Transformations: Coordinate Rules

Statement

In coordinate geometry, transformations change the position of points or shapes on the plane. The most common transformations are reflections and rotations. Each has a precise coordinate rule describing how the point \((x,y)\) moves:

• Reflection in x-axis: \((x,y) \to (x,-y)\)
• Reflection in y-axis: \((x,y) \to (-x,y)\)
• Rotation 90° anticlockwise about the origin: \((x,y) \to (-y,x)\)
• Rotation 180° about the origin: \((x,y) \to (-x,-y)\)
• Rotation 270° anticlockwise about the origin: \((x,y) \to (y,-x)\)

Why it’s true

• Reflections: Flipping across an axis reverses the sign of the coordinate perpendicular to that axis. Reflection in x-axis changes \(y\) to \(-y\), while reflection in y-axis changes \(x\) to \(-x\).
• Rotations: A 90° anticlockwise turn swaps \(x\) and \(y\) and changes the new \(x\) to its negative. Rotating 180° changes both signs. A 270° anticlockwise rotation is the same as 90° clockwise: swap \(x,y\) and change the new \(y\) to negative.
• These rules can be derived from rotation matrices in coordinate geometry.

Recipe (how to use it)

1. Write down the original coordinates \((x,y)\).
2. Select the transformation type (reflection or rotation).
3. Apply the corresponding rule to obtain the new coordinates.
4. Plot the new coordinates or use them in calculations as required.

Spotting it

Look for instructions like “reflect in the x-axis”, “rotate 90° anticlockwise about the origin”. These signal direct application of the coordinate rules above.

Common pairings

• Translations (\((x,y) \to (x+a, y+b)\)) often appear alongside reflections and rotations in exam questions.
• Symmetry problems usually combine reflections with axis identification.

Mini examples

1. Given: Point (3,4), reflect in x-axis. Answer: (3,-4).
2. Given: Point (-2,5), rotate 90° anticlockwise. Answer: (-5,-2).

Pitfalls

• Mixing up clockwise vs anticlockwise rotations.
• Forgetting that reflections flip only one coordinate.
• Confusing 270° anticlockwise with 90° anticlockwise (they give different results).

Exam strategy

• Write the rule before applying it to avoid mistakes.
• Check whether rotation is clockwise or anticlockwise.
• If unsure, sketch axes and quickly plot the effect.

Summary

Transformations are described by coordinate rules. Remember the five key ones: reflections (x or y axis) and rotations (90°, 180°, 270°). These are frequently tested in GCSE exams and form the building blocks of more advanced transformations.