Transformations

Module 4 · Lesson 4.05 · Spec G7, G8 · ⭐ High Priority

Rotation, reflection and translation — and the exact parameters AQA requires for each full description.

Why "Fully" Costs Full Marks

AQA describes transformation questions as one of the highest mark-loss topics. You either list every required parameter, or you score zero for the description — even with a perfect drawing.

The three transformations

Rotation needs four things.

Reflection needs two things.

Translation needs two things.

The phrase "describe fully"

A mark-scheme trigger. It means "name the type AND list every parameter". Miss one → full description not earned → usually B0.

AQA Evidence

Nov 2024 Paper 1 Q14 on transformations: "not answered well; one third scored for correct size; one eighth made no attempt". The most common loss: rotation without a centre.

Three Transformations: The Required Parameters

TransformationRequired parametersExample phrase

RotationType + angle + direction + centre (4 things)"Rotation 90° clockwise about (2, 3)"

ReflectionType + mirror line (2 things)"Reflection in the line y = x"

TranslationType + column vector (2 things)"Translation by vector (3, −2)ᵀ"

What changes, what stays the same

Rotation, reflection, translation all preserve shape and size. The image is congruent to the object.

Orientation flips under reflection; stays the same under translation; rotates under rotation.

Rotation: Four Required Parameters

Miss any one and you score nothing for the description — even if your drawn image is perfect.

The four

1

Type: the word "rotation" (not "rotate" or "turn").

2

Angle: usually 90°, 180°, or 270°. Units always in degrees.

3

Direction: clockwise or anti-clockwise. Only skipped for 180° — a half-turn is the same in both directions.

4

Centre: the fixed point the rotation spins around. Usually a coordinate: (a, b).

AQA Mark Scheme

"Rotation 90°" alone = 0 marks (missing direction and centre). "Rotation 90° clockwise" = 0 marks (missing centre). You need all four.

Rotation: The Tracing Paper Trick

AQA explicitly provides tracing paper on every exam. Use it — it's the fastest route to the correct answer.

Procedure

Trace the pre-image onto the tracing paper.

Mark the centre of rotation on the tracing paper with a sharp dot.

Hold the centre in place with the pencil tip. Rotate the paper by the required angle in the required direction.

Mark the new vertex positions on the grid underneath. Join them up.

Finding the Centre of Rotation

If the centre isn't given: draw a line from each vertex to its image. Construct the perpendicular bisector of each of those lines (from 4.04!). They all intersect at the centre of rotation.

Common Angles & Their Coordinate Rules

90° CW about (0,0): (x, y) → (y, −x)

90° anti-clockwise about (0,0): (x, y) → (−y, x)

180° about (0,0): (x, y) → (−x, −y)

270° CW = 90° anti-CW and 270° anti-CW = 90° CW. Convert to whichever is simpler.

Reflection: Name the Mirror Line

One parameter. Name the line of reflection, using its equation.

The four mirror lines you must recognise instantly

x-axis = the line y = 0 . Reflects vertically: (x, y) → (x, −y).

y-axis = the line x = 0 . Reflects horizontally: (x, y) → (−x, y).

y = x (diagonal, ↗). Swaps coordinates: (x, y) → (y, x).

y = −x (diagonal, ↘). Swaps and negates: (x, y) → (−y, −x).

Also Watch For: Non-axis Horizontal & Vertical Mirrors

x = a is a vertical line (all points have x-coordinate a). Reflection in x = a: (x, y) → (2a − x, y). Example: reflection in x = 3 sends (1, 4) to (5, 4).

y = b is a horizontal line. Reflection in y = b: (x, y) → (x, 2b − y). Example: reflection in y = 2 sends (3, 5) to (3, −1).

Axis Names

The y-axis is the line x = 0. The x-axis is the line y = 0. AQA accepts either form, but be sure you name the correct axis.

The Diagonal Mirrors: y = x and y = −x

These are the two mirror lines that trip students up most. Get them straight and you'll score reliably.

y = x goes ↗

Up and to the right. Through (0,0), (1,1), (2,2), etc. Reflection in y = x swaps coordinates: (3, 7) → (7, 3).

y = −x goes ↘

Down and to the right. Through (0,0), (1,−1), (2,−2). Reflection in y = −x: (3, 7) → (−7, −3) — swap and negate both.

Quick Self-Check

Plot one corner of your shape, reflect mentally, and check it's equidistant from the mirror on the opposite side. Every reflected point sits at the same perpendicular distance from the mirror line.

Translation: A Column Vector

A translation slides every point by the same amount — no rotation, no flip, no resize. The parameter is a column vector.

Translation by (a, b)ᵀ means: every point (x, y) → (x + a, y + b)

Reading a column vector

Top number (a) = horizontal shift. Positive = right. Negative = left.

Bottom number (b) = vertical shift. Positive = up. Negative = down.

Example: (3, −2)ᵀ means 3 right, 2 down.

Reading the Vector

Column vectors are written vertically in the exam paper. In this lesson we use (a, b)ᵀ with the superscript T to mean the same thing. The TOP number is always horizontal, the BOTTOM number always vertical.

Worked Example — Describe Fully (Reflection)01

Triangle A has vertices at (1, 1), (3, 1) and (1, 4). Triangle B has vertices at (−1, 1), (−3, 1) and (−1, 4). Describe fully the single transformation that maps triangle A onto triangle B. [2 marks]

M

Method: Compare corresponding coordinates. Identify the pattern in the change.

C

Calculate: (1, 1) → (−1, 1); (3, 1) → (−3, 1); (1, 4) → (−1, 4). Each x-coordinate is negated; each y-coordinate is unchanged. That pattern is reflection in the y-axis (the line x = 0).

E

Evaluate / Answer:Reflection in the y-axis (equivalently, reflection in the line x = 0).

Mark split: B1 for "reflection"; B1 for "in the y-axis" (or "x = 0"). Writing just "reflection" without the mirror line loses the second mark.

Worked Example — Describe Fully (Rotation)02

Triangle P has vertices at (2, 1), (5, 1) and (2, 3). Triangle Q has vertices at (−1, 2), (−1, 5) and (−3, 2). Describe fully the single transformation that maps P onto Q. [3 marks]

M

Method: Check congruence first. Triangle P has sides of length 3, 2 and √13; triangle Q has the same side lengths, so they are congruent. Same size + different orientation ⇒ rotation or reflection. Check the coordinate pattern: (x, y) → (−y, x). That's the rule for 90° anti-clockwise rotation about the origin.

C

Calculate: Test each vertex. (2, 1) → (−1, 2) ✓. (5, 1) → (−1, 5) ✓. (2, 3) → (−3, 2) ✓. All three match the 90° anti-clockwise rule about (0, 0).

E

Evaluate / Answer:Rotation 90° anti-clockwise about (0, 0). All four parameters present: type, angle, direction, centre.

Mark split: B1 for "rotation"; B1 for "90° anti-clockwise" (angle and direction together); B1 for "about (0, 0)". Missing the direction OR the centre each loses a mark. All three required for full marks.

Worked Example — Describe Fully (Translation)03

Triangle C has vertices at (1, 2), (4, 2) and (1, 5). Triangle D has vertices at (5, −1), (8, −1) and (5, 2). Describe fully the single transformation that maps C onto D. [2 marks]

M

Method: Subtract coordinates of one vertex pair. If every vertex is shifted by the same vector, it's a translation.

C

Calculate: (1, 2) → (5, −1): shift is (+4, −3). (4, 2) → (8, −1): same shift (+4, −3). (1, 5) → (5, 2): same (+4, −3). Consistent vector.

E

Evaluate / Answer:Translation by vector (4, −3)ᵀ (i.e. 4 right, 3 down).

Mark split: B1 "translation"; B1 for the correct vector. Writing "translate by 4 right and 3 down" is accepted, but the column vector form is clearer and preferred.

Worked Example — Performing a Rotation04

Triangle T has vertices at (1, 1), (4, 1), (1, 3). Rotate T by 180° about the point (0, 0). State the coordinates of the image. [3 marks]

M

Method: 180° rotation about (0, 0) maps (x, y) → (−x, −y). Apply to each vertex. (Direction doesn't matter for 180° — a half-turn is the same either way.)

C

Calculate: (1, 1) → (−1, −1). (4, 1) → (−4, −1). (1, 3) → (−1, −3).

E

Evaluate / Answer: the image has vertices at (−1, −1), (−4, −1) and (−1, −3).

Mark split: M1 for using (x, y) → (−x, −y) OR correctly using tracing paper; A1 A1 for each correct coordinate pair (lose one A1 for an arithmetic slip).

Finding the Centre (or Mirror Line)

When key parameters aren't given, use these methods (which rely on constructions from 4.04):

Finding the Centre of Rotation

Pick any vertex on the object and its corresponding vertex on the image.

Join them with a straight line.

Construct the perpendicular bisector of that line.

Repeat for a second vertex pair.

The two perpendicular bisectors meet at the centre of rotation.

Finding the Mirror Line of a Reflection

Pick any vertex on the object and its corresponding vertex on the image.

Construct the perpendicular bisector of the line joining them.

That perpendicular bisector IS the mirror line — every point and its reflection are equidistant from it.

Read off the equation (e.g. y = 3, x = −1, y = x).

Why They Work

For rotation: every point and its image are equidistant from the centre, so the centre is on the perpendicular bisector of the line joining them. Two such bisectors fix one point — the centre.

For reflection: every point and its image are equidistant from the mirror, and the line joining them is perpendicular to the mirror. So the mirror line IS that perpendicular bisector.

Shortcut: Tracing Paper

Trace the object. For rotation, place a pencil tip on a candidate centre and rotate. If the shape lands on the image, that's the centre. Try the origin first — it's the answer more often than not.

What Changes, What Stays the Same

Knowing what each transformation preserves helps you identify which one was used.

PropertyRotationReflectionTranslation

Shape and sizePreservedPreservedPreserved

Orientation (clockwise order of vertices)PreservedFlippedPreserved

PositionChanged (rotated)Changed (flipped)Changed (slid)

Fixed pointsCentre onlyEntire mirror lineNone

Identification Trick

If orientation is flipped (the vertices read clockwise on the object but anti-clockwise on the image), you're looking at a reflection. If orientation is preserved but the shape has moved, it's a rotation or translation.

Identifying a Single Transformation

When a question says "describe the single transformation", try the types in this order:

1. Is it a translation?

Is every vertex shifted by the same vector (x, y) → (x + a, y + b)? If yes → translation. Quick test: orientation is unchanged, and the corresponding vertices all move in the same direction by the same amount.

2. Is it a reflection?

Is orientation flipped? Draw a line between one vertex and its image; the mirror line is the perpendicular bisector of that line. Common mirror lines: x = 0, y = 0, y = x, y = −x, or simple verticals/horizontals like x = 2.

3. Is it a rotation?

If orientation is preserved but it's not a translation, try rotation. Test the 90° / 180° / 270° coordinate rules about (0, 0) first. If none match, find the centre using perpendicular bisectors.

"Describe the single transformation"

If the question says "single", a single transformation MUST exist — don't default to "reflection then translation". Check orientation and distance carefully.

Examiner Alerts

Alert 1 — Rotation without a centre

The single most common mark loss on transformations. "Rotation 90° clockwise" is INCOMPLETE. Add the centre: "Rotation 90° clockwise about (0, 0)". Always.

Alert 2 — Wrong mirror line name

"Reflection in the line y = x" and "reflection in the x-axis" (y = 0) are DIFFERENT reflections. Mix them up and the final drawing is in a completely different place.

Alert 3 — Two transformations when one suffices

"Reflection then translation" is almost never the correct answer to "describe the single transformation". Check for a single rotation or reflection first.

Alert 4 — Forgetting direction for rotation

"Rotation 90° about (0,0)" without "clockwise" or "anti-clockwise" is INCOMPLETE. Exception: a 180° rotation doesn't need direction — a half-turn is the same either way.

Alert 5 — Wrong column vector orientation

(3, −2)ᵀ is 3 right and 2 DOWN, not 2 up. The negative sign on the bottom means downward. Write it out: "3 right, 2 down" in rough working to avoid slips.

Summary

The three transformations

Rotation: type + angle + direction + centre (4 parameters).

Reflection: type + mirror line (2 parameters).

Translation: type + column vector (2 parameters).

Key coordinate rules (about the origin)

Reflection in x-axis: (x, y) → (x, −y).

Reflection in y-axis: (x, y) → (−x, y).

Reflection in y = x: (x, y) → (y, x).

Rotation 90° CW: (x, y) → (y, −x). 90° anti-CW: (x, y) → (−y, x). 180°: (x, y) → (−x, −y).

Translation by (a, b)ᵀ: (x, y) → (x + a, y + b).

All three preserve

Shape and size (image is congruent to object).

Orientation is flipped only by reflection.

Never Forget

Rotation needs a centre.Reflection needs a line.Translation needs a vector. Missing any of these and the description mark is lost.

Checkpoint

You should now be able to

Describe a rotation fully with type, angle, direction and centre.

Describe a reflection fully with type and mirror-line equation.

Describe a translation fully with type and column vector.

Perform each transformation on a given shape.

Identify the correct single transformation from a pair of images.

Find the centre of rotation using perpendicular bisectors or tracing paper.

Test yourself in under two minutes

1

Triangle A has vertices (2, 0), (4, 0), (2, 3). Apply (x, y) → (−x, y). What single transformation is this?

2

What column vector maps (3, 4) to (−1, 7)?

3

Describe fully: triangle M (1,1), (3,1), (1,2) → triangle N (−1,−1), (−3,−1), (−1,−2).

4

Which mirror line reflection sends (5, 2) to (2, 5)?

Answers: 1) x is negated, y unchanged → reflection in the y-axis (x = 0). 2) (−1, 7) − (3, 4) = (−4, 3)ᵀ. 3) (x, y) → (−x, −y) — rotation 180° about (0, 0). 4) Coordinates swapped → y = x.

Next Lesson (4.06)

We tackle enlargement — the fourth transformation, which changes size. Positive, fractional and negative scale factors, and centre of enlargement.