POP

G3.17 Transformations

Image · 914 × 270

01 — Explanation

What must I state to fully describe a translation?

The type (Translation) and the column vector $\binom{x}{y}$ , where $x$ is the horizontal shift (positive = right) and $y$ is the vertical shift (positive = up).

A' = A + \binom{x}{y}

Common Misconception: Treating the column vector as an absolute destination coordinate rather than a displacement. $\binom{-3}{2}$ means "move 3 left and 2 up from wherever you currently are" — not "move to the point $(-3,2)$ ."

02 — Explanation

What must I state to fully describe a reflection?

The type (Reflection) and the equation of the mirror line.

$x=a$ is a vertical line (fixes the $x$ -coordinate).
$y=b$ is a horizontal line (fixes the $y$ -coordinate).
Reflecting in $y=x$ swaps coordinates: $(x,y)\to(y,x)$ . Reflecting in $y=-x$ swaps and negates: $(x,y)\to(-y,-x)$ .

Every point stays the same perpendicular distance from the mirror line, just on the opposite side; points lying exactly on the line don't move at all.

Common Misconception: Swapping $x=a$ and $y=b$ — remember, the variable named in the equation is the one held constant, so $x=3$ is vertical (every point has $x=3$ ), not horizontal.

03 — Explanation

What must I state to fully describe a rotation?

Three things: Rotation, the angle ( $90°$ , $180°$ , or $270°$ at Foundation), the direction (clockwise/anticlockwise — not needed for $180°$ , since both directions land in the same place), and the centre of rotation.

For a $180°$ rotation, the centre is simply the midpoint of any object–image vertex pair:

\text{Centre} = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

For $90°$ / $270°$ rotations, construct the perpendicular bisector of at least two object–image vertex pairs; where they cross is the centre.

Exam Tip: Tracing paper is allowed and is the most reliable way to both perform and check a rotation — pin it at the centre with your pencil tip and physically turn it.

04 — Explanation

What must I state to fully describe an enlargement?

Enlargement, the scale factor ( $\text{SF} = \dfrac{\text{image length}}{\text{object length}}$ ), and the centre of enlargement. A scale factor between $0$ and $1$ shrinks the shape; a scale factor above $1$ enlarges it.

To draw an enlargement: find each vertex's displacement vector from the centre, multiply both components by the scale factor, then add that scaled vector back onto the centre to plot the image point.

Common Misconception: Enlarging about the origin by mistake when a different centre is given. Always measure displacement from the stated centre, not from $(0,0)$ .

05 — Explanation

How do I check whether a transformation changes a shape's size?

Translations, reflections, and rotations always produce a congruent image — exactly the same size and shape, just moved. Only enlargement changes the size, producing a similar shape where angles stay the same but lengths scale by the factor SF. If you've correctly identified an enlargement with $\text{SF}\ne1$ , the image lengths must all be proportionally bigger or smaller than the object's — if they're identical, you've actually got one of the other three transformations instead.

Common Misconception: Describing a complex single transformation as a sequence of two simpler ones (e.g. "reflect, then translate"). Exam questions specifically ask for the one transformation that maps the object directly onto the image — a combined description scores no marks, even if each individual step is correct.

06 — Worked example

Worked Example

Triangle $A$ has vertices $(1,1)$ , $(3,1)$ , $(1,4)$ . Describe fully the single transformation that maps $A$ onto its image $A'$ with vertices $(2,2)$ , $(6,2)$ , $(2,8)$ , given that the centre is the origin.

Compare corresponding side lengths: the horizontal side has length $2$ in $A$ and $4$ in $A'$ , so $\text{SF}=4\div2=2$ .
Check the centre: $(1,1)\times2=(2,2)$ ✓ — consistent with every vertex.
Answer: Enlargement, scale factor 2, centre $(0,0)$ .