Skip to content
Cambridge IGCSE0607

Transformations

International Mathematics 0607 Chapter Notes

What this chapter covers

Transformations
ShareWhatsAppPost
Transformations notes

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Transformations notes as text: skim, search, and jump between subtopics.

~8 min read

In this chapter

  1. 1Translation by a vector
  2. 2Reflection in a line
  3. 3Rotation about a centre
  4. 4Enlargement from a centre
  5. 5Describing a single transformation fully

1. Translation by a vector

A translation moves every point of a shape the same distance in the same direction.

The translation is given by a vector beginpmatrixx yendpmatrix. The top number is the move to the right (to the left if negative). The bottom number is the move up (down if negative).

Use the same move for every vertex. A ruler must be used for all straight edges.

(a, b) translated by beginpmatrixx yendpmatrix is (a + x, b + y)

reverse of beginpmatrixx yendpmatrix is beginpmatrix-x -yendpmatrix

Examiner insight

“The word translate, or translation, was required with words such as translocation, shift and move not accepted. A vector was also required, with answers given in coordinate form not accepted.” “Errors included reversing the components, sign errors, and components with the wrong magnitudes.”

Worked example 14 marks

Do not use a calculator.

A triangle has vertices (1, 1), (4, 1) and (1, 3). It is translated by the vector beginpmatrix-3 2endpmatrix.

(a) Write down the vertices of the image.

(b) Describe fully the single transformation that maps the image back onto the original triangle.

  1. 1

    add -3 to each x and 2 to each y [1]

  2. 2

    image: (-2, 3), (1, 3), (-2, 5) [1]

  3. 3

    translation [1]

  4. 4

    by the vector beginpmatrix3 -2endpmatrix (the signs are reversed) [1]

2. Reflection in a line

A reflection is described by its mirror line. Each point and its image are the same distance from the mirror line, on opposite sides of it.

Core lines are vertical, x = a, or horizontal, y = b. Give the line as an equation such as y = 0, or name the axis, not as a pair of coordinates. Give one answer for the line, not two. A ruler must be used for all straight edges.

Extended: the line may be any straight line, such as y = x. The reverse of a reflection is a reflection in the same line.

in x = a: (x, y) → (2a - x, y)

in y = x: (x, y) → (y, x)

Examiner insight

“Many gave a pair of coordinates instead of describing a line.” “Some gave two answers for the mirror line with one being correct and the other incorrect, for example, x-axis and x = 0 or y = 0 and y-axis.”

Worked example 14 marks

Do not use a calculator.

A triangle has vertices (1, 1), (4, 1) and (1, 3). It is reflected in the line x = 2.

(a) Write down the vertices of the image.

(b) Describe fully the single transformation that maps the image back onto the original triangle.

  1. 1

    (1, 1) is 1 to the left of x = 2, so its image is 1 to the right: (3, 1) [1]

  2. 2

    (4, 1) → (0, 1) and (1, 3) → (3, 3) [1]

  3. 3

    reflection [1]

  4. 4

    in the line x = 2 (the reverse is the same reflection) [1]

3. Rotation about a centre

A rotation needs three things: the angle, the direction (clockwise or anticlockwise) and the centre.

Core: the angle is a multiple of 90° and the centre is the origin, a vertex or the midpoint of an edge. Extended: the centre can be any point.

To rotate about a point that is not the origin, find each vertex's position relative to the centre, turn that, then add the centre back.

90^(°) clockwise about O: (x, y) → (y, -x)

90^(°) anticlockwise about O: (x, y) → (-y, x)

180^(°) about O: (x, y) → (-x, -y)

Examiner insight

“Most other candidates earned one mark for rotating the triangle through the correct angle but using the wrong centre, usually the origin or (1, 1) or for rotating the triangle clockwise rather than anticlockwise.” “Some candidates realised that the transformation was a rotation by 90, and didn’t give the direction of the rotation.”

Worked example 15 marks

Do not use a calculator.

A triangle has vertices (2, 1), (5, 1) and (2, 3). It is rotated through 90^(°) clockwise about the origin.

(a) Write down the vertices of the image.

(b) Describe fully the single transformation that maps the image back onto the original triangle.

  1. 1

    clockwise rule: (x,y) → (y, -x) [1]

  2. 2

    image: (1, -2), (1, -5), (3, -2) [1]

  3. 3

    rotation [1]

  4. 4

    90° anticlockwise [1]

  5. 5

    centre (0, 0) [1]

4. Enlargement from a centre

An enlargement needs a scale factor and a centre. Every length in the image is the scale factor times the original length.

Core: the scale factor is positive, and may be a fraction. A fractional scale factor gives an image smaller than the original, but it is still an enlargement. Extended: the scale factor may be negative.

A negative scale factor puts the image on the opposite side of the centre.

P' = C + k (P - C)

reverse of scale factor k is 1/k, same centre

Examiner insight

“First, the transformation is still called an enlargement even though the image is smaller than the original shape.” “Candidates were able to enlarge the triangle by a scale factor 2, and either used the wrong centre of enlargement, or did not consider the negative associated with the scale factor.”

Worked example 15 marks

Do not use a calculator.

A triangle has vertices (2, 1), (4, 1) and (2, 2). It is enlarged with scale factor 2 and centre (1, 0).

(a) Write down the vertices of the image.

(b) Describe fully the single transformation that maps the image back onto the original triangle.

  1. 1

    (2, 1) is (1, 1) from the centre, so its image is (2, 2) from the centre: (3, 2) [1]

  2. 2

    (4, 1) → (7, 2) and (2, 2) → (3, 4) [1]

  3. 3

    enlargement [1]

  4. 4

    scale factor 1/2 [1]

  5. 5

    centre (1, 0) [1]

5. Describing a single transformation fully

A question that says single transformation wants one name and every detail of it. One piece of information is needed for each mark.

Reflection: the equation of the mirror line. Rotation: angle, direction and centre. Enlargement: scale factor and centre. Translation: the vector.

Extended: two transformations in a row may equal one. Work out where a point goes at each stage, then compare start and finish.

Examiner insight

“Candidates must only give one transformation as the question says single transformation. A good indication to what transformation is required, is that one piece of information is needed for each mark.”

Worked example 14 marks

The transformation P is a reflection in the line x = 0.

The transformation Q is a rotation of 90^(°) clockwise about the origin.

(a) Find the image of the point with coordinates (4, 2) under the transformation P.

(b) Find the image of the point with coordinates (1, 5) under the transformation Q.

(c) Describe fully the single transformation equivalent to transformation P followed by transformation Q.

  1. 1

    P: (x,y) → (-x, y), so (4, 2) → (-4, 2) [1]

  2. 2

    Q: (x,y) → (y, -x), so (1, 5) → (5, -1) [1]

  3. 3

    P then Q: (x,y) → (-x,y) → (y, x)

  4. 4

    reflection [1]

  5. 5

    in the line y = x [1]

Recap

  • Use the exact name: reflection, rotation, enlargement or translation. ‘Flip’, ‘move’ and ‘shift’ score nothing.
  • Single transformation means one name only, with one detail per mark.
  • Translation: a vector, top number right, bottom number up.
  • Reflection: the equation of the mirror line, such as x = 5 or y = 0.
  • Rotation: angle, direction, centre.
  • Enlargement: scale factor, centre. A fractional factor is still an enlargement.
  • Extended: any mirror line, any centre, negative scale factors, and combinations of transformations.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. A shape is moved 4 units to the left and 3 units up. It does not turn and its size does not change. Describe fully the single transformation.2 marks
  2. Do not use a calculator. Find the image of the point (7, 3) when it is reflected in the line x = 4.2 marks
  3. Do not use a calculator. Find the image of the point (3, -2) under a rotation of 90^(°) clockwise about the origin.2 marks
  4. A rectangle measures 12 cm by 9 cm. It is enlarged with scale factor 1/3. (a) Write down the mathematical name of this transformation. (b) Work out the measurements of the image.3 marks
  5. Do not use a calculator. The point P(3, 2) is enlarged with scale factor 3 and centre (1, 1). Find the image of P.3 marks
  6. Do not use a calculator. A triangle has vertices A(2, 1), B(6, 1) and C(2, 4). It is rotated through 90^(°) clockwise about A. Find the image of B.3 marks
  7. Do not use a calculator. Triangle P has vertices (1, 1), (3, 1) and (1, 2). Triangle Q has vertices (-1, 1), (-3, 1) and (-1, 2). Describe fully the single transformation that maps P onto Q.2 marks
  8. Do not use a calculator. Find the image of the point (5, 4) under a rotation of 90^(°) anticlockwise about the point (2, 1).3 marks
  9. Do not use a calculator. A machine prints a pattern along a strip. Triangle T has vertices (1, 1), (3, 1) and (1, 4). T is reflected in the line x = -1 to give triangle U. U is then reflected in the line x = 4 to give triangle V. (a) Write down the vertices of U. (b) Write down the vertices of V. (c) Describe fully the single transformation that maps V back onto T.6 marks
  10. Do not use a calculator. A pinhole camera forms an image of an object by an enlargement with a negative scale factor. The pinhole is the centre of enlargement. On a coordinate grid the object is triangle ABD with A(6, 3), B(6, 7) and D(10, 3). The image of A is A'(0, 0), the image of B is B'(0, -2) and the image of D is D'(-2, 0). (a) Find the scale factor of the enlargement. (b) Find the coordinates of the pinhole. (c) E(8, 5) is the midpoint of BD. Find the image of E. (d) Describe fully the single transformation that maps the image back onto the object.9 marks

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Types of numberSetsPowers and rootsFractions, decimals and percentagesOrderingThe four operations