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Enlarging with negative scale factors – GCSE Maths

Higher · Grade 6 · Non-calculator

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Enlarge a shape with a negative scale factor so the image appears on the opposite side of the centre, upside down. It is Higher only, aimed at grade 6. Free questions with answers are on this page. Worked examples show each line before you try the questions.

A negative scale factor enlarges and rotates about the centre: the image is on the opposite side of the centre, and the shape is upside-down relative to the original. Scale factor means twice as far from the centre, other side, twice the size. Count squares from the centre to a vertex, then the same count the other way, times .

Describe a given enlargement with centre, scale factor (with the minus) and, if needed, the word enlargement. The image and object are similar; corresponding sides are parallel if is positive, and parallel-but-opposite if is negative (still parallel as lines).

Negative scale factor: the image is on the other side of the centre, and rotated . Vector from centre scale factor, then add the centre back. Scale factor , centre O, is a half-turn.

Objectives

Key points

Worked examples

Example 1

Enlarge the point (3, 1) with scale factor -2 and centre (0, 0). Find the image.

2 marks

  1. Multiply x-coordinate by -2: 3 × -2 = -6.
  2. Multiply y-coordinate by -2: 1 × -2 = -2.
  3. Combine coordinates into image point: (-6, -2).

Answer: (-6, -2)

Common slips
  • Slip: Add -2 to x-coordinate: 3 + (-2) = 1. Enlargement multiplies coordinates, not adds.
  • Slip: Leave y unchanged as 1. Both coordinates must be multiplied by the scale factor.
  • Slip: Keep signs positive: (6, 2). A negative factor reverses signs through the origin.

Example 2

The enlargement maps (3, 1) to (-6, -2). Find the scale factor.

2 marks

  1. Divide image x by original x: -6 ÷ 3 = -2.
  2. Verify with y-coordinates: -2 ÷ 1 = -2.
  3. State the negative scale factor: -2.

Answer: -2

Common slips
  • Slip: Subtract coordinates: -6 - 3 = -9. Scale factor is the ratio of coordinates, not difference.
  • Slip: Divide original by image: 3 ÷ -6 = -0.5. Scale factor is image divided by object.
  • Slip: State positive factor 2. A change in direction means the factor is negative.

Example 3

The point (3, 1) is enlarged with scale factor -2 and centre (0, 0). Find the distance from the origin to the image. Give your answer as a simplified surd.

3 marks

  1. Calculate object distance to origin: √(3^2 + 1^2) = √10.
  2. Multiply the distance by 2: 2 × √10 = 2√10.
  3. State exact simplified surd distance: 2√10.

Answer: 2√10

Common slips
  • Slip: Add coordinates: 3 + 1 = 4. Origin distance uses Pythagoras, not addition.
  • Slip: Square the factor: 4 × √10. Distance is multiplied by the size of the factor, 2, not by its square.
  • Slip: Round to decimal 6.32. Leave exact surd form unless asked to round.

Questions and answers

GCSE Maths enlarging with negative scale factors questions and answers. Try each one, then open the answer. Easy questions come first.

Question 1 Easy2 marks

Enlarge the point (1, 4) with scale factor -2 and centre (0, 0). Write down the image.

Show answer

Answer: (-2, -8)

  1. Multiply 1 by -2 to get -2.
  2. Multiply 4 by -2 to get -8.
  • M1 — Multiply 1 by -2 to get -2.
  • A1 — Multiply 4 by -2 to get -8.

Question 2 Easy2 marks

Enlarge the point (4, -2) with scale factor -1 and centre (0, 0). Write down the image.

Show answer

Answer: (-4, 2)

  1. Multiply 4 by -1 to get -4.
  2. Multiply -2 by -1 to get 2.
  • M1 — Multiply 4 by -1 to get -4.
  • A1 — Multiply -2 by -1 to get 2.

Question 3 Medium2 marks

An enlargement with centre (0, 0) maps (4, 2) to (-12, -6). Find the scale factor.

Show answer

Answer: -3

  1. Divide -12 by 4 to get -3.
  2. Verify -6 / 2 = -3.
  • M1 — Divide -12 by 4 to get -3.
  • A1 — Verify -6 / 2 = -3.

Question 4 Medium2 marks

An enlargement with centre (0, 0) maps (2, 5) to (-2, -5). Find the scale factor.

Show answer

Answer: -1

  1. Divide -2 by 2 to get -1.
  2. Verify -5 / 5 = -1.
  • M1 — Divide -2 by 2 to get -1.
  • A1 — Verify -5 / 5 = -1.

Question 5 Hard2 marks

The point (6, 8) is enlarged with scale factor -2 and centre (0, 0). Find the distance from the origin to the image.

Show answer

Answer: 20

  1. Origin distance is √(6^2 + 8^2) = 10.
  2. Multiply 10 by 2 to get 20.
  • M1 — Origin distance is √(6^2 + 8^2) = 10.
  • A1 — Multiply 10 by 2 to get 20.

Question 6 Hard2 marks

The point (5, 12) is enlarged with scale factor -2 and centre (0, 0). Find the distance from the origin to the image.

Show answer

Answer: 26

  1. Origin distance is √(5^2 + 12^2) = 13.
  2. Multiply 13 by 2 to get 26.
  • M1 — Origin distance is √(5^2 + 12^2) = 13.
  • A1 — Multiply 13 by 2 to get 26.

Question 7 Exam-style4 marks

The point A is (4, 3). It is enlarged with scale factor -2 and centre (0, 0) to give A'. Work out the length AA'.

Show answer

Answer: 15

  1. The image A' is (4 × -2, 3 × -2) = (-8, -6).
  2. The change in x is 12 and the change in y is 9.
  3. AA' = √(12^2 + 9^2) = √225 = 15.
  • M1 — Image (-8, -6)
  • M1 — Differences 12 and 9
  • M1 — √(12^2 + 9^2)
  • A1 — 15

Common mistakes

Most lost marks on this topic are one slip.

  1. Keeping the signs when the scale factor is negative

    Multiply by the negative number

  2. Giving a positive scale factor when the image is in the opposite quadrant

    Include the minus sign

  3. Adding the coordinates instead of using Pythagoras for the distance

    Square, add, and square-root, then multiply

Spot the slip

The point (3, 1) is enlarged with scale factor -2 and centre (0, 0). Ada keeps both signs and writes (6, 2). What is the image?

  1. (-6, -2)
  2. (6, 2)
  3. (-3, -1)
Show the slip

(-6, -2) Multiply both coordinates by -2.

Exam tips

Printable worksheet

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Questions students ask

Is Enlarging with negative scale factors Foundation or Higher?

Enlarging with negative scale factors is Higher only, aimed at grade 6.

What is a common slip on Enlarging with negative scale factors?

Keeping the signs when the scale factor is negative. Multiply by the negative number.

Where can I get Enlarging with negative scale factors questions and answers?

The questions on this page are free. Open Show answer under each one, or download the worksheet PDF. Start free to practise this topic in the app.

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Last updated 9 November 2026.

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