Transformations

The mirror, the spin, the zoom, the slide

• Computer graphics use transformations 变换 to animate characters — reflections 反射 for mirror images, rotations 旋转 for spinning, enlargements 放大 for zooming.
• Every transformation needs a complete description to earn full marks.

Reflection

• Flips a shape over a mirror line 对称轴. Give the line's equation.
• Reflect $(3,2)$ in the $y$-axis → $(-3, 2)$ (only $x$'s sign changes).
• Reflect in the $x$-axis → $(3, -2)$ (only $y$'s sign changes).

Reflection in the $y$-axis: flip the sign of $x$. The shape stays the same size — it's congruent.

Rotation

• Turns a shape about a centre. Give centre, angle, and direction.
• Rotate $(3,1)$ by $90^{\circ}$ anticlockwise about $O$: $(x,y) \to (-y, x) \to (-1, 3)$.

Geometric tiles are built by reflecting, rotating and translating shapes

Enlargement

• Changes size by a scale factor 比例因子 $k$ from a centre. Give centre and $k$.
• Enlarge $(1,2)$ from $O$ by $k=2$ → $(2, 4)$.
• Core includes positive fractional factors: $k=\dfrac12$ makes it smaller. Extended also includes negative factors, such as $k=-1$, which puts the image on the opposite side of the centre.

An enlargement multiplies every distance from the centre by the scale factor (here 2)

A rotation turns the shape about a fixed centre, here $90^\circ$ anticlockwise about the origin

Translation 平移

• Slides every point by the same amount. Describe with a column vector 列向量 $\begin{pmatrix} x \\ y \end{pmatrix}$.
• Translate $(5,3)$ by $\begin{pmatrix} -2 \\ 4 \end{pmatrix}$ → $(5-2, 3+4) = (3, 7)$.
• Translation preserves everything: size, shape, and orientation.

A translation slides every point by the same column vector, with no turning

The centre need not be the origin

• Enlarge $P=(3,2)$ by factor 2 about $C=(1,1)$. Displacement $P-C=(2,1)$ doubles to $(4,2)$, so $P'=C+2(P-C)=(5,3)$.
• To find a centre from a shape and image, rule lines through matching vertices and extend them; they meet at the centre. A factor $1/2$ puts each image point halfway from the centre to its original. Core transformations are single operations, not combinations.

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