Describe and perform single transformations
Map points and shapes by reflection, rotation, enlargement and translation while stating each transformation precisely.
Section 8 · Transformations and vectors
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What stays the same length under a rotation?
What you will learn
- Reflect shapes in allowed lines and identify the mirror line
- Rotate about a stated centre through a multiple of 90°
- Enlarge from a centre with a signed scale factor or translate by a vector
Follow each point from its original position
A reflection places each point the same perpendicular distance on the other side of the mirror line. Core specifies vertical or horizontal lines; Extended allows any straight mirror line.
A rotation needs a centre, direction and angle. Core restricts the centre to the origin, a vertex or an edge midpoint; Extended uses a stated centre. Use multiples of 90° and preserve each point's distance from the centre.
Use centre and scale for enlargement
For scale factor k about centre C, the image point P′ lies on ray CP and CP′ = k·CP. A positive factor keeps the direction; a negative factor places the image on the opposite ray. Core uses positive factors; Extended also includes negative factors.
A translation adds the same displacement vector to every vertex. Use a ruler for straight edges, keep corresponding vertices in order and state the centre, mirror line, angle or vector in the description.
Pause and explain
What does a translation do to every vertex?
Worked example
Rotate point A(2, 1) 90° anticlockwise about the origin.
Show the worked solution
- A 90° anticlockwise rotation about the origin maps (x, y) to (−y, x).
- Substitute x = 2 and y = 1.
- A′ = (−1, 2).
Answer A′ = (−1, 2).
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Reflect point P(3, −2) in the x-axis.
Give me a hint
Reflection in the x-axis keeps x and changes the sign of y.
Compare my reasoning
- The x-coordinate remains 3.
- The y-coordinate changes from −2 to 2.
- P′ = (3, 2).
P′ = (3, 2).
Look for these in your work
- Mirror line identified
- Coordinate rule applied
- Image point labelled
Enlarge point P(4, 3) by scale factor 2 about centre C(1, 1).
Give me a hint
Find the centre-to-point vector, scale it, then add it back to the centre.
Compare my reasoning
- CP = (4−1, 3−1) = (3, 2).
- Scale by 2: (6, 4).
- P′ = (1, 1) + (6, 4) = (7, 5).
P′ = (7, 5).
Look for these in your work
- Centre-to-point vector found
- Scale applied to both coordinates
- Centre restored
A map point at (−3, 2) is translated 5 units east and 4 units south. Find its image and describe the vector.
Give me a hint
East increases x; south decreases y.
Compare my reasoning
- The displacement is (5, −4).
- Add it to (−3, 2): (−3+5, 2−4).
- The image is (2, −2), under vector (5, −4).
(2, −2); translation vector (5, −4).
Look for these in your work
- Direction translated to signed coordinates
- Vector added
- Image and vector stated
Common mistakes
- Giving an angle without a centre of rotation
- Reversing clockwise and anticlockwise
- Enlarging from the origin when another centre is specified
- Using a negative scale factor for a Core-only question
What can you explain now?
What information fully describes a rotation?
Compare with the explanation
Centre, direction and angle
The image depends on all three.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Which details let another learner reconstruct a reflection, rotation, enlargement or translation?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.