TransformationsEdexcel IGCSE Maths: Revision notes

A translation slides every point of a shape the same distance in the same direction, without turning or resizing it. You describe it fully with a column vector $\begin{pmatrix} x \\\\ y \end{pmatrix}$, where the top number is the horizontal shift (positive = right) and the bottom number is the vertical shift (positive = up).

Example: triangle with vertices $A(1,1)$, $B(3,1)$, $C(1,4)$ translated by $\begin{pmatrix} 2 \\\\ -3 \end{pmatrix}$ gives $A'(3,-2)$, $B'(5,-2)$, $C'(3,1)$ — add the vector to every vertex.

A translation is congruent to the original: same size, same shape, same orientation, just in a different place.

A reflection flips a shape over a mirror line, so every point and its image are the same perpendicular distance from the line, on opposite sides. To describe a reflection fully, state the equation of the mirror line, e.g. $x = 2$, $y = -1$, $y = x$, or $y = -x$.

Key rules for finding images:

• Reflection in $y = x$: swap the coordinates, $(a,b) \rightarrow (b,a)$.
• Reflection in $y = -x$: swap and negate both, $(a,b) \rightarrow (-b,-a)$.
• Reflection in the $x$-axis ($y=0$): $(a,b) \rightarrow (a,-b)$.
• Reflection in the $y$-axis ($x=0$): $(a,b) \rightarrow (-a,b)$.

Example: point $P(3,5)$ reflected in $x = 1$. The distance from $P$ to the line is $3-1=2$, so the image is $2$ units on the other side: $P'(-1,5)$.

A reflection is congruent to the original but has opposite orientation (it is a mirror image — think of a shape you could not rotate to match, only flip).

A rotation turns a shape about a fixed point called the centre of rotation, through a given angle (e.g. $90°$, $180°$, $270°$) in a given direction (clockwise or anticlockwise). All three — centre, angle, direction — must be stated for full marks. Note $180°$ rotations do not need a direction (clockwise and anticlockwise give the same result).

Quick rules for rotations about the origin $(0,0)$:

• $90°$ clockwise: $(a,b) \rightarrow (b,-a)$
• $90°$ anticlockwise: $(a,b) \rightarrow (-b,a)$
• $180°$ (either direction): $(a,b) \rightarrow (-a,-b)$

To rotate about a centre that is not the origin, e.g. $(2,1)$: subtract the centre from each vertex, apply the rule above, then add the centre back on.

Example: rotate $(5,3)$ by $90°$ clockwise about $(2,1)$. Shift: $(5-2, 3-1)=(3,2)$. Apply $90°$ CW rule: $(2,-3)$. Shift back: $(2+2,-3+1)=(4,-2)$.

A rotation is congruent to the original and keeps the same orientation (unlike a reflection).

An enlargement resizes a shape from a centre of enlargement by a scale factor ($k$). To describe it fully, state both the centre and the scale factor.

To enlarge a point: find the vector from the centre to the point, multiply it by $k$, then add it back to the centre.

Scale factor rules:

• $k > 1$: shape gets bigger, image on the same side as the object.
• $0 < k < 1$ (fractional): shape gets smaller, image on the same side as the object.
• $k < 0$ (negative): shape is resized and flipped through the centre — the image appears on the opposite side of the centre from the object, upside down/inverted.

Example: enlarge point $(4,2)$ by scale factor $3$ from centre $(1,1)$. Vector from centre: $(3,1)$. Multiply by $3$: $(9,3)$. Add to centre: $(1+9,1+3)=(10,4)$.

Example with negative scale factor: enlarge point $(3,1)$ by scale factor $-2$ from centre $(1,1)$. Vector from centre: $(2,0)$. Multiply by $-2$: $(-4,0)$. Add to centre: $(1-4,1+0)=(-3,1)$ — the image lands on the opposite side of the centre.

Unlike the other three transformations, an enlargement (with $k \neq \pm 1$) is not congruent — it is similar, with sides scaled by $k$ and area scaled by $k^2$.

Given an object and image, work through this checklist:

1. Same size and same orientation (not flipped)? → translation (find the vector joining corresponding vertices) or rotation (check if it's turned — if not turned, it's a translation).
2. Same size but mirror-image (flipped)? → reflection (find the line exactly midway between corresponding points).
3. Same size, turned, same orientation sense? → rotation (find the centre by seeing which point doesn't move, or use perpendicular bisectors of lines joining a vertex to its image).
4. Different size? → enlargement (compare corresponding side lengths to find $k$; if the image is upside down relative to the object, $k$ is negative).

Combined transformations may be asked for as two steps described separately — describe each one fully and in order (order can matter for marks even though the final image is a single shape).