Negative Scale Factor Enlargement –

Negative scale factor enlargement is a Higher-tier GCSE Maths topic tested on AQA, Edexcel, and OCR papers. A negative scale factor produces an image that is inverted (upside down and flipped left-to-right) and appears on the opposite side of the centre of enlargement. This guide explains the concept, shows you how to draw negative enlargements step by step, and provides worked examples and practice questions.

What Is a Negative Scale Factor Enlargement?

An enlargement transforms a shape by multiplying all distances from a fixed centre of enlargement by a scale factor.

• A positive scale factor keeps the image on the same side of the centre.
• A negative scale factor places the image on the opposite side of the centre and inverts it.

The magnitude (absolute value) of the scale factor determines the size:

• |SF| > 1: the image is larger than the original.
• |SF| = 1: the image is the same size (but inverted for negative).
• |SF| < 1: the image is smaller than the original.

Key Principle

Step-by-Step Method

Drawing a Negative Scale Factor Enlargement

1. Plot the centre of enlargement (O).
2. For each vertex of the original shape, draw a line from O through the vertex.
3. Measure the distance from O to the vertex.
4. Multiply the distance by the absolute value of the scale factor.
5. Since the scale factor is negative, measure this new distance from O in the opposite direction (through O and out the other side).
6. Plot the new point. Repeat for all vertices.
7. Join the new points to form the image.

Describing a Negative Enlargement

1. State the transformation type: "Enlargement."
2. Give the scale factor (including the negative sign).
3. Give the centre of enlargement as coordinates.

Worked Example 1 — Foundation Level

This topic is Higher only, but this example uses simple coordinates.

Question: Triangle A has vertices at (2, 1), (4, 1), and (2, 3). Enlarge it by scale factor −1 about the origin (0, 0).

Working:

Step 1 — Scale factor −1 means each point moves to the opposite side of the origin at the same distance.

Step 2 — (2, 1) becomes (−2, −1). (4, 1) becomes (−4, −1). (2, 3) becomes (−2, −3).

Step 3 — The image is the same size as the original but inverted through the origin.

Answer: The image vertices are (−2, −1), (−4, −1), and (−2, −3).

Worked Example 2 — Higher Level

Question: Enlarge triangle B with vertices (1, 2), (3, 2), and (1, 5) by scale factor −2 about the centre (1, 1).

Working:

Step 1 — For each vertex, find the vector from the centre (1, 1):

• (1, 2): vector (0, 1).
• (3, 2): vector (2, 1).
• (1, 5): vector (0, 4).

Step 2 — Multiply each vector by −2:

• (0, 1) × −2 = (0, −2).
• (2, 1) × −2 = (−4, −2).
• (0, 4) × −2 = (0, −8).

Step 3 — Add each result to the centre (1, 1):

• (1 + 0, 1 − 2) = (1, −1).
• (1 − 4, 1 − 2) = (−3, −1).
• (1 + 0, 1 − 8) = (1, −7).

Answer: The image vertices are (1, −1), (−3, −1), and (1, −7).

Worked Example 3 — Exam Style

Question: Shape P is mapped onto shape Q by an enlargement. P has a vertex at (3, 4) that maps to (−1, 0) on Q. The centre of enlargement is (1, 2). Find the scale factor.

Working:

Step 1 — Vector from centre to original point: (3 − 1, 4 − 2) = (2, 2).

Step 2 — Vector from centre to image point: (−1 − 1, 0 − 2) = (−2, −2).

Step 3 — Scale factor = image vector / original vector = (−2)/(2) = −1.

Step 4 — Check with y-component: (−2)/(2) = −1. Consistent.

Answer: The scale factor is −1.

Common Mistakes

• Forgetting to go through the centre to the other side. With a negative scale factor, the image point is on the opposite side of the centre from the original — not on the same side.
• Ignoring the negative sign when calculating size. The image size depends on |SF|. A scale factor of −2 means the image is twice as large, not smaller.
• Describing the transformation without all three details. You must state: (1) enlargement, (2) scale factor (including sign), and (3) centre of enlargement to earn full marks.

Exam Tips

• Use the vector method: find the vector from the centre to each vertex, multiply by the scale factor, then add back to the centre. This avoids mistakes with direction.
• If a question shows the original and image on opposite sides of a point, suspect a negative scale factor.
• The image is always similar to the original (same angles, proportional sides). Negative SF also inverts orientation.
• On coordinate grids, count squares carefully to find the centre — it lies on the line joining each original vertex to its image.

Practice Questions

Q1 (Higher): A square has vertices at (2, 2), (4, 2), (4, 4), and (2, 4). Enlarge by scale factor −1 about (3, 3). What are the image coordinates?

Q2 (Higher): Triangle C at (0, 0), (4, 0), (0, 3) is enlarged by SF = −½ about (0, 0). Find the image.

Q3 (Higher): An enlargement maps point (2, 3) to (−4, −3) with centre (0, 1). Find the scale factor.

Practise negative enlargement questions with instant feedback free on GCSEMathsAI.

Related Topics

• Transformations: Enlargement and Scale Factor — positive scale factor enlargements.
• Transformations: Reflection, Rotation, Translation — other transformations.
• Congruence and Similarity — similar shapes from enlargement.

Summary

Negative scale factor enlargement places the image on the opposite side of the centre of enlargement and inverts it. The magnitude of the scale factor controls the size change. Use the vector method for accuracy: find the vector from the centre to each vertex, multiply by the scale factor (which reverses direction because it is negative), and add the result back to the centre. Always describe the transformation with all three details: type (enlargement), scale factor (including the negative sign), and centre of enlargement.