📚 IGCSE Mathematics: Mastering Transformations | IGCSE数学：掌握变换
Transformations is a core topic in IGCSE Mathematics Extended. It tests your ability to visualise, describe and perform changes to shapes on a coordinate grid. A strong grasp of translations, reflections, rotations and enlargements is essential for both Paper 2 and Paper 4.
变换是 IGCSE 数学（Extended）的核心考点，考查你在坐标网格中直观想象、准确描述并完成图形变化的能力。熟练掌握平移、反射、旋转和放大，对 Paper 2 和 Paper 4 都至关重要。
1. Translation | 平移
A translation slides every point of a shape by the same vector. The vector has two components: the top number is the displacement in the x-direction, and the bottom number is the displacement in the y-direction. A positive value means move right or up; a negative value means move left or down.
平移是将图形的每个点沿同一个向量滑动。向量有两个分量：上面的数表示沿 x 方向的位移，下面的数表示沿 y 方向的位移。正数代表向右或向上，负数代表向左或向下。
(x, y) → (x + a, y + b)
For example, if point A(2, 3) is translated by the column vector (5, −1)ᵀ, the image is A′(7, 2). The x-coordinate increases by 5, while the y-coordinate decreases by 1.
例如，点 A(2, 3) 沿列向量 (5, −1)ᵀ 平移后，对应点 A′ 为 (7, 2)。x 坐标增加 5，y 坐标减少 1。
A translation does not change the size, shape or orientation of the object. The original shape and its image are congruent.
平移不改变图形的大小、形状或方向。原图形与像图形全等。
2. Reflection | 反射
A reflection produces a mirror image. Every point is reflected across a line called the mirror line or line of reflection. The mirror line is always the perpendicular bisector of the segment joining a point to its image.
反射产生镜像。每个点都关于一条称为“镜面线”或“反射线”的直线对称。镜面线总是连接原点和像点的线段的中垂线。
The most important reflection rules are shown below:
最重要的反射规则如下：
| Mirror line | 镜面线 | Coordinate rule | 坐标规则 |
| x-axis | x 轴 | (x, y) → (x, −y) |
| y-axis | y 轴 | (x, y) → (−x, y) |
| y = x | (x, y) → (y, x) |
| y = −x | (x, y) → (−y, −x) |
| vertical line x = k | 竖直线 x = k | (x, y) → (2k − x, y) |
| horizontal line y = k | 水平线 y = k | (x, y) → (x, 2k − y) |
the line x = 2 maps the point (1, 4) to (3, 4). Notice that 2 is the midpoint of 1 and 3.
例如，关于直线 x = 2 反射，点 (1, 4) 将变为 (3, 4)。注意到 2 是 1 和 3 的中点。
Reflection preserves lengths and angle sizes, but it reverses orientation: a clockwise order becomes anticlockwise in the image.
反射保持线段长度和角的大小不变，但会改变方向：原本顺时针的顶点顺序，在像中会变为逆时针。
3. Rotation | 旋转
A rotation turns a shape about a fixed point called the centre of rotation. You must give three pieces of information: the centre, the angle, and the direction.
旋转是图形围绕一个固定点转动，这个固定点称为旋转中心。描述一个旋转必须给出三个要素：旋转中心、旋转角度和旋转方向。
For rotations about the origin, the following rules are essential.
对于绕原点旋转，以下规则至关重要。
| Rotation about the origin | 绕原点旋转 | Coordinate rule | 坐标规则 |
| 90° anticlockwise | 逆时针 90° | (x, y) → (−y, x) |
| 180° | 180° | (x, y) → (−x, −y) |
| 90° clockwise | 顺时针 90° | (x, y) → (y, −x) |
A 90° clockwise rotation is equivalent to a 270° anticlockwise rotation. For a rotation about a point other than the origin, draw a grid or use the method of translate, rotate, and translate back.
顺时针旋转 90° 等价于逆时针旋转 270°。如果旋转中心不是原点，可以借助方格纸，或者使用“先平移、再旋转、最后平移回去”的方法。
For example, rotating the point P(2, 5) by 90° anticlockwise about the origin gives P′(−5, 2). Rotation preserves size, shape and orientation.
例如，将点 P(2, 5) 绕原点逆时针旋转 90°，得到 P′(−5, 2)。旋转保持图形的大小、形状和方向不变。
4. Enlargement | 放大
An enlargement changes the size of a shape by a scale factor k, from a fixed centre of enlargement. Every point moves along the line joining it to the centre, and its distance from the centre is multiplied by k.
放大是通过放大中心，按照比例因子 k 改变图形的大小。每个点都沿着它与中心的连线移动，到中心的距离乘以 k。
(x′, y′) = (a + k(x − a), b + k(y − b))
Here (a, b) is the centre of enlargement and k is the scale factor.
其中 (a, b) 是放大中心，k 是比例因子。
(1, 1) and scale factor 2. Then P′ has coordinates (1 + 2(2 − 1), 1 + 2(3 − 1)) = (3, 5).
例如，将点 P(2, 3) 以中心 (1, 1) 和比例因子 2 放大。P′ 的坐标为 (1 + 2(2 − 1), 1 + 2(3 − 1)) = (3, 5)。
If k is positive, the image is on the same side of the centre as the object. If k is negative, the image is on the opposite side and is also rotated through 180° about the centre. If k is a proper fraction, the image is smaller than the original.
当 k 为正数时，像在原图形相对于中心的同侧。当 k 为负数时，像在中心的另一侧，并且相当于绕中心旋转了 180°。当 k 是真分数时，像比原图形小。
s by factor k and areas by factor k². If the original area is A, the image area is A × k².
按比例因子 k 放大后，长度变为原来的 k 倍，面积变为原来的 k² 倍。如果原面积为 A，则像的面积为 A × k²。
5. Combined Transformations | 复合变换
In many IGCSE questions you must apply two transformations in order. The order matters: reflecting first and then translating can give a completely different image from translating first and then reflecting.
在许多 IGCSE 题目中，你需要按顺序完成两个变换。顺序非常重要：先反射再平移，与先平移再反射，得到的像可能完全不同。
For example, start with A(1, 2). First reflect in the y-axis, then translate by (2, 0)ᵀ.
例如，从点 A(1, 2) 开始。先关于 y 轴反射，再沿向量 (2, 0)ᵀ 平移。
(1, 2) → (−1, 2) → (1, 2)
But if you translate first and then reflect, the result is different.
但如果先平移再反射，结果会不同。
(1, 2) → (3, 2) → (−3, 2)
ion matrices, if transformation T₁ has matrix M₁ and transformation T₂ has matrix M₂, then “T₂ after T₁” ined matrix M₂ × M₁. Matrix multiplication is not commutative, so do not switch the order.
在使用变换矩阵时，如果变换 T₁ 的矩阵为 M₁，变换 T₂ 的矩阵为 M₂，那么“先 T₁ 后 T₂”的复合矩阵为 M₂ × M₁。矩阵乘法不满足交换律，所以不能随意交换顺序。
6. Invariant Points and Lines | 不变点与不变线
An invariant point is a point that maps to itself under a transformation. A line of invariant points is a line where every single point on that line stays fixed.Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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