < 1$: smaller - Negative sf: also rotates 180° (image on opposite side of centre). - **Centre of enlargement (CoE)** determines position. - **Area scale factor** = $(\text{length sf})^2$. - **To describe**: state "enlargement", sf, CoE coordinates. - **Reverse**: sf becomes reciprocal (keep sign), same CoE. ### Common Errors - Translation vector is the movement, not the gap between shapes. - For reflection, don't confuse $x = k$ (vertical) with $y = k$ (horizontal). - For negative enlargement, count distances on opposite side of CoE. ### Quick Tips - Use tracing paper for rotations and to find centres. - For enlargement, draw lines from CoE through vertices to check. - Always label images clearly (e.g., A', B').

Key terms

Translation: A transformation that moves every point of a shape by the same distance in the same direction, described by a column vector.
Reflection: A transformation that flips a shape across a line (mirror line), producing a mirror image.
Rotation: A transformation that turns a shape about a fixed point (centre of rotation) by a given angle and direction.
Enlargement: A transformation that changes the size of a shape by a scale factor, with respect to a centre of enlargement.
Scale factor: The multiplier applied to distances from the centre of enlargement; determines size change. Negative values also rotate the shape 180°.
Centre of enlargement: The fixed point from which distances are measured in an enlargement; lines through corresponding vertices meet here.
Invariant point: A point that does not move during a transformation (e.g., points on the mirror line in a reflection, or the centre of rotation).
Object: The original shape before a transformation.
Image: The shape after a transformation; vertices are often labelled with primes (e.g., A').

Slides

What are Transformations?

Transformations change the position, orientation, or size of a shape.
The original shape is called the object; the transformed shape is the image.
There are four types: translations, reflections, rotations, and enlargements.
Corresponding vertices are labelled A, B, C... and A', B', C'...

Which of the following is NOT a type of transformation?

Translation
Reflection
Rotation
Expansion

Translations

A translation moves a shape without changing its size or orientation.
The movement is described by a column vector (x y).
x: horizontal movement $(positive =$ right, negative $= left)$ .
y: vertical movement $(positive = up$ , negative $= down)$ .

To translate a shape by vector (3 -2), you move $it ____$ units right $and ____$ units down.

Describing a Translation

State that the transformation is a translation.
Give the column vector that moves the object to the image.
Find the vector by counting horizontal and vertical distances from a point on the object to its corresponding point on the image.

Shape A is translated to shape B. A point on A is (1,2) and its image on B is (4,6). What is the translation vector?

(3 4)
(5 8)
(3 2)
(4 6)

Reflections

A reflection flips a shape across a mirror line (line of reflection).
The image is the same size and shape, but reversed.
Each point and its image are equidistant from the mirror line.
Points on the mirror line are invariant.

Which statements about reflections are true?

The image is congruent to the object.
The mirror line can be horizontal, vertical, or diagonal.
Points on the mirror line move to a new position.
The perpendicular distance from a point to the mirror line equals the distance from its image to the mirror line.

Describing a Reflection

State that the transformation is a reflection.
Give the equation of the mirror line.
Common lines: $x = k (vertical), y = k (horizontal), y = x, y = -x$ .

A reflection in the line ____ flips a shape over the vertical line through $x = 2$ .

Rotations

A rotation turns a shape around a fixed point called the centre of rotation.
The image is the same size and shape, but oriented differently.
You need the centre, angle (90°, 180°, 270°), and direction (clockwise or anticlockwise).
For 180°, direction is not needed.

What is the image of point (2,3) after a 90° clockwise rotation about the origin?

(3,-2)
(-3,2)
(-2,-3)
(2,-3)

Describing a Rotation

State that it is a rotation.
Give the centre of rotation as coordinates.
Give the angle and direction (unless 180°).
Use tracing paper to find the centre if needed.

Which are correct for a rotation of 90° anticlockwise about (0,0)?

Point (1,0) maps to (0,1).
Point (0,1) maps to (-1,0).
Point (1,1) maps to (-1,1).
The centre of rotation is invariant.

Enlargements

An enlargement changes the size of a shape by a scale factor.
Scale factor $> 1$ makes it larger; $0 <$ scale factor $< 1$ makes it smaller.
The centre of enlargement is a fixed point.
Negative scale factors also rotate the shape by 180°.

A shape is enlarged with scale factor 3 and centre (0,0). A vertex at (1,2) maps to:

(3,6)
$(\frac{1}{3},\frac{2}{3})$
(3,2)
(1,6)

Describing an Enlargement

State that it is an enlargement.
Give the scale factor.
Give the coordinates of the centre of enlargement.
To find scale factor: divide a side length of the image by the corresponding side length of the object.

To reverse an enlargement of scale factor 2 with centre (1,1), use an enlargement of scale factor ____ with the same centre.

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