Learning Objectives
  • Recognise, describe and draw reflections, rotations, translations and enlargements.
  • State every parameter needed to describe a transformation fully.
  • Use positive, fractional and negative enlargement scale factors.
  • Apply combinations of transformations in the correct order.
  • Use coordinates and vectors to determine images and pre-images.
Key Terms
Image
The transformed position of an object.
Object
The original shape before transformation.
Invariant
A property or point unchanged by a transformation.
Centre of rotation
The fixed point around which a rotation occurs.
Centre of enlargement
The fixed point from which distances are scaled.
Scale factor
The multiplier relating image lengths to object lengths.
Translation vector
A column vector showing horizontal and vertical movement.
What This Chapter Covers
  • Reflection in a straight line
  • Rotation about a centre through multiples of 90^\circ
  • Translation by a vector
  • Enlargement with positive, fractional and negative scale factors
  • Combined transformations
Describing A Transformation Completely
Transformation Required description
Translation The translation vector
Reflection The equation or clear description of the mirror line
Rotation Centre, angle and direction
Enlargement Centre and scale factor

Statements such as “turned” or “moved left” are incomplete. A transformation must be described with the exact parameters.

Translation

A translation moves every point by the same vector. Shape, size and orientation are preserved.

\begin{pmatrix}x\\y\end{pmatrix}\mapsto\begin{pmatrix}x+a\\y+b\end{pmatrix}\quad\text{for translation }\begin{pmatrix}a\\b\end{pmatrix}
Worked Example: Translate A Point

Question: Point P is (-3,5). Translate it by \begin{pmatrix}7\\-4\end{pmatrix}.

  1. Add 7 to the x-coordinate.
  2. Add -4 to the y-coordinate.
  3. P\prime=(-3+7,5-4).

Answer: P\prime=(4,1).

Reflection

A reflection flips a shape across a mirror line. Each point and its image lie on a perpendicular to the mirror line and are the same distance from it. Common coordinate rules can speed up simple cases.

Mirror line Coordinate rule
x-axis (x,y)\mapsto(x,-y)
y-axis (x,y)\mapsto(-x,y)
y=x (x,y)\mapsto(y,x)
y=-x (x,y)\mapsto(-y,-x)
Worked Example: Reflection

Question: Reflect (6,-2) in the line y=x.

  1. Swap the coordinates.

Answer: (-2,6).

Rotation

A rotation preserves shape and size but changes orientation. For coordinate-grid questions, tracing paper may help. Direction must be stated for 90^\circ rotations; 180^\circ has the same result clockwise or anticlockwise.

Rotation about origin Coordinate rule
90^\circ anticlockwise (x,y)\mapsto(-y,x)
90^\circ clockwise (x,y)\mapsto(y,-x)
180^\circ (x,y)\mapsto(-x,-y)
Worked Example: Rotation About Origin

Question: Rotate (4,-1) through 90^\circ anticlockwise about the origin.

  1. Use (x,y)\mapsto(-y,x).
  2. (4,-1)\mapsto(1,4).

Answer: (1,4).

Enlargement

For centre C and scale factor k, the image lies on the line from C through the object point, with vector from C multiplied by k. If 0\lt k\lt1, the image is smaller and lies between the centre and object. If k is negative, the image lies on the opposite side of the centre and orientation is reversed.

\overrightarrow{CP\prime}=k\overrightarrow{CP}
Worked Example: Enlargement From Origin

Question: Enlarge point P=(3,-2) about the origin by scale factor -2.

  1. Multiply both coordinates by -2.
  2. P\prime=(-6,4).
  3. The negative factor places the image on the opposite side of the centre.

Answer: (-6,4).

Finding The Centre Of Enlargement

Join corresponding object and image vertices with straight lines. Extend the lines; their intersection is the centre of enlargement. Then compare corresponding lengths, including sign from the relative positions, to find the scale factor.

Combinations Of Transformations

The order matters. A reflection followed by a translation usually gives a different result from the translation followed by the reflection. Work one transformation at a time and label the intermediate image.

Worked Example: Combined Mapping

Question: Point A is (2,1). First reflect it in the y-axis, then translate by \begin{pmatrix}3\\4\end{pmatrix}.

  1. Reflection in y-axis gives (-2,1).
  2. Add the translation vector to obtain (1,5).

Answer: Final image (1,5).

Properties Preserved
Transformation Lengths Angles Orientation Area
Translation Preserved Preserved Preserved Preserved
Rotation Preserved Preserved Preserved Preserved
Reflection Preserved Preserved Reversed Preserved
Enlargement factor k Multiplied by |k| Preserved Reversed if k is negative Multiplied by k^2
Examination Guidance
  • Give every required parameter when describing a transformation.
  • For a translation, use a column vector.
  • For rotation, include centre, angle and direction.
  • For enlargement, include centre and signed scale factor.
  • In a combined transformation, apply the operations in the stated order.
Common Mistakes
  • Describing a translation as “left 3 and up 2” when a vector is expected.
  • Forgetting the centre of rotation or enlargement.
  • Treating a negative scale factor as merely a reduction.
  • Applying combined transformations in reverse order.
  • Using coordinate rules for rotations about a point other than the origin without first translating relative to the centre.
Knowledge Check And Practice

1. Translate (1,-4) by \begin{pmatrix}-3\\6\end{pmatrix}.

Answer: (-2,2).

2. Reflect (5,2) in the y-axis.

Answer: (-5,2).

3. Rotate (3,1) 180^\circ about the origin.

Answer: (-3,-1).

4. Enlarge (2,-5) about the origin by factor \frac12.

Answer: (1,-2.5).

5. What four details may be needed to describe a rotation?

Answer: Transformation type, centre, angle and direction.

6. What happens to area under enlargement scale factor -3?

Answer: It is multiplied by (-3)^2=9.
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