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Cambridge IGCSE Mathematics — 0580 Extended

Topic 7.5: Transformations — Combined Transformations

A combined transformation is two (or more) transformations applied in order. The first-named one happens first. The result can often be rewritten as a single transformation — that is a standard 0580 ask.

Order matters

Translation then reflection is not, in general, the same as reflection then translation. Draw the intermediate image; do not skip it.

Method

  1. Copy the object. Apply the first transformation; label the intermediate \(A_1B_1C_1\).
  2. Apply the second transformation to that intermediate, not to the original object.
  3. The final image is what the question means by “\(A''\)” or “the image after both”.

Compare “translate by \(\binom{4}{0}\), then reflect in the \(x\)-axis” with the reverse order.

Two side-by-side grids showing translate-then-reflect versus reflect-then-translate
The green images are not in the same place. Always apply the first-named transformation first.

A single equivalent transformation

Two reflections in perpendicular mirrors through a point \(P\) are equivalent to a \(180^\circ\) rotation about \(P\). Always check by tracking one vertex, then describe the single map fully.

Reflect in the \(x\)-axis, then in the \(y\)-axis. Name the single equivalent transformation.

Reflection in the x-axis then the y-axis equivalent to a 180 degree rotation about O
The single equivalent transformation is a rotation of \(180^\circ\) about the origin.

Object \(ABC\) maps to \(A'B'C'\) after two transformations. Describe the single equivalent transformation.

Object and image related by a 180 degree rotation about the origin
Single description: rotation \(180^\circ\) about \((0, 0)\).

Inverse transformations

The inverse maps the image back onto the object.

TransformationInverse
Reflection in a lineThe same reflection (it undoes itself)
Rotation about \(C\), angle \(\theta\) ACWRotation about \(C\), angle \(\theta\) CW (or \(360^\circ - \theta\) ACW)
Translation \(\binom{p}{q}\)Translation \(\binom{-p}{-q}\)
Enlargement SF \(k\), centre \(C\)Enlargement SF \(1/k\), same centre

What enlargement undoes “scale factor 2, centre \(O\)”?

Scale factor 2 and scale factor 1/2 about O are inverse enlargements
\(k = 2\) and \(k = \tfrac{1}{2}\) with the same centre undo each other. For \(k = -2\) the inverse is \(k = -\tfrac{1}{2}\).

Invariant points and lines

An invariant point maps to itself. An invariant line maps to itself as a set (points on it may slide along it).

Invariant points: mirror line, rotation centre, enlargement centre, none for a translation
A non-zero translation has no invariant points.
TransformationInvariant points
ReflectionEvery point on the mirror (the whole line)
Rotation (angle not a multiple of \(360^\circ\))Only the centre
Enlargement (\(k \neq 1\))Only the centre
Translation (non-zero)None

Paper 2 (non-calculator)

Scale factor \(k = 1\) is the identity (every point invariant). Scale factor \(k = -1\) about \(C\) has the same image as a \(180^\circ\) rotation about \(C\) — only \(C\) is invariant.

Try this

A triangle is reflected in \(x = 2\), then rotated \(180^\circ\) about \(O\). Does the order matter? Name the inverse of “rotation \(90^\circ\) anticlockwise about \(O\)”.

Show answer
Answer

  1. A reflection and a rotation about a different centre do not commute

    order matters

  2. Undo a \(90^\circ\) anticlockwise turn with a \(90^\circ\) clockwise turn about the same centre

    \(90^\circ\) clockwise about \(O\)

0/10

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