Position and Transformations

A coordinate is a pair $(x, y)$: along the x-axis first, then up the y-axis. The two axes cross at the origin $(0, 0)$ and split the grid into four quadrants.

This year, coordinates do more than mark a point — they let you describe a transformation exactly. Instead of saying "the triangle moved over there", you give precise corner positions.

The golden rule still holds: read along the corridor, then up the stairs. A negative $x$ goes left of the origin; a negative $y$ goes down.

When you transform a shape, write down the before and after coordinates. Comparing the two often reveals exactly what the transformation did.

A reflection flips a shape across a mirror line. This year, "describe the transformation" means you must name the mirror line — for example $x = 3$, $y = -2$, or $y = x$. A point on the mirror line does not move.

A rotation turns a shape about a fixed point. A full description gives three things: the centre of rotation, the angle of turn, and the direction (clockwise or anticlockwise).

Both reflections and rotations keep the image congruent — exactly the same size and shape, just facing a new way or sitting in a new spot.

A reliable check: pick one corner of the original, apply the rule, and confirm the matching corner of the image lands where you expect. Tracing paper helps with rotations — trace the shape, pin the centre, and turn.

A translation slides a shape without turning or flipping it. Last year you described a slide in words; this year you can write it as a column vector.

A column vector is two numbers stacked in a bracket: $\begin{pmatrix} x \\ y \end{pmatrix}$ The top number is the move left or right (right is positive); the bottom number is the move up or down (up is positive).

So the slide above — 6 right and 3 down — is written: $\begin{pmatrix} 6 \\ -3 \end{pmatrix}$

To translate a point, add the vector to its coordinates. The point $(2, 5)$ translated by $\begin{pmatrix} 6 \\ -3 \end{pmatrix}$ moves to $(2 + 6,\ 5 - 3) = (8, 2)$.

A translation is the gentlest transformation — the image is congruent and the same way up. Nothing about the shape itself changes, only its position.

Reflection, rotation and translation all keep a shape the same size. An enlargement is different — it changes the size, using two pieces of information: a scale factor and a centre of enlargement.

The scale factor tells you how many times bigger every length becomes:

• Scale factor $2$ — every length doubles.
• Scale factor $3$ — every length is three times as long.
• Scale factor $\frac{1}{2}$ — every length halves, so the shape gets smaller.

The centre of enlargement is the fixed point the shape grows away from. To enlarge, measure the distance from the centre to each corner, multiply that distance by the scale factor, and plot the new corner.

Two important facts: the angles stay the same, so the image is the same shape (mathematicians say the shapes are similar). But the area changes more than you might expect — with scale factor $2$, the area becomes $4$ times bigger, because area depends on length twice over.

Often a shape is transformed twice — first one transformation, then another. The result of the two together is called a combined transformation.

Here is the interesting part: two transformations done in a row can frequently be described as just one transformation.

A few useful results to explore:

• Two reflections in parallel mirror lines combine to make a single translation.
• Two reflections in mirror lines that cross combine to make a single rotation.
• A rotation followed by another rotation about the same centre is just one bigger rotation — add the angles.

The order can matter. Reflecting then translating may give a different result from translating then reflecting, so always do the steps in the stated order. When you have finished, compare the first shape with the last and ask: what single transformation would get me from one to the other?

Getting confident with position and transformations sets you up well:

• Similarity and scale — enlargement leads directly into similar shapes, scale drawings and maps.
• Vectors — column vectors here are your first step towards the wider study of vectors.
• Graphs and functions — translating and reflecting graphs uses exactly these ideas.
• In everyday life, transformations appear in computer graphics, animation, design, photography and architecture.

If a later geometry or graphing topic feels tricky, it is often a transformation or coordinate skill underneath. Keep this guide handy — describing and combining transformations gets quicker every time you practise.