📚 IGCSE Mathematics Transformations | IGCSE数学变换
Transformations are one of the most visual topics in IGCSE Mathematics. They describe how a shape moves, flips, turns or changes size on a coordinate grid. You need to know four core types: translation, reflection, rotation and enlargement.
变换是 IGCSE 数学中最具视觉性的主题之一。它描述一个图形在坐标网格上如何移动、翻转、旋转或改变大小。你需要掌握四种核心变换：平移、反射、旋转和放大（伸缩）。
1. Translation | 平移
A translation slides every point of a shape by the same vector. The shape does not turn, flip or change size. If a point (x, y) is translated by vector (a, b), the image point is (x + a, y + b).
平移是指图形上每一个点都沿同一个向量滑动。图形不会旋转、翻转或改变大小。若点 (x, y) 按向量 (a, b) 平移，则像点为 (x + a, y + b)。
(x, y) → (x + a, y + b)
The column vector is written as [a above b], where a is the horizontal movement and b is the vertical movement. A positive a moves right, a negative a moves left; a positive b moves up, a negative b moves down.
列向量写作 [上方 a，下方 b]，其中 a 表示水平移动，b 表示垂直移动。a 为正向右移，a 为负向左移；b 为正向上移，b 为负向下移。
- Translation by [4, -2] means move 4 units right and 2 units down.
- 平移 [4, -2] 表示向右移动 4 个单位，再向下移动 2 个单位。
- The original shape and its image are congruent.
- 原图形与像图形全等。
2. Reflection | 反射（对称）
A reflection maps a point to its mirror image across a given line, called the mirror line. The perpendicular distance from a point to the mirror line equals the perpendicular distance from its image to the mirror line.
反射将一个点关于给定直线（称为镜面线）映射到其镜像。一个点到镜面线的垂直距离等于其像点到镜面线的垂直距离。
Common reflection equations: 常见反射的镜面线有：
- x-axis: (x, y) → (x, -y)
- x 轴：(x, y) → (x, -y)
- y-axis: (x, y) → (-x, y)
- y 轴：(x, y) → (-x, y)
- y = x: (x, y) → (y, x)
- 直线 y = x：(x, y) → (y, x)
- y = -x: (x, y) → (-y, -x)
- 直线 y = -x：(x, y) → (-y, -x)
k, keep the y-coordinate unchanged. The x-coordinate changes according to: new x = 2k - x.
在直线 x = k 上作反射时，y 坐标不变，x 坐标按公式变化：新 x = 2k − x。
3. Rotation | 旋转
Rotation turns every point of a shape around a fixed centre by a given angle and direction. Unless the angle is a multiple of 360°, the shape changes its orientation but not its size.
旋转是指图形上每一点绕固定中心按给定角度和方向转动。除非旋转角度是 360° 的整数倍，否则图形方向改变但大小不变。
Rotation facts you must remember: 必须记住的旋转要点：
- Rotation by 90° anticlockwise about the origin: (x, y) → (-y, x)
- 绕原点逆时针旋转 90°：(x, y) → (-y, x)
- Rotation by 90° clockwise about the origin: (x, y) → (y, -x)
- 绕原点顺时针旋转 90°：(x, y) → (y, -x)
- Rotation by 180° about the origin: (x, y) → (-x, -y)
- 绕原点旋转 180°：(x, y) → (-x, -y)
For rotations about a point other than the origin, first translate the centre to the origin, apply the rotation rule, then translate back. Alternatively, use tracing paper in an exam.
绕非原点旋转时，可以先把旋转中心平移到原点，应用旋转规则，再平移回去。考试中也可使用描图纸辅助。
4. Enlargement | 放大（伸缩）
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 the centre to the point, and its distance from the centre is multiplied by k.
放大（伸缩）以固定的放大中心为基准，按比例因子 k 改变图形大小。每个点都沿着中心与点的连线移动，其到中心的距离乘以 k。
new distance = k × original distance
0) and a point is (x, y), the image is (kx, ky). For a centre C = (a, b), the formula is:
如果放大中心在原点 (0, 0)，点 (x, y) 的像是 (kx, ky)。若中心为 C = (a, b)，公式为：
(x, y) → (a + k(x − a), b + k(y − b))
A negative scale factor produces an image on the opposite side of the centre. A fractional scale factor between 0 and 1 produces a smaller shape.
负比例因子得到的像位于中心的另一侧。比例因子为 0 到 1 之间的分数时，图像会变小。
5. Scale Factor and Area Scale Factor | 比例因子与面积比例因子
When a shape is enlarged by scale factor k, every length is multiplied by k. The area is multiplied by k². If you know the original area and the image area, the area scale factor equals the image area divided by the original area.
当图形按比例因子 k 放大时，每个长度都乘以 k，面积乘以 k²。若知道原面积和像面积，面积比例因子等于像面积除以原面积。
Area of image = k² × Area of original
For example, an enlargement with scale factor 2 creates an image with 4 times the area. A scale factor of ½ creates an image with ¼ of the original area.
例如，比例因子为 2 的放大得到的像面积是原来的 4 倍；比例因子为 ½ 时，像面积是原来的 ¼。
| Scale Factor k | Length change | Area change |
| 2 | × 2 | × 4 |
| 3 | × 3 | × 9 |
| ½ | × ½ | × ¼ |
6. Describing Transformations | 描述变换
In IGCSE exams you are often asked to describe a transformation completely. A complete description must include all key details, or you will lose marks.
在 IGCSE 考试中，经常要求你完整描述一个变换。完整描述必须包含所有关键信息，否则会扣分。
- Translation: give the column vector.
- 平移：给出列向量。
- Reflection: give the equation of the mirror line.
- 反射：给出镜面线的方程。
- Rotation: give the centre, angle and direction (clockwise or anticlockwise).
- 旋转：给出旋转中心、角度和方向（顺时针或逆时针）。
- Enlargement: give the scale factor and the centre of enlargement.
- 放大：给出比例因子和放大中心。
To find the centre of enlargement, draw lines joining corresponding vertices of the original shape and the image. The point where these lines meet is the centre.
要找放大中心，可连接原图形与像图形的对应顶点，延长线交会处就是放大中心。
7. Combined Transformations | 复合变换
When two transformations are applied in order, the result is a combined transformation. Order matters: applying a translation then a rotation usually gives a different image from applying the rotation first.
当按顺序施加两个变换时，结果就是复合变换。顺序很重要：先平移再旋转通常与先旋转再平移得到不同的图像。
For example, reflect shape A in the y-axis to get shape B, then translate B by [3, -1] to get shape C. You may be asked to describe the single transformation that maps A directly to C, or to draw the final image.
例如，将图形 A 关于 y 轴反射得到图形 B，然后将 B 按 [3, -1] 平移得到图形 C。题目可能要求描述从 A 直接到 C 的单一变换，或要求画出最终图像。
When reversing transformations, use inverse operations: the inverse of a translation by [a, b] is a translation by [-a, -b]; the inverse of a rotation by 90° clockwise is a rotation by 90° anticlockwise; the inverse of an enlargement by scale factor k is an enlargement by scale factor 1/k with the same centre.
当需要逆转变换时，使用逆操作：平移 [a, b] 的逆变换是平移 [-a, -b]；顺时针旋转 90° 的逆变换是逆时针旋转 90°；比例因子 k 的放大的逆变换是以同一中心按 1/k 放大。
8. Invariant Points and Lines | 不变点与不变线
An invariant point is a point that stays in the same position after a transformation. An invariant line is a line where every point on the line maps to another point on the same line (the line as a whole is unchanged).
不变点是指经过变换后位置不变的点。不变线是指线上每个点仍映射到同一条线上的线（整条线保持不变）。
- For a reflection in the x-axis, every point on the x-axis is invariant.
- 关于 x 轴反射时，x 轴上的每一点都是不变点。
- For a rotation by 180° about the origin, only the origin is an invariant point.
- 绕原点旋转 180° 时，只有原点是不变点。
- For a translation by a non-zero vector, there are no invariant points.
- 非零向量平移没有不变点。
In enlargements, the centre of enlargement is always invariant, because its distance from itself multiplied by k is still 0.
在放大中，放大中心总是不变点，因为它到自身的距离乘以 k 仍为 0。
9. Using Coordinate Grids and Ray Method | 坐标网格与射线法
To perform an enlargement accurately, use the ray method. Mark the centre of enlargement C. Draw a ray from C through each vertex of the shape. Then measure or count the distance from C to the vertex and multiply it by the scale factor to place the image vertex on the same ray.
要准确完成放大，可使用射线法。标出放大中心 C，从 C 经过每个顶点作射线。然后测量或数出 C 到顶点的距离，乘以比例因子，在该射线上放置像顶点。
mple, if k = -2 and a vertex is 3 units right of C, the image vertex is 6 units left of C.
对于负比例因子，像顶点位于 C 的另一侧，沿同一直线。例如，若 k = −2，某顶点在 C 右侧 3 个单位，则像顶点在 C 左侧 6 个单位。
10. Congruent and Similar Shapes | 全等与相似图形
Translations, reflections and rotations all produce congruent shapes: same size and same angles, but possibly different orientation or position. Enlargements produce similar shapes: same angles but sizes changed by the scale factor.
平移、反射和旋转产生的都是全等图形：大小和角度相同，但方向或位置可能改变。放大产生相似图形：角度相同，但大小按比例因子改变。
If two shapes are similar and you know one side length in each shape, you can find the scale factor by dividing the image side length by the original side length. This scale factor also applies to all corresponding sides.
若两个图形相似，并且知道每个图形中一条边的长度，则像边长除以原边长即可得到比例因子。该比例因子适用于所有对应边。
k = image length ÷ original length
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many students lose marks because of small errors in transformations. Here are the most common pitfalls and how to avoid them.
许多学生因为小错误在变换题目中失分。以下是最常见的陷阱及避免方法。
- Mistake: Forgetting to state the centre of rotation or enlargement.
- 错误：忘记说明旋转中心或放大中心。
- Fix: Always include centre, angle, direction and scale factor in every description.
- 对策：在每次描述中始终包含中心、角度、方向和比例因子。
- Mistake: Reflecting in the wrong line, for example confusing y = x with y = -x.
- 错误：在错误的镜面线中反射，例如混淆 y = x 与 y = -x。
- Fix: Test one point, such as (1, 2), to check the image position.
- 对策：用一个点测试，例如 (1, 2)，检查像的位置。
- Mistake: Using the wrong direction for rotation.
- 错误：旋转方向用错。
- Fix: Clockwise is the same direction as a clock's hands; anticlockwise is opposite.
- 对策：顺时针与钟表指针方向相同；逆时针则相反。
- Mistake: Mixing up the order in combined transformations.
- 错误：在复合变换中搞乱顺序。
- Fix: Read the question carefully and apply transformations in the stated order, drawing intermediate shapes if needed.
- 对策：仔细读题，按给定顺序执行变换，必要时画出中间图形。
12. Worked Example | 综合例题
in the line y = x to form triangle C. Find the coordinates of the vertices of triangle C.
三角形 A 的顶点为 (1, 1)、(3, 1) 和 (1, 4)。它以原点 (0, 0) 为中心、比例因子 2 被放大得到三角形 B。然后将 B 关于直线 y = x 反射得到三角形 C。求三角形 C 各顶点坐标。
Step 1: Enlarge A by scale factor 2 about the origin. Multiply each coordinate by 2.
第一步：以原点为中心按比例因子 2 放大 A。将每个坐标乘以 2。
(1, 1) → (2, 2), (3, 1) → (6, 2), (1, 4) → (2, 8).
Step 2: Reflect B in y = x. In this reflection, (x, y) becomes (y, x).
第二步：将 B 关于 y = x 反射。此反射中 (x, y) 变为 (y, x)。
(2, 2) → (2, 2), (6, 2) → (2, 6), (2, 8) → (8, 2).
Therefore the vertices of triangle C are (2, 2), (2, 6) and (8, 2).
因此三角形 C 的顶点为 (2, 2)、(2, 6) 和 (8, 2)。
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