📚 072 Transformations - Reflections, Rotations, Translations and Enlargements | 072 变换——反射、旋转、平移和放大
In the Edexcel IGCSE Mathematics syllabus, transformations form a key part of geometry. You will need to perform translations, reflections, rotations, and enlargements on the coordinate plane, both individually and in combination. You must also learn to describe a given transformation fully, specifying all necessary details such as the vector, mirror line, centre of rotation, angle, or scale factor. This article breaks down each type, explains the precise language required in the exam, and highlights common mistakes. All examples follow the Edexcel style, ensuring you can tackle Paper 1 and Paper 2 questions confidently.
在爱德思 IGCSE 数学教学大纲中，变换是几何学中的关键部分。你需要学会在坐标平面上进行平移、反射、旋转和放大，既包括单一变换，也包括复合变换。你还必须学会完整地描述给定的变换，详细说明所有必要细节，例如向量、对称镜线、旋转中心、角度或比例因子。本文逐一分析每种变换类型，解释考试中所需的精确语言，并指出常见错误。所有示例均遵循爱德思风格，确保你能够自信应对试卷一和试卷二中的题目。
1. Introduction to Transformations | 变换简介
A transformation maps an original shape (often called the object) onto its image after applying a rule. The four transformations you must master are: translation (sliding), reflection (flipping), rotation (turning) and enlargement (resizing). In all cases except enlargement by a scale factor other than 1, the object and image are congruent – meaning they have the same size and shape. Enlargement produces a similar shape, where angles stay the same but side lengths change by the scale factor.
变换是将原始图形（常称为原像）按照某种规则映射到它的像上。你必须掌握的四类变换是：平移（滑动）、反射（翻转）、旋转（转动）和放大（改变尺寸）。除了比例因子不为 1 的放大以外，其他变换中原像和像都是全等的——意味着它们大小形状完全相同。放大则产生相似图形，角度保持不变，但边长按比例因子变化。
Edexcel questions often present a shape on a coordinate grid and ask you to draw the image after a given transformation. Alternatively, you may be shown the object and image and asked to describe the transformation fully. When describing, always mention the type, together with its defining details. For a translation, that is the column vector; for a reflection, the equation of the mirror line; for a rotation, the centre, angle and direction; for an enlargement, the centre and scale factor.
爱德思考题经常在坐标网格上给出一个图形，要求你画出经过给定变换后的像。或者，也可能给出原像和像，请你完整描述该变换。在描述时，始终要说明类型及其定义细节。对于平移，需要给出列向量；对于反射，要给出对称镜线的方程；对于旋转，要给出中心、角度和方向；对于放大，要给出中心和比例因子。
2. Translation – Moving Without Turning | 平移——无转动移动
A translation moves every point of a shape by the same amount in the same direction. It is fully described by a column vector, written as ( x above y ), for example (³₋₂). The top number tells you how many units to move right (positive) or left (negative). The bottom number tells you how many units to move up (positive) or down (negative). The shape’s orientation and size do not change – it simply slides to a new position.
平移将图形上的每一个点按相同距离和相同方向移动。它完全由一个列向量来描述，写为（上 x 下 y）的形式，例如 (³₋₂)。上方的数字表示向右（正数）或向左（负数）移动多少个单位。下方的数字表示向上（正数）或向下（负数）移动多少个单位。图形的方向与大小都不变——它只是滑动到了新位置。
on the image. Subtract the object’s coordinates from the image’s coordinates: vector = (ximage − xobject , yimage − yobject) written vertically. For example, if A(2, 1) moves to A'(5, −3), the translation vector is (₅₋₂₋₁) = (³₋₄).
当给出原像和像时，要找到平移向量，需在原像上选取一个顶点，并在像上找到对应的顶点。用像的坐标减去原像的坐标：向量 =（x像 − x原像 , y像 − y原像）竖直书写。例如，若点 A(2, 1) 移动到 A'(5, −3)，则平移向量为 (₅₋₂₋₁) = (³₋₄)。
On the grid, you must apply the vector to every vertex of the shape. The Edexcel exam often expects you to draw the translated shape accurately. Always check that corresponding sides remain parallel and equal in length.
在网格上，你必须将该向量应用于图形的每个顶点。爱德思考试通常要求你准确地画出平移后的图形。一定要检查对应边是否保持平行且长度相等。
3. Reflection – Mirror Images | 反射——镜像
A reflection creates a mirror image of the shape across a given line called the mirror line. The mirror line can be horizontal, vertical or diagonal. Every point on the object is the same perpendicular distance from the mirror line as its image, but on the opposite side. The orientation of the shape is reversed, but length and angles are preserved.
反射会沿着一条给定的线（称为对称镜线）创建图形的镜像。镜线可以是水平的、垂直的或斜的。原像上的每一点到镜线的垂直距离与它的像到镜线的距离相等，但位于相反一侧。图形的方向会反转，但长度和角度保持不变。
Common mirror lines include the axes: x = 0 (the y‑axis), y = 0 (the x‑ they are invariant). To draw a reflection, work vertex by vertex. Draw a perpendicular line from the vertex to the mirror line, extend it the same distance beyond, and mark the image point.
常见的镜线包括坐标轴：x = 0（y 轴），y = 0（x 轴），以及诸如 y = x、y = −x 的直线，或者任意垂直或水平线，例如 x = 2、y = −1。当图形被反射时，位于镜线上的点保持不动（它们是不变的）。要绘制反射图形，需要逐点作图。从顶点向镜线作垂线，将其延长相同距离到另一侧，并标记像点。
Remember to describe a reflection using the full equation of the mirror line. Writing 'reflection in the y‑axis' is acceptable, but 'reflection in the line x = 0' is more precise and expected for higher marks.
描述反射时，记得使用镜线的完整方程。写“关于 y 轴反射”是可以接受的，但“关于直线 x = 0 反射”更为精确，也是高分题目的要求。
4. Rotation – Turning Around a Point | 旋转——绕一点转动
A rotation turns a shape about a fixed centre by a given angle in a specified direction (clockwise or anticlockwise). The shape remains congruent – all lengths and angles are unchanged. To fully describe a rotation you must give three things: the centre of rotation, the angle of rotation and the direction.
旋转是将图形绕一个固定中心按给定角度向指定方向（顺时针或逆时针）转动。图形保持全等——所有长度和角度都不变。要完整描述一次旋转，必须给出三个要素：旋转中心、旋转角度和旋转方向。
The centre of rotation can be a coordinate point, often the origin (0,0), but it can also be a named point such as (3,1). Angles that appear most frequently are 90°, 180° and 270°. A rotation of 180° is the same in both clockwise and anticlockwise directions. For 90° and 270°, you must state the direction.
旋转中心可以是一个坐标点，通常是原点 (0,0)，但也可以是命名点，如 (3,1)。最常见的角度是 90°、180° 和 270°。180° 的旋转在顺时针和逆时针方向上效果相同。对于 90° 和 270°，则必须说明方向。
r 180°: (x, y) → (−x, −y). These algebraic rules apply only when the centre is the origin.
进行旋转时，使用描图纸或从中心数垂直网格步数会很有帮助。例如，将一点绕原点顺时针旋转 90°，可以交换 x 和 y 坐标并改变新 y 的符号：(x, y) → (y, −x)。逆时针 90°：(x, y) → (−y, x)。180°：(x, y) → (−x, −y)。这些代数规则仅当旋转中心为原点时适用。
5. Enlargement – Changing Size | 放大——改变尺寸
e image is identical. If the scale factor is negative, the image appears on the opposite side of the centre, reversed. Enlargement produces a shape that is mathematically similar to the original – all angles stay the same, but side lengths change in proportion to k.
放大是将图形上所有点到固定中心的距离乘以一个比例因子 k。当 k > 1 时，像变大；当 0 < k < 1 时，像变小（分数缩小）；当 k = 1 时，像与原件重叠。如果比例因子为负数，像会出现在中心的另一侧，方向反转。放大产生的形状与原图形数学上相似——所有角度保持不变，但边长按 k 的比例变化。
image side ÷ length of corresponding object side. If the image is inverted, k is negative.
要完整描述一次放大，必须给出放大中心和比例因子。要找到中心，可连接原像和像的对应顶点画射线；这些射线的交点即为中心。比例因子由 k = 像的边长 ÷ 对应原像的边长求得。如果像是倒立的，则 k 为负值。
,1) maps to (6,2). With centre (1,2) and scale factor 3, point (2,3) maps to (1 + 3×(2−1), 2 + 3×(3−2)) = (4,5).
在坐标网格上绘制放大图形时，首先标出中心。对于每个顶点，数出从中心出发的水平和垂直距离，将每个距离乘以 k，然后绘制新点。连接像点即完成图形。例如，中心为 (0,0)、比例因子为 2，点 (3,1) 映射到 (6,2)。中心为 (1,2)、比例因子为 3，点 (2,3) 映射到 (1 + 3×(2−1), 2 + 3×(3−2)) = (4,5)。
6. Describing Transformations Fully | 完整描述变换
Edexcel markschemes are very strict about the wording you use to describe a transformation. You must mention the type of transformation, and then give the specific details in the correct order. A full description follows these templates:
- Translation: 'Translation by vector (³₋₂)' – the vector must be written as a column.
- Reflection: 'Reflection in the line y = x' – give the equation of the mirror line.
- Rotation: 'Rotation, centre (2,−1), 90° clockwise' – state centre, angle, direction.
- Enlargement: 'Enlargement, centre (−3,1), scale factor ½' – state centre and scale factor; if negative, write scale factor −2.
爱德思评分方案对你描述变换所使用的措辞非常严格。必须说明变换类型，然后按正确顺序给出具体细节。完整描述可参考以下模板：
- 平移：“通过向量 (³₋₂) 平移”——向量必须写成列向量形式。
- 反射：“关于直线 y = x 反射”——给出镜线方程。
- 旋转：“旋转，中心 (2,−1)，90° 顺时针”——说明中心、角度和方向。
- 放大：“放大，中心 (−3,1)，比例因子 ½”——说明中心和比例因子；若为负数，写比例因子 −2。
A common pitfall is to write 'move' instead of 'translation', or to miss out the direction for a 90° rotation. Even if the diagram makes it obvious, you will lose the mark if the description is incomplete. Always assume the examiner expects the full formal phrase.
常见的失误是写“移动”而不是“平移”，或者遗漏 90° 旋转的方向。即使示意图一目了然，如果描述不完整就会被扣分。一定要假定考官要求的是完整的规范表述。
7. Combining Transformations | 复合变换
Edexcel frequently asks you to apply two successive transformations to a shape. You might be asked to transform shape A to B using a transformation P, and then shape B to C using transformation Q, and finally describe the single transformation that maps A directly to C. This single transformation is called the equivalent transformation. Understanding combinations helps you solve problems involving inverse transformations as well.
爱德思经常要求你对一个图形连续施加两次变换。可能会要求你使用变换 P 将图形 A 变为 B，再用变换 Q 将 B 变为 C，最后描述将 A 直接映射到 C 的单一变换。这种单一变换称为等效变换。理解复合变换还有助于解决涉及逆变换的问题。
When combining translations, you simply add the column vectors. For reflections, two reflections in parallel lines produce a translation; two reflections in intersecting lines produce a rotation. For rotations about the same centre, angles add. Enlargements with the same centre multiply scale factors. You do not need to prove these properties at IGCSE level, but recognising them can save time in multiple-choice or structured questions.
复合平移时，只需将列向量相加。对于反射，关于平行线的两次反射产生一次平移；关于相交直线的两次反射产生一次旋转。对于同一中心的旋转，角度相加。同一中心的放大，比例因子相乘。在 IGCSE 阶段你无需证明这些性质，但识别出来可在选择题或结构化问题中节省时间。
Always perform transformations in the order given. Applying transformation P then Q may not be the same as Q then P. Draw each intermediate image accurately, and label shapes to avoid confusion. If the question asks for the single transformation, compare the initial object and the final image and describe it using the appropriate method.
务必按给定顺序执行变换。先 P 后 Q 的结果可能与先 Q 后 P 不同。要准确地画出每一个中间像，并为图形命名以避免混淆。如果题目要求写出单一变换，需比较初始原像和最终像，并使用适当方法描述。
8. Invariant Points and Lines | 不变点与不变线
variant. For a rotation, only the centre of rotation is invariant (unless angle is a multiple of 360°). For an enlargement, the centre of enlargement is invariant; when k = 1, all points are invariant.
不变点是指在给定变换下不移动的点。识别不变点有助于你判断变换类型。对于平移，除非平移向量为零向量，否则不存在任何不变点。对于反射，镜线上的每个点都是不变的。对于旋转，只有旋转中心是不变的（除非角度是 360° 的倍数）。对于放大，放大中心是不变的；当 k = 1 时，所有点都不变。
ariant as a whole because points on it are reflected onto other points on the same line. At IGCSE, you usually only need to identify invariant points, but knowing about invariant lines can help with extension questions.
不变线是指映射到自身的直线，尽管该线上的个别点可能沿该线移动。例如，在关于直线 y = x 反射时，y = x 是逐点不变的，但 y = −x 作为整体也是不变的，因为其上的点被反射到同一直线的其他点上。在 IGCSE 中，你通常只需要找出不变点，但了解不变线有助于解答拓展题。
9. Transformations and Coordinate Geometry | 变换与坐标几何
Transformations are often combined with coordinate geometry. You must be able to transform points given as coordinates, find midpoints, distances, or gradients of lines in images, and solve problems involving area ratios after enlargements. The area of the image after an enlargement with scale factor k is k² times the area of the object. Similarly, if an enlargement reduces a shape, the area scales by the square of the fractional factor.
变换常与坐标几何相结合。你必须能够对以坐标形式给出的点进行变换，求出像中直线的中点、距离或斜率，并解决放大后的面积比问题。经过比例因子为 k 的放大后，像的面积是原像面积的 k² 倍。同样，如果放大缩小了图形，面积按照分数因子的平方变化。
For example, if shape A is enlarged by scale factor 3 to give shape B, the area of B is 9 times the area of A. If shape B is the result of a fractional enlargement by ½, the area of B is ¼ of the area of A. This relationship is frequently tested in questions that combine enlargement with perimeter and area calculations.
例如，若图形 A 按比例因子 3 放大得到图形 B，则 B 的面积是 A 的 9 倍。若图形 B 是由比例因子 ½ 缩小所得，则 B 的面积是 A 面积的 ¼。这种关系经常在放大与周长、面积计算相结合的题目中考查。
You may also need to find the equation of a mirror line given two corresponding points. The mirror line is the perpendicular bisector of the segment joining a point and its image. For rotation, you can locate the centre by finding the intersection of the perpendicular bisectors of segments joining corresponding points.
你可能还需要根据两个对应点求出镜线方程。镜线是连接一点与其像的线段的垂直平分线。对于旋转，可以通过寻找对应点连线的垂直平分线的交点来确定旋转中心。
10. Common Exam Mistakes and Tips | 常见考试错误与技巧
Mistake 1: Confusing the effects of different transformations. Remember that translation, reflection and rotation preserve lengths and angles, so the image is congruent. Only enlargement changes size. Mistake 2: Missing the direction in a 90° rotation description – always add 'clockwise' or 'anticlockwise'. Mistake 3: Writing the translation vector as a row or using words like 'move 3 right, 2 down'. Always use column vector form (³₋₂). Mistake 4: Using the wrong centre for rotation or enlargement. Double-check that the centre is exactly where required, especially when the centre is not the origin. Mistake 5: Forgetting the negative sign for a scale factor when the image is inverted. A negative enlargement produces an image that looks rotated 180° as well.
错误一：混淆不同变换的效果。请记住，平移、反射和旋转保持长度和角度不变，因此像是全等的。只有放大会改变尺寸。错误二：在 90° 旋转的描述中遗漏方向——一定要加上“顺时针”或“逆时针”。错误三：将平移向量写成行向量，或使用“向右移 3，向下移 2”等文字。一定要用列向量形式 (³₋₂)。错误四：旋转或放大的中心使用错误。务必核对中心是否准确位于指定位置，尤其是当中心并非原点时。错误五：当图像倒立时忘记比例因子的负号。负放大产生的图像看起来也相当于旋转了 180°。
To avoid mistakes, always use a ruler and a sharp pencil for constructions on the grid. Label your object and image clearly. Write down the transformation rule or vector before you plot points. In describing a transformation, check that your description maps the shape exactly as shown on the diagram. If the image is in a different orientation, it is likely a rotation or reflection; if it is the same way up but shifted, it is a translation; if it is the same orientation but a different size, it is an enlargement.
为避免错误，在网格上作图时始终使用直尺和削尖的铅笔。清晰标明原像和像。在描点之前先写下变换规则或向量。在描述变换时，检查你的描述是否恰好将图形映射为图中所示的样子。如果像的方向不同，很可能是旋转或反射；如果方向不变但位置移动，则是平移；如果方向相同但大小不同，则是放大。
Finally, practice with past Edexcel IGCSE papers. The more transformations you sketch, the more confident you become. Pay close attention to the markscheme wording so you know exactly how to phrase your answers. Mastering transformations will also help you in topics like vectors, trigonometry and graph transformations, making it a valuable area to strengthen.
最后，用爱德思 IGCSE 历年真题进行练习。你画的变换越多，就会越自信。要密切注意评分方案中的措辞，以便确切地知道如何表述答案。掌握变换还有助于你学习向量、三角学和图像变换等主题，因此这是一个非常值得强化的领域。
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