Combined Transformations [H]
Combined Transformations
Two or more transformations can be applied in sequence to produce a combined effect. The key skills are: applying each transformation in turn to find the final image, identifying the single equivalent transformation that maps the original directly to the final image, and understanding which properties are invariant (unchanged) under each transformation.
Order Matters
Transformations are generally not commutative — applying transformation \( T_1 \) then \( T_2 \) usually gives a different result from applying \( T_2 \) then \( T_1 \). Always apply in the stated order.
Invariance
| Transformation | Invariant properties | Not invariant |
|---|---|---|
| Translation | Shape, size, orientation, angles | Position |
| Reflection | Shape, size, angles; points on mirror line | Orientation (sense reversed) |
| Rotation | Shape, size, angles; centre point | Orientation of diagram |
| Enlargement | Shape, angles; centre point | Size, position |
Common Combined Results
- Two reflections in parallel lines → translation (distance = twice the gap between lines).
- Two reflections in intersecting lines → rotation (angle = twice the angle between the lines, about the intersection point).
- Rotation followed by rotation about the same centre → single rotation (angles add).
- Any combination of isometric transformations → single isometric transformation.
Worked Examples
Triangle at \( (1,1),(3,1),(1,4) \) is reflected in \( x=0 \), then translated by \( \begin{pmatrix}2\-3\end{pmatrix} \). Find the final image.
Step 1 — reflect in \( x=0 \) (the \( y \)-axis): \( (x,y)\to(-x,y) \).
Image after step 1: \( (-1,1),(-3,1),(-1,4) \).
Step 2 — translate by \( \begin{pmatrix}2\-3\end{pmatrix} \): add 2 to \( x \), subtract 3 from \( y \).
Final image: \( (1,-2),(-1,-2),(1,1) \).
A shape is reflected in \( y=1 \) then reflected in \( y=4 \). Describe the single equivalent transformation.
Two reflections in parallel horizontal lines distance 3 apart → translation of \( 2\times 3=6 \) upward.
Single equivalent: translation by \( \begin{pmatrix}0\6\end{pmatrix} \).
A triangle is rotated 90° anticlockwise about the origin then reflected in the \( x \)-axis. Is area invariant? Is orientation invariant?
Area: yes — both rotation and reflection are isometric transformations that preserve size.
Orientation: no — reflection reverses orientation (sense). The combined transformation reverses the sense of the shape.
Key Takeaways
- Always apply transformations in the stated order — order matters.
- Two reflections in parallel lines → translation by twice the perpendicular distance between them.
- Two reflections in intersecting lines → rotation by twice the angle between them, about the intersection.
- Translation, reflection and rotation preserve size and angles (isometric). Enlargement preserves shape and angles only.
- Reflection reverses orientation; translation and rotation preserve it.