Combination of Transformations (OCR GCSE Maths)

What do I need to know about combined transformations?

• Combined transformations are when more than one transformation is performed one after the other
• It is often the case that two transformations can be equivalent to one alternative single transformation and you will be expected to describe the single transformation
• You will need to have a clear understanding of the following three transformations and their properties to do this

• Rotation
• • Requires an angle, direction and centre of rotation
  • It is usually easy to tell the angle from the orientation of the image
  • Use some instinct and a bit of trial and error to find the centre of enlargement.

• Reflection
  • A reflection will be in a mirror line which can be vertical (x = k), horizontal (y = k) or diagonal (y = mx + c)
  • Points on the mirror line do not move
  • Double reflections are possible if the mirror line passes through the object

• Translation
  • A translation is a movement which does not change the direction or size of the shape
  • A translation is described by a vector in the form 
    • This represents a movement of x units to the right and y units vertically upwards

• Note that the transformation enlargement changes the size of the shape and so cannot be a part of a combined transformation 

What are common combinations of transformations?

• A combination of two reflections can be the same as a single rotation
  • One reflection using the line and the other using the line 
  • This is the same as a 180° rotation about the centre 
• A combination of a 180° rotation about a centre and an enlargement of scale factor k about the same centre is the same as a single enlargement
  • This enlargement would have the same centre but the scale factor would be -k
• The order of the combination can be important to the overall effect
  • A reflection in the line y = x followed by a reflection in the x-axis is the same as a 90° rotation clockwise about the origin
  • A reflection in the x-axis followed by a reflection in the line y = x is the same as a 90° rotation anticlockwise about the origin

What are invariant points?

• Invariant points are points that do not change position when a transformation, or combination of transformations, has been performed
  • Invariant points return back to their original position

How do I undo a transformation to get back to the original shape?

• After transforming shape A to make shape B you might be asked to describe the transformation that maps B to A

Transformation from A to B	Transformation from B to A
Translation by vector	Translation by vector
Reflection in a given line	Reflection in the same line
Rotation by θ° in a direction about the centre	Rotation by θ° in the opposite direction about the centre
Enlargement of scale factor  about the centre	Enlargement of scale factor  about the centre