Measure each corner from the centre of enlargement, multiply by \(3\), and then go that far the other way. That last step is what the minus sign does.

\((6, 3)\) is 2 right and 1 up from \((4, 2)\). Multiplying by \(3\) gives 6 and 3, so instead go 6 left and 3 down: the image of that corner is \((-2, -1)\).

Doing the same with the other two corners gives \((-8, -1)\) and \((-2, -10)\).

The image is \(3\) times as big and it is upside down — a negative scale factor turns the shape through half a turn as well as scaling it.

A scale factor of \(-1\) leaves every length unchanged — \(1\) times as long — but sends each corner the same distance the other side of the centre.

\((6, 6)\) is 3 right and 2 up from \((3, 4)\), so its image is 3 left and 2 down: \((0, 2)\).

The other two corners give \((-3, 2)\) and \((0, 0)\).

B is the same size as A and upside down.

B is the same size as A, so the transformation could be a rotation. Every corner has gone to the opposite side of \((3, 4)\) at the same distance from it, which is exactly a half turn about that point.

A rotation of \(180^\circ\) needs no direction — clockwise and anticlockwise land in the same place — but it does need the centre, and \((3, 4)\) is it.

This is why "describe fully the single transformation" is never asked about a scale factor of \(-1\): the enlargement and the rotation are both right, and there would be no way to choose between them.

\((7, 6)\) is 5 right and 2 up from the centre \((2, 4)\).

Multiply both by \(2\): 10 and 4. The scale factor is negative, so count them the other way — 10 left and 4 down from the centre.

That lands on \((-8, 0)\).

Going back undoes both halves of what was done. The lengths were multiplied by \(2\), so now they are divided by \(2\); the shape crossed to the other side of the centre, so it crosses back.

The scale factor is \(-\frac{1}{ 2 }\), about the same centre \((2, 4)\) — a negative enlargement is undone by another negative enlargement, never by a positive one.

The size of the scale factor comes from a pair of matching sides: the horizontal side of A is 3 squares and the matching side of B is 9, so \(9 \div 3 = 3\).

B is on the other side of the centre from A and it is upside down, so the scale factor is negative.

The scale factor is \(-3\).

Follow any one of the dashed lines. It starts at a corner of A, runs into the cross at \((2, 2)\), and carries on out the far side to the matching corner of B.

With a positive scale factor the image would be on the same side of the centre as the object and the same way up. Here the rays pass through the centre and B is upside down, and that is what a negative scale factor looks like.

Two things are happening at once here: the shape gets smaller, because the size of the scale factor is less than 1, and it goes to the other side of the centre, because the scale factor is negative.

\((6, 4)\) is 4 right and 2 up from \((2, 2)\). Divide both by \(2\) to get 2 and 1, then count them the other way — 2 left and 1 down — to reach \((0, 1)\).

The other two corners give \((-2, 1)\) and \((0, -2)\).

The image is \(\frac{1}{ 2 }\) of the size of A and upside down.

B is a different size from A, so the transformation is an enlargement.

Comparing a pair of matching sides gives the size of the scale factor: the horizontal side of A is 3 squares and the matching side of B is 6, so the scale factor is \(6 \div 3 = 2\) in size.

B is upside down compared with A, so the scale factor is negative: it is \(-2\).

For the centre, join each corner of A to the matching corner of B and continue the lines until they cross. They meet at \((2, 3)\). Check it: \((5, 5)\) is 3 right and 2 up from there, and \((-4, -1)\) is \(2\) times as far in the opposite direction.

All three parts are needed. "Enlargement, scale factor \(2\)" scores nothing extra without the minus sign and the centre.

Measure each corner from the centre of enlargement, multiply by \(2\), and then go that far the other way. That last step is what the minus sign does.

\((5, 5)\) is 2 right and 3 up from \((3, 2)\). Multiplying by \(2\) gives 4 and 6, so instead go 4 left and 6 down: the image of that corner is \((-1, -4)\).

Doing the same with the other two corners gives \((-7, -4)\) and \((-1, -8)\).

The image is \(2\) times as big and it is upside down — a negative scale factor turns the shape through half a turn as well as scaling it.

A scale factor of \(-1\) leaves every length unchanged — \(1\) times as long — but sends each corner the same distance the other side of the centre.

\((5, 5)\) is 2 right and 2 up from \((3, 3)\), so its image is 2 left and 2 down: \((1, 1)\).

The other two corners give \((-1, 1)\) and \((1, -2)\).

B is the same size as A and upside down.

B is the same size as A, so the transformation could be a rotation. Every corner has gone to the opposite side of \((3, 3)\) at the same distance from it, which is exactly a half turn about that point.

A rotation of \(180^\circ\) needs no direction — clockwise and anticlockwise land in the same place — but it does need the centre, and \((3, 3)\) is it.

This is why "describe fully the single transformation" is never asked about a scale factor of \(-1\): the enlargement and the rotation are both right, and there would be no way to choose between them.

\((8, 7)\) is 4 right and 3 up from the centre \((4, 4)\).

Multiply both by \(2\): 8 and 6. The scale factor is negative, so count them the other way — 8 left and 6 down from the centre.

That lands on \((-4, -2)\).

Going back undoes both halves of what was done. The lengths were multiplied by \(2\), so now they are divided by \(2\); the shape crossed to the other side of the centre, so it crosses back.

The scale factor is \(-\frac{1}{ 2 }\), about the same centre \((4, 4)\) — a negative enlargement is undone by another negative enlargement, never by a positive one.

The size of the scale factor comes from a pair of matching sides: the horizontal side of A is 3 squares and the matching side of B is 9, so \(9 \div 3 = 3\).

B is on the other side of the centre from A and it is upside down, so the scale factor is negative.

The scale factor is \(-3\).

Follow any one of the dashed lines. It starts at a corner of A, runs into the cross at \((2, 2)\), and carries on out the far side to the matching corner of B.

With a positive scale factor the image would be on the same side of the centre as the object and the same way up. Here the rays pass through the centre and B is upside down, and that is what a negative scale factor looks like.