Overview
What you’ll learn
- Perform and describe reflections, rotations, translations and enlargements, naming all parameters.
- Handle negative and fractional scale factors, placing images on the correct side of the centre (Extended).
- Add, subtract and scale column vectors, and compute magnitudes exactly (Extended).
- Write routes such as AB = b − a from position vectors and prove parallelism or collinearity (Extended).
Syllabus checklist (0580)
Statements are paraphrased and grouped by tier. Core is studied by everyone; Extended content is examined on Papers 2 and 4 only.
Core — every candidate
- 7.1Reflect in horizontal and vertical lines, rotate about the origin through multiples of 90°, translate by a vector, and enlarge from a centre with positive and fractional scale factors.
Extended — Extended candidates only
- 7.1Reflect in any straight line, enlarge with negative scale factors, and describe combined transformations. (Extended)
- 7.2Use column-vector notation, adding and subtracting vectors and multiplying by scalars. (Extended)
- 7.3Calculate the magnitude of a vector as √(x² + y²). (Extended)
- 7.4Use directed line segments and position vectors, express vectors in terms of two others, and prove lines parallel or collinear. (Extended)
In 30 seconds
- 1Every transformation description needs its full set of parameters: direction and distance (translation), mirror line (reflection), centre, angle and direction (rotation), centre and scale factor (enlargement).
- 2Rotation reminders about O: 90° anticlockwise (x, y) → (−y, x); 90° clockwise (x, y) → (y, −x); 180° (x, y) → (−x, −y).
- 3Enlargement by k from O sends (x, y) → (kx, ky); a negative k lands the image on the opposite side of the centre, upside-down.
- 4The area scale factor is k²: SF 3 multiplies areas by 9 (Core for the fact, Extended for linking to volume work later).
- 5Vectors add tip-to-tail and commute; 2a − b means double a, then subtract b component-wise (Extended).
- 6|v| = √(x² + y²) — the length of the vector (Extended).
- 7AB = OB − OA = b − a (end minus start); a common factor shows parallel vectors, and a shared point upgrades that to collinear (Extended).
Notes · 1 of 4
The four transformations
Flip, spin, slide, stretch — described precisely.
A reflection needs its mirror line: the x-axis flips (3, 5) to (3, −5), the y-axis to (−3, 5), and the diagonal y = x swaps the coordinates: (3, 5) → (5, 3). A translation needs a column vector: adding (3, −2) slides (1, 4) to (4, 2) — right 3, down 2. A rotation needs centre, angle and direction: about O, 90° anticlockwise maps (2, 1) → (−1, 2); 180° maps (4, −2) → (−4, 2).
An enlargement needs a centre and a scale factor. From centre O with SF 2, (3, 1) triples nothing — it doubles to (6, 2). A fractional SF ½ shrinks: (6, 4) → (3, 2). Every side of the image is k times the original, and the area is k² times: SF 3 gives nine times the area. Describing a transformation in words earns the same marks as drawing it — and “enlargement, SF 2, centre (0, 0)” is complete only when all three parts appear.
Notes · 2 of 4
Negative and fractional scale factors
Same distances, opposite direction, inverted image. (Extended)
Extended. A negative scale factor pushes the image through the centre to the other side. With SF −2 from O, the point (1, 1) maps to (−2, −2): twice as far, diagonally opposite. A whole triangle flips and inverts — vertices (1, 2), (3, 2), (1, 5) land at (−2, −4), (−6, −4), (−2, −10). Construction tip: draw the ray from the centre through each vertex and continue it the same multiple beyond.
Fractional factors work inside the unit interval: SF ½ from O maps (10, −6) to (5, −3), compressing toward the centre. Distances from the centre scale by |k|; the shape keeps its proportions. Area still obeys k²: (−2)² and (½)² behave like any other square — negative factors do not flip area signs.
Notes · 3 of 4
Column vectors: add, scale, measure
Arithmetic component-wise; length by Pythagoras. (Extended)
Extended. A column vector (x over y) records horizontal then vertical change. Addition is component-wise: (3, 1) + (−1, 4) = (2, 5). Scalars multiply through: 2(3, 1) = (6, 2). Mixed arithmetic like 2a − b with a = (3, 1), b = (−1, 4): double a to (6, 2), subtract b to get (7, −2). Order does not matter for a + b, but keep signs disciplined for subtraction.
The magnitude |v| is the vector’s length: |(5, 12)| = √(25 + 144) = 13. Surd answers stay exact: for a = (4, −2), |a| = √(16 + 4) = √20 = 2√5 ≈ 4.47. Two vectors are parallel when one is a scalar multiple of the other: (6, 9) = 3 × (2, 3). If they also share a point, the three points are collinear — parallel plus a common point is the standard proof shape.
Notes · 4 of 4
Vector geometry
Routes as end-minus-start; proving parallel and collinear. (Extended)
Extended. Write journeys with position vectors: if OA = a and OB = b, then AB = OB − OA = b − a (end minus start). Midpoints halve routes: with M the midpoint of AB, OM = ½(a + b) — the average of the endpoints. Parallelogram diagrams (OACB with OA = a, OC = c) give AC = c − a, and a midpoint M of OA turns MC into c − ½a.
To prove parallelism, factor: if v = (6, 15) and u = (2, 5), then v = 3u, so the vectors are parallel. To prove collinear points, show two routes parallel and share a point: PQ = 3·QR with Q common forces P, Q, R onto one straight line. Scalars may be negative — a reverse direction still counts as parallel.
Revise
Key terms
The words examiners expect you to use precisely.
- Translation
- A slide of every point by the same vector.(1, 4) + (3, −2) → (4, 2).
- Reflection line
- The mirror a figure flips across.y = x swaps coordinates: (3, 5) → (5, 3).
- Rotation
- A turn about a centre through an angle in a direction.90° anticlockwise about O: (2, 1) → (−1, 2).
- Enlargement
- A stretch from a centre by a scale factor.SF 2 from O: (3, 1) → (6, 2).
- Negative scale factor (Extended)
- An enlargement through the centre to the opposite side.SF −2: (1, 1) → (−2, −2).
- Column vector (Extended)
- A pair (x over y) recording horizontal and vertical change.AB from (1, 2) to (4, 6) is (3 over 4).
Show 2 more key termsShow fewer key terms
- Magnitude (Extended)
- The length of a vector, √(x² + y²).|(5, 12)| = 13.
- Collinear (Extended)
- Points on one straight line: parallel routes plus a shared point.PQ = 3·QR with Q common.
Revise
Common mistakes
Where students most often lose marks in this topic — and what to do instead.
Mistake: Describing a rotation without the direction.
90° clockwise and anticlockwise give different images.
Instead: State centre, angle and direction: “90° anticlockwise about O”.
Mistake: Swapping the 90° rotation rules.
Anticlockwise maps (x, y) → (−y, x), not (y, −x).
Instead: Anticlockwise: (2, 1) → (−1, 2); clockwise: (2, 1) → (1, −2).
Show 10 more mistakesShow fewer mistakes
Mistake: Enlarging distances from the wrong centre.
The scale factor applies from the centre, not from the origin by default.
Instead: Rays run centre → vertex, extended k times.
Mistake: Treating a negative scale factor as a reflection only.
It enlarges as well: distances grow by |k|.
Instead: SF −2 = half-turn + enlargement by 2 about the centre.
Mistake: Adding vectors coordinate-mixed: a + b = (3 + 4, 1) ?
Both components combine.
Instead: (3, 1) + (−1, 4) = (2, 5).
Mistake: Subtracting b from 2a in the wrong order: 2a − b as b − 2a.
Vector subtraction is not commutative.
Instead: 2a − b = (6, 2) − (−1, 4) = (7, −2).
Mistake: Forgetting to square the negative component in a magnitude.
(−4)² = 16, not −16.
Instead: |(3, −4)| = √(9 + 16) = 5.
Mistake: Writing AB as a − b instead of b − a.
Routes run end minus start.
Instead: AB = OB − OA = b − a.
Mistake: Claiming parallel vectors prove collinear points.
Parallel routes can sit on different lines.
Instead: Parallel plus a shared point gives collinearity.
Mistake: Reflecting in the y-axis by flipping the y-coordinate.
The mirror line is vertical, so the horizontal gap is what flips.
Instead: (x, y) → (−x, y): (−2, 6) lands on (2, 6).
Mistake: Reading a column vector bottom-entry first.
The top entry is the horizontal step.
Instead: (3, −2) means three right, two down.
Mistake: Describing an enlargement without its centre.
The scale factor alone leaves infinitely many possible images.
Instead: A full description names type, scale factor and centre (or mirror line).
Check yourself
Quick quiz
24 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Reflect a point in the x-axis
The image of (3, 5) after reflection in the x-axis is
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Practise
Topical questions
14 exam-style questions with model answers and mark schemes.
A set of structured and calculation questions matched to this topic, in three tiers, each with a model answer, mark scheme and examiner tip.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a Cambridge paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Topic 14 Practice Paper
30 marks · 40 min · 4 pages
10 multiple-choice questions and structured questions with answer lines. Topic 14: reflections, rotations, translations and enlargements including negative scale factors, column vectors, vector arithmetic, magnitudes, and vector routes in parallelogram diagrams.
Mark scheme
PROTopic 14 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Which papers examine this topic?
Paper 1 / 3
Core: 80 marks each, 1 h 30 each
Paper 1 non-calculator, Paper 3 calculator
Paper 2 / 4
Extended: 100 marks each, 2 h each
Paper 2 non-calculator, Paper 4 calculator
Accuracy
3 s.f., angles to 1 d.p.
Unless a question states otherwise
Want a full paper? Try the Paper 1-style Core, Paper 2-style non-calculator or Paper 4-style calculator sets.
Questions
Frequently asked questions
How do I remember the rotation rules without mixing them up?
Test each rule on the point (1, 0). A 90° anticlockwise turn must carry it up to (0, 1): the rule (x, y) → (−y, x) does exactly that. Clockwise carries it down to (0, −1): the rule (x, y) → (y, −x). One test point pins both rules for the exam.
What does a negative scale factor do to a shape?
It enlarges by |k| and rotates the image a half-turn about the centre, placing it on the opposite side. Distances from the centre multiply by |k|; areas still scale by k². Construct it by extending each centre-to-vertex ray the multiple |k| beyond the centre.
Why does AB = b − a and not a − b?
Routes run from start to end: AB = AO + OB = −a + b = b − a. A quick direction check saves marks — the vector should point the way you would walk from A to B.
What is the full recipe for proving points are collinear?
Two ingredients: show one route is a scalar multiple of another (parallel), and show the routes share a point (e.g. PQ = 3·QR with Q common). Parallel alone is not enough — the lines could be different.
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