Transformations · 变换
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| transformations/trænsfɔːˈmeɪʃnz/ | 变换 | biàn huàn |
| reflection/rɪˈflekʃn/ | 反射 | fǎn shè |
| rotation/rəʊˈteɪʃn/ | 旋转 | xuán zhuǎn |
| enlargement/enˈlɑːdʒmənt/ | 放大 | fàng dà |
| mirror line/ˈmɪrə laɪn/ | 对称轴 | duì chèn zhóu |
| scale factor/skeɪl ˈfæktə/ | 比例因子 | bǐ lì yīn zi |
| translation/trænˈsleɪʃn/ | 平移 | píng yí |
| column vector/ˈkɒlʌm ˈvektə/ | 列向量 | liè xiàng liàng |
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.
Transformations · 变换
(x, y) → (x', y')
Translate, reflect, rotate or enlarge a shape — and see what stays the same.
Reflect the point (3, 2) in the y-axis. The image is (a, 2). What is a?
Reflecting in the y-axis flips the sign of x: a = −3.
Rotation
- Turns a shape about a centre. Give centre, angle, and direction.
- Rotate $(3,1)$ by $90^{\circ}$ anticlockwise about $O$: $(x,y) \to (-y, x) \to (-1, 3)$.
Quick rules about the origin: $90^{\circ}$ clockwise: $(x,y) \to (y, -x)$. $90^{\circ}$ anticlockwise: $(x,y) \to (-y, x)$. $180^{\circ}$: $(x,y) \to (-x, -y)$.
Geometric tiles are built by reflecting, rotating and translating shapes
Rotate (3, 1) by 90° anticlockwise about the origin. The rule is (x, y) → (−y, x). The image is (a, 3). What is a?
(3, 1) → (−1, 3). So a = −1.
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.
Describe fully. "It got bigger" is not enough. You must give: centre of enlargement AND scale factor. Both are needed for full marks.
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
Which transformation changes the size of a shape?
Only enlargement changes size; the other three keep it congruent.
Enlarge the point (4, 6) from the origin with scale factor 3. The image is (12, b). What is b?
Multiply both coordinates by 3: (4×3, 6×3) = (12, 18).
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
Translate (5, 3) by the vector (−2, 4). The image is (3, b). What is b?
Move 4 up: 3 + 4 = 7.
A translation changes the size and shape of an object.
Translation only changes position — the shape stays congruent (same size and shape).
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.
Enlarge P = (3,2) by factor 2 about C = (1,1). Find the image x-coordinate.
x′ = 1 + 2(3−1) = 5.
You've got it
- reflection (mirror line), rotation (centre, angle, direction)
- enlargement (centre, scale factor $k$), translation (column vector)
- describe each fully — that's what earns the marks