Ch 1 · 7.1 Transformations · IGCSE Mathematics 0580
Topic 1 of 2 in Transformations and Vectors — IGCSE Mathematics 0580 — 58 lessons.
Introduction
Flip a shape, turn it, slide it or resize it: these are the four transformations. Computer games and animation use exactly these moves. Exam questions ask you to draw them and to describe them fully, which means giving every required detail. (Syllabus section 7.1.)
Key ideas — Describing a transformation fully
Use one transformation name only, never "moved and turned".
🧑🏫 Teacher's explanation
Imagine you are playing a video game where you can manipulate objects on the screen. A transformation is simply a rule that moves or changes the size of a shape. If you want to tell a friend exactly how to move a shape so they get it right every time, you can't just say 'move it a bit' or 'turn it'. You need a precise set of instructions.
📖 Definitions
🔍 Worked example — Easy
Describe fully the transformation that maps shape A onto shape B, where shape B is a mirror image of shape A across the line x = 3.
🔍 Worked example — Standard
Shape P is rotated 90^° clockwise about the origin (0,0) to become shape Q. Describe the transformation fully.
🔍 Worked example — Exam-style
An enlargement maps a triangle with area 4 cm^2 to a new triangle with area 36 cm^2. The centre of enlargement is (2, 2). Describe the enlargement fully.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Reflections
Each point and its image are the same distance from the mirror line, on opposite sides, along a perpendicular. Useful rules: reflection in the x-axis (x, y) → (x, -y); in the y-axis (x, y) → (-x, y); in y = x, (x, y) → (y, x); in y = -x, (x, y) → (-y, -x).
🧑🏫 Teacher's explanation
Imagine you are standing in front of a mirror. Every part of your body that is 5 ext cm away from the glass will appear to be 5 ext cm behind the glass. This is the essence of a reflection. It is a 'flip' that preserves the size and shape of an object, but reverses its orientation.
📖 Definitions
🔍 Worked example — Easy
Point A has coordinates (3, 4). Describe and find the coordinates of the image A' when A is reflected in the x-axis.
🔍 Worked example — Standard
A triangle has vertices at P(1, 2), Q(4, 2), and R(1, 5). Find the coordinates of the image triangle P'Q'R' after a reflection in the line y = x.
🔍 Worked example — Exam-style
A shape is reflected in the line y = -x. If the image of point B is (4, -2), find the coordinates of the original point B.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Triangle ABC is reflected in the line x = 4 to produce triangle A'B'C'. State the coordinates of A', B', and C'. (b) Describe fully the single transformation that maps triangle ABC onto triangle A'B'C'. [3 marks]
Key ideas — Rotations
Use tracing paper, or rules about the origin: 90^° clockwise (x, y) → (y, -x); 90^° anticlockwise (x, y) → (-y, x); 180^° (x, y) → (-x, -y). To find a centre, find the point that stays the same distance from each point and its image (perpendicular bisectors meet there).
🧑🏫 Teacher's explanation
Imagine you are holding a transparent sheet over a drawing. A rotation is like pinning that sheet down with a single finger at one specific point and then spinning the sheet around that finger. The point under your finger (the centre) is the only part that doesn't move; every other point travels along a circular path around it.
📖 Definitions
🔍 Worked example — Easy
A point A has coordinates (3, 4). It is rotated 90^° clockwise about the origin (0, 0). State the coordinates of the image A'.
🔍 Worked example — Standard
A triangle with vertices P(1, 1), Q(3, 1), and R(1, 4) is rotated 180^° about the origin. Find the coordinates of the image P'Q'R'.
🔍 Worked example — Exam-style
A shape is rotated 90^° anticlockwise about a centre C(2, 2). The point A(4, 2) is mapped to A'. Find the coordinates of A'.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Enlargements
Draw rays from the centre through each vertex; multiply each distance from the centre by the scale factor k.
🧑🏫 Teacher's explanation
Imagine you are using a projector to cast a shadow of a small object onto a wall. If you move the projector closer to the object, the shadow gets bigger; if you move it further away, the shadow gets smaller. This is the essence of an enlargement. We aren't just 'making things bigger'; we are scaling everything relative to a fixed point called the 'centre of enlargement'.
📖 Definitions
🔍 Worked example — Easy
Triangle ABC has vertices at A(2, 2), B(4, 2), and C(4, 4). It is enlarged from the origin (0, 0) by a scale factor of k=3. Find the coordinates of the image A'B'C'.
🔍 Worked example — Standard
A square has a side length of 4 units. It is enlarged by a scale factor of k = 1/2. Find the side length of the image.
🔍 Worked example — Exam-style
Triangle XYZ is enlarged to X'Y'Z' with a scale factor of k = -2. If X is at (1, 2) and the centre of enlargement is (0, 0), find the coordinates of X'.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) State the scale factor k. (b) Find the coordinates of B' and C'. [4 marks]
Key ideas — Translations and combinations
A translation by beginpmatrix 3 \\ -2 endpmatrix moves every point 3 right and 2 down. For a combination, do the first transformation, then apply the second to the image. Often a single transformation describes the result (e.g. two reflections in parallel lines make a translation).
🧑🏫 Teacher's explanation
Imagine you are playing a board game where you move a piece across the grid. A translation is like sliding that piece from one square to another without rotating it or changing its size. You aren't turning the piece; you are simply shifting its position using a 'map' called a vector. If the vector is beginpmatrix x \ y endpmatrix, you move x units horizontally and y units vertically.
📖 Definitions
🔍 Worked example — Easy
A point A has coordinates (2, 5). Describe and perform a translation by the vector beginpmatrix 4 \ -3 endpmatrix.
🔍 Worked example — Standard
Triangle T is translated by beginpmatrix -2 \ 4 endpmatrix and then translated by beginpmatrix 5 \ -1 endpmatrix. Find the single resulting translation vector.
🔍 Worked example — Exam-style
A square with vertices at (1,1), (3,1), (3,3), (1,3) is translated by beginpmatrix 2 \ 2 endpmatrix and then reflected in the line y=0 (the x-axis). Describe the final position of the vertex originally at (1,1).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
The point (2, 5): reflection in y = x gives (5, 2); rotation 90^° clockwise about the origin gives (5, -2); enlargement scale factor -2, centre the origin, gives (-4, -10).
🔍 Worked example — Easy
A point P has coordinates (3, -4). Describe the image of P under: (i) a reflection in the x-axis; (ii) a rotation of 90^° clockwise about the origin.
🔍 Worked example — Standard
Triangle ABC has vertices A(1, 1), B(3, 1), and C(1, 4). Triangle A'B'C' is an enlargement of ABC with centre (0,0) and scale factor k = -2. Find the coordinates of A', B', and C'.
🔍 Worked example — Exam-style
Shape S is translated by vector beginpmatrix 3 \ -2 endpmatrix and then reflected in the line y = x. If the original position of a point is (2, 5), find its final position.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) (-1, 3). (2) (-1, 4). (3) 5 × 3^2 = 45 cm². (4) a translation by the vector beginpmatrix 3 \\ -2 endpmatrix.