Transformations

Translation (slide). Every point moves the same distance in the same direction.

Describe with a vector: $\begin{pmatrix} x \\ y \end{pmatrix}$ means '$x$ right, $y$ up'.

Example: a translation by $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ moves every point 3 right and 2 down.

Rotation (turn). A shape rotates about a CENTRE by a fixed angle.

Three pieces of info needed:

• Angle (e.g. $90°$).
• Direction (clockwise or counter-clockwise / anti-clockwise).
• Centre of rotation.

Common rotations about the origin:

• $90°$ counter-clockwise: $(x, y) \to (-y, x)$.
• $180°$: $(x, y) \to (-x, -y)$.
• $270°$ counter-clockwise (or $90°$ clockwise): $(x, y) \to (y, -x)$.

Reflection (flip). A shape flips over a LINE OF REFLECTION.

Common reflection lines and their effect on $(x, y)$:

• x-axis (line $y = 0$): $(x, y) \to (x, -y)$.
• y-axis (line $x = 0$): $(x, y) \to (-x, y)$.
• $y = x$: $(x, y) \to (y, x)$.
• $y = -x$: $(x, y) \to (-y, -x)$.

Enlargement (resize). A shape is scaled by a SCALE FACTOR from a CENTRE of enlargement.

• Scale factor 2: every point moves to twice its distance from the centre.
• Scale factor $-1$: rotation by $180°$ (reflection through the centre).
• Scale factor $0 < k < 1$: reduction.

These four cover almost every geometric transformation you'll see at MYP Extended Level.

For full marks (criterion C), a transformation description must include:

Transformation	What to state
Translation	Vector $\begin{pmatrix} x \\ y \end{pmatrix}$
Rotation	Angle + direction + centre
Reflection	Equation of the line of reflection
Enlargement	Scale factor + centre of enlargement

Worked example. Triangle $A$ at $(2, 1), (4, 1), (4, 4)$ is mapped to triangle $A'$ at $(5, 1), (7, 1), (7, 4)$.

• Compare a point: $(2, 1) \to (5, 1)$. Change: $+3$ in $x$, $0$ in $y$.
• Check all points show same change: yes.
• Translation by $\begin{pmatrix} 3 \\ 0 \end{pmatrix}$.

Worked example. Triangle with vertices $(1, 0), (3, 0), (3, 2)$ is mapped to $(0, 1), (0, 3), (-2, 3)$.

• Try rotation about origin. $90°$ counter-clockwise: $(x, y) \to (-y, x)$.
• Check: $(1, 0) \to (0, 1)$ ✓. $(3, 0) \to (0, 3)$ ✓. $(3, 2) \to (-2, 3)$ ✓.
• It's a rotation: $90°$ counter-clockwise about the origin.

For mixed real-context problems (e.g. a logo design), recognising the transformation is the criterion-D skill.