GCSE · Maths · AQA · Spec 8300 · Higher

Negative scale factors for enlargement (Higher)

Negative sounds like 'less', so scale factor −2 should shrink a shape. It doesn't. It doubles it and throws it through the centre, upside down. Try it below.

Geometry · Enlargement

Where does A′ belong?

P to A 2.0. P to A′ 4.0. AC 2.0. A′C′ 8.9. angle A 63°. angle A′ 17°. Classification: Enlargement · scale factor −2 · centre P. Relationship: Scale factor −2 about P: A′ is on the other side of P, twice as far away (P to A′ = 2 × P to A). Only then is A′C′ = 2 × AC, and angle A′ = angle A.

P to A2.0P to A′4.0AC2.0A′C′8.9angle A63°angle A′17°drag A′ through P

Enlargement · scale factor −2 · centre P

Scale factor −2 about P: A′ is on the other side of P, twice as far away (P to A′ = 2 × P to A). Only then is A′C′ = 2 × AC, and angle A′ = angle A.

Triangle ABC is being enlarged by scale factor −2, centre P. B′ and C′ are already in the right place. A′ is sitting where you'd put it if you forgot the minus sign: twice as far from P, but on the same side. Drag A′ along the line, through P and out the other side, until A′B′C′ is a proper copy of ABC.

Construct it on a grid

Problem

Triangle DEF has vertices D(6, 5), E(10, 5) and F(6, 9). Enlarge it by scale factor −½, centre (2, 1).

Spot the slip

One line of this answer goes wrong

Enlarge the point J(3, 2) by scale factor −2, centre (1, 1).

A student's answer — which line goes wrong?

The centre is (1, 1) and the scale factor is −2.
J is at (3, 2), so the step to J is right 3, up 2.
Multiply the step by −2: left 6, down 4.
Count that from the centre (1, 1): J′ = (−5, −3).

Now go backwards

Describe the enlargement

Triangle T has vertices (3, 2), (5, 2) and (3, 3). Its image T′ has vertices (−1, −2), (−7, −2) and (−1, −5). Describe fully the single transformation that maps T onto T′.

Join each vertex of T to its matching vertex on T′: (3, 2) to (−1, −2), (5, 2) to (−7, −2), and (3, 3) to (−1, −5).Matching vertices: the corners that play the same part in each triangle.
missing step

Which line is step 2?

All three lines cross at (2, 1), so that's the centre of enlargement.The centre is (1, 0), halfway between (3, 2) and its image (−1, −2).The centre is (0, 0), because that's where the axes cross.

What do you really think?

Smaller? Rotated? Or one enlargement?

Triangle T is enlarged by scale factor −½, centre (0, 0). Its image T′ is half the size of T, on the other side of the origin, and upside down.

Which of these is closest to what you think right now?

It's half the size because the scale factor is negative. Negative scale factors make shapes smaller.It's really two transformations: a rotation of 180° about (0, 0), then an enlargement by scale factor ½.It's one enlargement. The ½ makes it smaller, and the minus sign puts it on the other side of the centre, upside down.

How sure are you?

I'm sureI think soI'm guessing

It's half the size because the scale factor is negative. Negative scale factors make shapes smaller.

This is a common misconception

The minus sign isn't what shrank it. The ½ did that. Think back to the triangle at the top of the page: scale factor −2 made it twice as big, and still sent it to the other side. Size comes from the number without its sign. The sign only decides the side.

It's really two transformations: a rotation of 180° about (0, 0), then an enlargement by scale factor ½.

This is a common misconception

It does look like that, and those two moves together would land it in the same place. But you don't need two moves. One enlargement by −½ does the whole job, turn and shrink together, because multiplying each step by a negative number turns it round and scales it at the same time. So describe it as one transformation: enlargement, scale factor −½, centre (0, 0).

It's one enlargement. The ½ makes it smaller, and the minus sign puts it on the other side of the centre, upside down.

This matches the evidence

Exactly right. One number, two jobs: its size sets how big, its sign sets which side. And because a single enlargement does everything, that's how you describe it: enlargement, scale factor −½, centre (0, 0).

A negative scale factor is one enlargement, not a rotation plus an enlargement, and the minus sign never decides the size.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The size of the scale factor decides how big the image is. The sign decides where it goes: negative means through the centre of enlargement and out the other side, upside down.

What you need to know

To enlarge a shape you need two things: the centre of enlargement and the scale factor.
Each vertex's distance from the centre is multiplied by the scale factor to give its image's distance from the centre. Every length in the shape is multiplied by the scale factor too.
A negative scale factor puts the image on the other side of the centre, turned through 180°.
The size of the scale factor, ignoring its sign, decides the size of the image. Between 0 and 1 gives a smaller image, whether the scale factor is positive or negative.
To describe an enlargement, state the centre of enlargement and the scale factor. A negative scale factor is described as one enlargement, not as a rotation followed by an enlargement.

The big picture

An enlargement multiplies every distance from the centre by the scale factor, so every length in the shape is multiplied too. A negative scale factor also reverses each step, so the image lands on the other side of the centre, turned through 180°. How big the image is depends only on the size of the number: between 0 and 1 gives a smaller image, positive or negative. Describe it as one enlargement, giving its centre and its scale factor.

Key points

1Measure every step from the centre of enlargement. Multiplying the coordinates themselves only works when the centre is (0, 0).

2With a negative scale factor, multiply the step and reverse it: right 2, up 1 with scale factor −3 becomes left 6, down 3.

3Scale factor −1 keeps the size the same but still puts the image on the other side of the centre, upside down.

4If the image has changed size, there has been an enlargement with a scale factor that isn't 1.

5To find the centre, join each vertex to its matching image vertex. The lines cross at the centre.

6The size of the scale factor is image length ÷ object length. The sign comes from which side of the centre the image is on.

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Worked example

Problem

The point A(7, 5) is enlarged by scale factor −2. Its image is A′(−5, −1). Find the centre of enlargement.

⚠ Watch out

Forgetting to reverse the direction. With scale factor −2, a step of right 3, up 1 from the centre becomes left 6, down 2, not right 6, up 2. If your image is on the same side of the centre as the object, the minus sign has gone missing.

🧠

Memory hook

Sign → which side. Size → how big. Never let one do the other's job.

✓

Check yourself

Enlarge the point (5, 3) by scale factor −3, centre (4, 2). Which side of the centre should the image be on, and what are its coordinates?

Flashcards

(12)

A vertex is 3 units from the centre. The scale factor is −2. How far is its image from the centre?

6 units: the distance is multiplied by 2, and the minus sign puts it on the other side.

What happens to each length in a shape when it is enlarged?

It is multiplied by the scale factor (by its size, ignoring any minus sign).

Where does the image go when the scale factor is negative?

Through the centre of enlargement to the other side, turned through 180°.

Which scale factors make the image smaller?

Ones whose size, ignoring the sign, is between 0 and 1, such as ½ or −½.

Does a scale factor of −4 make the image bigger or smaller?

Bigger: four times as big. The minus sign only affects the side, not the size.

What does an enlargement with scale factor −1 do?

Keeps the size the same, but puts the image on the other side of the centre, upside down.

What must you give to describe an enlargement fully?

Say it's an enlargement, then give the scale factor and the centre: the same two things you need to draw one.

Should an enlargement with a negative scale factor be described as a rotation then an enlargement?

No. One enlargement with a negative scale factor does the whole job, turn included, so describe it as that single transformation.

How do you find the centre of enlargement from a diagram?

Join each vertex to its matching image vertex. The lines cross at the centre.

How do you find the size of the scale factor from a diagram?

Divide a length on the image by the matching length on the object.

An image is on the opposite side of the centre and turned 180°. What does that tell you about the scale factor?

It's negative.

The centre is (2, 1). Where do you measure each step from?

From (2, 1), not from the origin.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Negative scale factors for enlargement (Higher)

One line of this answer goes wrong

What does each scale factor do?

Describe the enlargement

Smaller? Rotated? Or one enlargement?

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