Transformations

A transformation occurs when a shape is moved by flipping it (reflection)Orakei_Basin_Reflection.JPG, turning it (rotation) or sliding it (translation).

Other types of transformation include enlargement and stretch.

There are many examples of transformations in real life:

Transformations_01.jpg
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Transformations_03.jpg
Transformations_04.jpg
Reflectionn
Rotation
Enlargement
Translation

There are four main transformations in mathematics;

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Transformations_06.jpg
Transformations_07.gif
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Reflection
Rotation
Enlargement
Translation

When an object is transformed the result is called the image.

Below is a summary of the mathematical features of each type of transformation.

Reflection

When a shape or point is reflected its image is on the opposite side of a mirror line or axis of symmetry.

The mirror line is halfway between the shape and its image.

The object and the image are the same distance from the mirror line m.

The axis of symmetry is often shown by the letter m.

The object and the image are congruent, the same shape and size.

Length, angle size and shape are said to be invariant for reflection, which means they do not change.

Examples of reflections − Transformations_10.gif

 

 

Transformations_09.gif

The diagram shows a reflection in the mirror line m

Rotation

 

A rotation is a transformation where a point, or an object, is turned around a fixed point to a new position called the image.

The object and the image are the same shape and the same size but in different positions.

When a shape or point is rotated there is a centre of rotation which remains fixed.

The angle of rotation gives the number of degrees that the shape is rotated through.

An anti-clockwise rotation is said to be positive.
A clockwise rotation is said to be negative.

Common rotations are:

a quarter turn − 90o
a half turn − 180o
a three quarter turn − 270o

Examples of rotations − 

 

 

Transformations_11.gif

The diagram shows an anti-clockwise quarter turn rotation (90o ) about the centre of rotation, O.

Translation

A translation is a transformation where all points move the same distance and in the same direction.

The object and the image are the same shape and the same size.

Translations can be represented by vectors.

In general the vector Transformations_13.gif can represent a translation where x is the horizontal movement and y is the vertical movement.

Examples of translations − Transformations_10.gif
Transformations_12.gif

The diagram shows a translation ofTransformations_14.gif

Enlargement

An enlargement is a transformation where the size of an object changes.

Examples of enlargements − Transformations_10.gif

The object becomes larger or smaller.

If points on the object and the corresponding points on the image are joined with a straight line, these straight lines meet at the centre of enlargement.

The scale factor for an enlargement tells how much an object has been enlarged by.

 
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The diagram shows an enlargement,centre O with a scale factor of 2.