Enlargements and Combined Transformations

An enlargement multiplies every distance from a fixed centre by the scale factor $k$ . If $0<k<1$ the shape shrinks; if $k$ is negative the image is on the opposite side of the centre and upside down. Combine transformations by doing them in order.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Enlarge a point or shape about any centre.
Use fractional and negative scale factors.
Find the centre and scale factor of an enlargement.
Find the single transformation equivalent to two others.

Pause and think

A torch shines through a small cut-out shape onto a wall. Move the torch further away from the cut-out. What happens to the shadow?

Section 2 of 14

Core idea

Start hereA projector

A projector throws a small slide onto a big screen. Rays fan out from the lamp; every point on the screen is the same multiple of the distance from the lamp as its point on the slide. The lamp is the centre of enlargement.

The big idea

Centre and scale factor $k$ : image is $k$ times as far from the centre. $0<k<1$ shrinks; $k<0$ goes through the centre, upside down. Area × $k^2$ .

1. Drawing

Enlarging a triangle from O with scale factor k

scale factor k = -2.0

Rays from the centre pass through each point and its image. Negative

k

turns the image upside down on the far side of O.

\text{image}=C+k(P-C)

$C$: centre.
$P$: original point.
$k$: scale factor.

Worked exampleEnlarging $(2,1)$

$k=3$ about $(0,0)$ ; $k=3$ about $(1,1)$ ; $k=-2$ about $(0,0)$ .

Goal: Images.
Strategy: Vector from centre × $k$ .

$3(2,1)=(6,3)$
About the origin.
$(1,1)+3(1,0)=(4,1)$
Vector from $(1,1)$ is $(1,0)$ .
$-2(2,1)=(-4,-2)$
Negative: opposite side.

Answer

(6,3)

(4,1)

(-4,-2)

✓ CheckCentre is the only invariant point ✓.

Edge caseFractional k

k=\frac12

is still called an "enlargement", though it shrinks.

Pause and think

Area 5 enlarged by $k=-3$ . New area?

Summary

Distances from the centre × $k$ .
Area × $k^2$ .

Scale factor: Image length ÷ original length.

HistoryThe camera obscura. Light passing through a small hole into a dark room makes an upside-down image of the scene outside on the far wall. The Chinese philosopher Mozi described this in the 5th century BC, and Ibn al-Haytham studied it carefully around 1020. It is an enlargement with a negative scale factor, with the hole as the centre.

Fun factYour eye flips the world. Your eye's lens works like that hole: the image on your retina is upside down. Your brain turns it the right way up without you noticing.

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