Transformations

A transformation moves or resizes a shape according to a precise rule, and OCR GCSE Mathematics (J560) tests four of them: translations, reflections, rotations and enlargements. Two skills are assessed in every transformation question — carrying out a transformation accurately on a grid, and describing a given transformation fully using the correct language. The describing half is where marks are quietly won and lost, because a full description must name the type and give every defining detail. Performing a transformation is AO1; describing one precisely and completely is AO2; and combining transformations or solving an unfamiliar problem is AO3. Higher tier adds negative and fractional enlargement scale factors.

A useful way to organise the topic is to notice that three of the four transformations — translation, reflection and rotation — keep the shape exactly the same size, producing a congruent image. These are called rigid transformations because the shape neither grows nor shrinks; only its position or orientation changes. The fourth, enlargement, is the odd one out: it changes the size, producing a similar image whose angles are unchanged but whose lengths are scaled. Keeping this congruent-versus-similar distinction at the front of your mind helps you check answers, because if you carry out a translation, reflection or rotation and your image is a different size, you know something has gone wrong. Each transformation also has its own checklist of facts that a "describe fully" answer must contain, and learning those checklists is the surest route to full marks.

Key Vocabulary

Term	Meaning
Object / Image	The original shape / the shape after transformation
Translation	A slide, with no turning or resizing, described by a column vector
Reflection	A flip across a mirror line
Rotation	A turn about a fixed centre, through an angle, in a direction
Enlargement	A resizing about a centre by a scale factor
Column vector	$\begin{pmatrix} x \\ y \end{pmatrix}$(xy​): move $x$x right and $y$y up (negatives reverse)
Congruent / Similar	Same shape and size / same shape, different size

Translations

A translation slides every point of a shape the same distance in the same direction, with absolutely no turning and no resizing. Because every point moves by an identical amount, the shape simply glides to a new position while keeping its orientation, so the object and image are always congruent — identical in both shape and size. The movement is described by a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$(xy​), in which the top number is the movement in the $x$x-direction (positive for right, negative for left) and the bottom number is the movement in the $y$y-direction (positive for up, negative for down). To translate a point you add the vector to its coordinates; to find a vector from an object to an image you subtract the object's coordinates from the image's, always image minus object. A complete description of a translation needs only the single column vector — but it must be written the right way up and with correct signs, because that vector is the whole answer. The figure shows a triangle translated by $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$(42​), four units right and two up.

Worked Example 1

A point $P(3, 5)$P(3,5) is translated by $\begin{pmatrix} -2 \\ 4 \end{pmatrix}$(−24​). Work out the coordinates of its image.

Add the vector to the coordinates: $x$x: $3 + (-2) = 1$3+(−2)=1; $y$y: $5 + 4 = 9$5+4=9. The image is $(1, 9)$(1,9).

Answer: $(1, 9)$(1,9).

Worked Example 2

Triangle $A$A has a vertex at $(1, 2)$(1,2); its image $B$B has the matching vertex at $(6, -1)$(6,−1). Describe the translation as a column vector.

Right movement $= 6 - 1 = 5$=6−1=5; up movement $= -1 - 2 = -3$=−1−2=−3. So the vector is $\begin{pmatrix} 5 \\ -3 \end{pmatrix}$(5−3​).

Answer: $\begin{pmatrix} 5 \\ -3 \end{pmatrix}$(5−3​).

Common error: writing the vector upside down (movement up on top), or subtracting the coordinates the wrong way round (image minus object is correct).

Reflections

A reflection flips a shape across a mirror line to produce a mirror image. Every point of the object moves to a point exactly the same perpendicular distance on the other side of the line, so a point sitting on the mirror line does not move at all. The image is congruent to the object but "turned over", like a shape seen in a mirror, which means its sense (clockwise versus anticlockwise lettering) is reversed. The single most important part of describing a reflection is stating the equation of the mirror line, because without it the description is worthless. The lines you meet most often are the $x$x-axis (whose equation is $y = 0$y=0), the $y$y-axis ($x = 0$x=0), the horizontal and vertical lines such as $y = 3$y=3 or $x = -1$x=−1, and the two diagonals $y = x$y=x and $y = -x$y=−x. A reliable method when reflecting by hand is to measure the perpendicular distance from each vertex to the mirror line and step the same distance across to the other side. The figure shows a shape reflected in the $y$y-axis.

Worked Example 3

Reflect the point $(4, 3)$(4,3) in the $x$x-axis. Work out the image.

Reflecting in the $x$x-axis ($y = 0$y=0) keeps $x$x the same and changes the sign of $y$y: $(4, 3) \to (4, -3)$(4,3)→(4,−3).

Answer: $(4, -3)$(4,−3).

Worked Example 4

Reflect the point $(2, 5)$(2,5) in the line $y = x$y=x. Work out the image.

Reflecting in $y = x$y=x swaps the coordinates: $(2, 5) \to (5, 2)$(2,5)→(5,2).

Answer: $(5, 2)$(5,2).

Common error: reflecting in $y = x$y=x by changing signs instead of swapping the coordinates.

Rotations

A rotation turns a shape about a fixed point called the centre of rotation, through a given angle, in a given direction — clockwise or anticlockwise. The centre stays fixed while every other point sweeps round it on a circular arc, so the further a point is from the centre, the further it travels. A full description must state all three pieces of information: the centre written as a pair of coordinates, the size of the angle, and the direction. The one exception is a half-turn of $180^{\circ}$180∘, where the direction can be omitted because turning a shape half a revolution clockwise lands it in exactly the same place as turning it half a revolution anticlockwise. As with the other rigid transformations, a rotation keeps the object and image congruent. By far the most dependable practical tool is tracing paper: trace the object, hold your pencil firmly on the centre, and turn the paper through the required angle and direction to read off the image. The figure shows a triangle rotated $90^{\circ}$90∘ anticlockwise about the origin.

Worked Example 5

Rotate the point $(3, 1)$(3,1) by $90^{\circ}$90∘ anticlockwise about the origin. Work out the image.

For a $90^{\circ}$90∘ anticlockwise rotation about the origin, $(x, y) \to (-y, x)$(x,y)→(−y,x). So $(3, 1) \to (-1, 3)$(3,1)→(−1,3).

Answer: $(-1, 3)$(−1,3).

Worked Example 6

Rotate the point $(4, 2)$(4,2) by $180^{\circ}$180∘ about the origin. Work out the image.

A $180^{\circ}$180∘ rotation about the origin sends $(x, y) \to (-x, -y)$(x,y)→(−x,−y). So $(4, 2) \to (-4, -2)$(4,2)→(−4,−2).

Answer: $(-4, -2)$(−4,−2).

Common error: giving a direction for a $180^{\circ}$180∘ rotation (unnecessary) or muddling clockwise and anticlockwise for $90^{\circ}$90∘ turns. Tracing the turn on tracing paper avoids both.

Enlargements