Transformation Geometry: All Four Transformations

Every transformation an exam sets — reflection in the axes and y = x, translation, rotation about the origin, and enlargement — with the coordinate rule for each, worked examples and a worksheet.

A transformation changes the position (or size) of a shape on the Cartesian plane. Three of them — reflection, translation and rotation — keep the shape exactly the same size (the image is congruent); a fourth, enlargement, resizes it. Each has a coordinate rule you can apply to any point.

1The coordinate rules (learn these)

Reflect in the x-axis: (x, y) → (x, −y).
Reflect in the y-axis: (x, y) → (−x, y).
Reflect in the line y = x: (x, y) → (y, x).
Translation: add the shift to each coordinate, e.g. 3 right, 2 up → (x + 3, y + 2).
Rotate 180° about the origin: (x, y) → (−x, −y).
Enlargement by factor k from the origin: (x, y) → (kx, ky).

Reflecting a point in the x-axis: the x stays, the y flips sign. (3, 5) → (3, −5).

2Type 1: Reflection

Worked ExampleExample 1: reflect (3, 5) in the x-axis

1
Rule: (x, y) → (x, −y).
Reflecting in the x-axis keeps x, flips the sign of y.
2
So (3, 5) → (3, −5).
The point drops to the mirror position below the axis.

Worked ExampleExample 2: reflect (4, −2) in the line y = x

1
Rule: (x, y) → (y, x).
Reflecting in y = x swaps the coordinates.
2
So (4, −2) → (−2, 4).
Just swap the two numbers.

3Type 2: Translation

Worked ExampleExample 3: translate (2, 3) by 4 right and 1 down

1
4 right adds 4 to x; 1 down subtracts 1 from y.
Right/up are positive, left/down negative.
2
(2 + 4, 3 − 1) = (6, 2).
Apply the shift to each coordinate.

4Type 3: Rotation about the origin

Worked ExampleExample 4: rotate (3, 5) by 180° about the origin

1
Rule: (x, y) → (−x, −y).
A half-turn flips the sign of both coordinates.
2
So (3, 5) → (−3, −5).
The point moves to the opposite side of the origin.

5Type 4: Enlargement

An enlargement multiplies every coordinate by a scale factor k (from the origin). A factor greater than 1 enlarges; between 0 and 1 reduces.

Worked ExampleExample 5: enlarge (3, 4) by scale factor 2

1
Multiply each coordinate by 2: (2×3, 2×4).
Enlargement rule (x, y) → (kx, ky).
2
= (6, 8).
The shape is now twice as far from the origin in every direction.
3
The AREA becomes 2² = 4 times bigger.
Area scales by the factor squared.

💡

Reflection, translation and rotation don't change size or shape — the image is congruent to the original. Only enlargement changes the size.

⚠️

For enlargement, area scales by the factor squared. Doubling the sides (factor 2) makes the area 4 times bigger, not 2 times.

Transformations reappear in Grade 10–12 function graphs (shifts and reflections of curves), so the coordinate rules here are worth knowing cold. Practise every type below.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Reflect (2, 7) in the y-axis. (reflect)
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(−2, 7)
2
Reflect (4, −1) in the x-axis. (reflect)
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(4, 1)
3
Reflect (5, 2) in the line y = x. (reflect)
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(2, 5)
4
Reflect (−3, 6) in the y-axis. (reflect)
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(3, 6)
5
Translate (1, 4) by 5 right and 2 up. (translate)
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(1+5, 4+2) = (6, 6)
6
Translate (3, −2) by 2 left and 3 up. (translate)
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(3−2, −2+3) = (1, 1)
7
Rotate (2, 5) by 180° about the origin. (rotate)
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(−2, −5)
8
Rotate (−4, 1) by 180° about the origin. (rotate)
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(4, −1)
9
Enlarge (3, 4) by scale factor 2 from the origin. (enlarge)
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(6, 8)
10
Enlarge (4, 6) by scale factor 12 from the origin. (reduce)
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(2, 3)