A reflection sends every corner to the point the same number of squares from the y-axis, on the other side of it.

Reflecting in the y-axis keeps the y-coordinate of a point the same and changes the sign of its x-coordinate.

\((-1, -2) \to (1, -2)\), \((-3, -2) \to (3, -2)\) and \((-1, -4) \to (1, -4)\).

Plot those three points and join them up.

The top number of the vector is the movement across and the bottom number is the movement up, so every corner of shape A moves 5 squares right and 5 squares up.

The corner \((-1, 2)\) lands on \((4, 7)\), \((1, 2)\) lands on \((6, 7)\), and the top corner \((0, 4)\) lands on \((5, 9)\).

Move every corner by the same vector and the image is exactly the same size, shape and way up as A.

The top number of a column vector is the movement across and the bottom number is the movement up or down.

The top number is 5 with a positive sign, and the other number is 0, so the shape moves across (left or right) only.

A positive top number means right.

The mirror \(x = -1\) is vertical, so each corner moves straight across it and its \(y\)-coordinate does not change.

Count the squares from each corner to the mirror and go the same number the other side.

\((-3, 0)\) is 2 square(s) from the line, so it maps to \((1, 0)\). \((-6, 0)\) is 5 squares from the line, so it maps to \((4, 0)\).

The other two corners go the same way: \((-6, 2) \to (4, 2)\) and \((-3, 3) \to (1, 3)\).

The image is the same size and shape as A, turned over — the taller side is still the one nearer the mirror.

Trace shape A, put a pencil point on the origin and turn the tracing a quarter turn clockwise.

A quarter turn clockwise about the origin sends the corner \((1, 1)\) to \((1, -1)\): what was 1 across and 1 up ends up 1 and 1 the other way round.

The other corners follow: \((3, 1) \to (1, -3)\), \((3, 2) \to (2, -3)\) and \((1, 5) \to (5, -1)\).

The image is the same size and shape as A — a rotation only turns it.

A and B are the same size, the same shape and the same way up, so the transformation is a translation.

Pick a corner of A and find the matching corner of B. \((1, 0)\) on A matches \((4, -4)\) on B, which is 3 squares right and 4 squares down.

So the transformation is a translation by \(\begin{pmatrix} 3 \\ -4 \end{pmatrix}\).

A translation is fully described by its column vector alone — but the vector must be there, and written as a column. "Moved 3 right and 4 down" describes it in words but is not the column vector the mark is for.

Each corner moves at right angles to the mirror, to the point the same distance the other side of it.

In coordinates, reflecting in \(y = -x\) means the two coordinates of a point swap over and change sign:

\((3, -2) \to (2, -3)\), \((5, -2) \to (2, -5)\) and \((3, 1) \to (-1, -3)\).

Check one of them by counting: the mirror runs diagonally through the origin, and a corner and its image are the same distance from it, measured at right angles to the line.

A half turn sends each corner through the centre to the point the same distance the other side of it.

The corner \((4, 0)\) is 2 right and 1 up from \((2, -1)\), so its image is 2 left and 1 down from the centre, at \((0, -2)\).

\((7, 0)\) is 5 right and 1 up, so it maps to \((-3, -2)\); \((4, 2)\) is 2 right and 3 up, so it maps to \((0, -4)\).

A half turn is the same either way round, so no direction is needed.

Measure each corner from the centre of enlargement and multiply both measurements by 2.

\((0, 1)\) is 1 right and 2 up from \((-1, -1)\). Multiplying by 2 gives 2 right and 4 up, so the image of that corner is \((1, 3)\).

\((1, 1)\) is 2 right and 2 up, so it maps to \((3, 3)\), and \((0, 3)\) is 1 right and 4 up, so it maps to \((1, 7)\).

Every length on the image is 2 times the matching length on A, and the image is the same way up — an enlargement does not turn the shape.

A and B are the same size and shape, but B is the other way up — the longer of the two parallel sides has swapped ends. That is a reflection, not a translation and not a rotation.

The mirror line is halfway between a corner of A and the matching corner of B. \((0, 0)\) on A matches \((0, 2)\) on B, and halfway between \(0\) and \(2\) is \(y = 1\).

So the transformation is a reflection in the line \(y = 1\).

A reflection needs its mirror line, given as an equation. "Reflection" on its own, or "reflection in the horizontal line", does not describe it.