Algebra 8: Transformations and Symmetry | 代数8：变换与对称

📚 Algebra 8: Transformations and Symmetry | 代数8：变换与对称

Algebra 8 in the Edexcel IGCSE Mathematics syllabus introduces the fundamental concepts of transformations and symmetry. This topic is essential for understanding how shapes move, flip, rotate, and resize within a coordinate plane. Mastering these skills not only prepares you for exam questions but also builds spatial reasoning used in higher-level mathematics and real-world applications such as computer graphics and engineering design.

Edexcel IGCSE 数学课程中的”代数8″介绍了变换与对称的基本概念。这一主题对于理解图形如何在坐标平面内移动、翻转、旋转和缩放至关重要。掌握这些技能不仅为考试题目做好准备，还能培养用于更高层次数学以及计算机图形学、工程设计等实际应用的空间推理能力。

1. Translation | 平移

A translation moves every point of a shape by the same distance in the same direction. The shape retains its size, orientation, and shape; it simply slides to a new position. Translations are described using column vectors, written as \(\begin{pmatrix} x \\ y \end{pmatrix}\), where \(x\) represents the horizontal shift and \(y\) the vertical shift. Positive \(x\) moves right, negative \(x\) moves left, positive \(y\) moves up, and negative \(y\) moves down.

平移是将图形的每一个点沿相同方向移动相同距离。平移后图形的大小、方向和形状保持不变，只是滑动了位置。平移用列向量表示，写作 \(\begin{pmatrix} x \\ y \end{pmatrix}\)，其中 \(x\) 表示水平位移，\(y\) 表示垂直位移。正 \(x\) 向右移动，负 \(x\) 向左移动，正 \(y\) 向上移动，负 \(y\) 向下移动。

Column Vector: \(\binom{x}{y}\) → (x units right/left, y units up/down)

列向量：\(\binom{x}{y}\) → （向右/左移动 x 个单位，向上/下移动 y 个单位）

To translate a point \((a, b)\) by \(\binom{p}{q}\), the image is \((a+p, b+q)\).
将点 \((a, b)\) 按 \(\binom{p}{q}\) 平移，像为 \((a+p, b+q)\)。
Example: Translating \((2, 3)\) by \(\binom{-4}{5}\) gives \((-2, 8)\).
例：将 \((2, 3)\) 按 \(\binom{-4}{5}\) 平移得到 \((-2, 8)\)。
When translating a shape, apply the same vector to every vertex and then join the image points.
平移图形时，对每个顶点应用相同向量，然后连接像点。

2. Reflection | 反射

A reflection produces a mirror image of a shape across a given line, called the mirror line or axis of reflection. The image is the same distance from the mirror line as the original but on the opposite side. The shape’s size is unchanged, but its orientation is reversed; a clockwise figure becomes anticlockwise, and vice versa.

反射是图形关于一条给定直线（称为镜线或反射轴）产生的镜像。像与镜线的距离与原图形相同，但位于直线的另一侧。图形的大小不变，但方向相反；顺时针的图形变为逆时针，反之亦然。

Mirror Line (x-axis)	\((x, y) → (x, -y)\)
Mirror Line (y-axis)	\((x, y) → (-x, y)\)
Line \(y = x\)	\((x, y) → (y, x)\)
Line \(y = -x\)	\((x, y) → (-y, -x)\)
Vertical line \(x = a\)	\((x, y) → (2a – x, y)\)
Horizontal line \(y = b\)	\((x, y) → (x, 2b – y)\)

To find the line of reflection given an object and its image, the mirror line is the perpendicular bisector of the segment joining any point and its corresponding image point. Practise identifying reflections in the axes and in the diagonal lines \(y = x\) and \(y = -x\), which are common in IGCSE exams.

已知原图形和像求镜线时，镜线是连接任意点与其对应像点的线段的垂直平分线。练习识别关于坐标轴以及直线 \(y = x\) 和 \(y = -x\) 的反射，这些在 IGCSE 考试中很常见。

3. Rotation | 旋转

A rotation turns a shape about a fixed point, called the centre of rotation, by a given angle and in a given direction (clockwise or anticlockwise). The shape remains congruent to the original; every point moves along an arc of a circle centred at the centre of rotation. The angle of rotation is typically 90°, 180°, or 270°.

旋转是图形绕一个固定点（称为旋转中心）按给定角度和方向（顺时针或逆时针）转动。旋转后的图形与原图形全等；每个点沿着以旋转中心为圆心的圆弧移动。旋转角通常为 90°、180° 或 270°。

Rotation about origin 90° anticlockwise	\((x, y) → (-y, x)\)
Rotation about origin 90° clockwise	\((x, y) → (y, -x)\)
Rotation about origin 180°	\((x, y) → (-x, -y)\)
绕原点逆时针旋转 90°	\((x, y) → (-y, x)\)
绕原点顺时针旋转 90°	\((x, y) → (y, -x)\)
绕原点旋转 180°	\((x, y) → (-x, -y)\)

When the centre of rotation is not the origin, you must use tracing paper or carefully plot each vertex’s arc. In an exam, you may be asked to describe a rotation fully: state the centre, the angle, and the direction. For example, “Rotation of 90° clockwise about the point (1, 2)”.

当旋转中心不是原点时，需要使用描图纸或仔细绘制每个顶点的圆弧。在考试中，你可能被要求完整描述一个旋转：说明旋转中心、角度和方向。例如，”绕点 (1, 2) 顺时针旋转 90°”。

4. Enlargement | 放大与缩小

An enlargement changes the size of a shape by a scale factor \(k\), relative to a fixed centre of enlargement. If \(k > 1\), the image is larger; if \(0 < k < 1\), the image is smaller; if \(k\) is negative, the image is inverted and lies on the opposite side of the centre. The shape and image are mathematically similar, meaning corresponding angles are equal and corresponding side lengths are in the ratio \(k\).

缩放是图形相对于固定缩放中心按比例因子 \(k\) 改变大小。若 \(k > 1\)，像变大；若 \(0 < k < 1\)，像变小；若 \(k\) 为负数，像被反转并位于中心的对侧。原图形与像是数学相似的，即对应角相等，对应边长度之比为 \(k\)。

Enlargement: \((x, y) → (cx + k(x – cx), cy + k(y – cy))\)

缩放：\((x, y) → (cx + k(x – cx), cy + k(y – cy))\)

Simplify the formula when the centre is the origin:

当中心为原点时，公式简化为：

Scale factor \(k\) centred at origin: \((x, y) → (kx, ky)\)

以原点为中心的缩放 \(k\)：\((x, y) → (kx, ky)\)

To find the scale factor: \(k = \frac{\text{length of image side}}{\text{length of corresponding object side}}\).
求比例因子：\(k = \frac{像的边长}{对应的原图形边长}\)。
To find the centre of enlargement, draw straight lines connecting each object point to its corresponding image point; the lines intersect at the centre.
求缩放中心：连接每个原图形点与其对应像点的直线，这些直线的交点即为缩放中心。
A negative scale factor also rotates the shape by 180° about the centre.
负比例因子还会使图形绕中心旋转 180°。

5. Combined Transformations | 组合变换

Multiple transformations can be applied to a single shape in sequence. The order matters: applying a reflection followed by a translation usually gives a different result than a translation followed by a reflection. When describing a combined transformation, always specify the sequence clearly, for example, “a translation by \(\binom{3}{-2}\) followed by a reflection in the x-axis.”

可以对一个图形依次应用多个变换。顺序很重要：先反射再平移的结果通常与先平移再反射不同。描述组合变换时，必须清楚地说明顺序，例如”先按 \(\binom{3}{-2}\) 平移，再关于 x 轴反射”。

However, certain simple combinations can be replaced by a single transformation. For instance, two reflections in parallel lines are equivalent to a translation; two rotations about the same centre add their angles.

然而，某些简单组合可以用单个变换替代。例如，关于两条平行线的两次反射等价于一个平移；绕同一中心的两次旋转角度相加。

Two reflections in parallel lines: equivalent to a translation by twice the distance between the lines

关于两条平行线的两次反射：等价于按两条线之间距离的 2 倍的平移

6. Describing Transformations | 描述变换

Exam questions often provide an object and its image, asking you to describe the transformation fully. To gain full marks, your description must be precise and complete. For a translation, give the column vector. For a reflection, give the equation of the mirror line. For a rotation, give the centre, angle, and direction. For an enlargement, give the scale factor and centre.

考试题目通常给出原图形和像，要求你完整描述该变换。要获得满分，你的描述必须精确且完整。平移给出列向量；反射给出镜线方程；旋转给出中心、角度和方向；缩放给出比例因子和中心。

Useful verbs: “maps”, “transforms”, “is the image of”.
常用动词：”映射”、”变换”、”是……的像”。
Always mention “centre” for rotation and enlargement.
旋转和缩放必须提及”中心”。
Include “clockwise/anticlockwise” for rotations other than 180°.
除 180° 外的旋转需要说明”顺时针/逆时针”。
Include “scale factor” and “centre” for enlargement, even if the centre is the origin.
缩放必须说明”比例因子”和”中心”，即使中心是原点也要写。

7. Line Symmetry | 轴对称

Line symmetry (also called reflectional symmetry) exists when a shape can be folded along a line so that the two halves match exactly. The fold line is called the axis of symmetry. A shape may have one, several, or infinitely many lines of symmetry. For example, a square has 4 lines of symmetry, an equilateral triangle has 3, and a circle has infinitely many.

轴对称（也称为镜面对称）是指一个图形可以沿某条直线对折，使两部分完全重合。这条折痕称为对称轴。一个图形可以有一条、多条或无数条对称轴。例如，正方形有 4 条对称轴，等边三角形有 3 条，圆有无数条。

To find all lines of symmetry, look for lines that divide the shape into identical mirror halves. Try vertical, horizontal, and diagonal orientations. In coordinate geometry, check if reflecting a shape across a given line produces the identical set of points.

要找到所有对称轴，寻找能将图形分成完全相同的镜像两半的直线。尝试垂直、水平和对角方向。在坐标几何中，检查将图形沿给定直线反射是否得到相同的点集。

Number of lines of symmetry for regular n-gon: n

正 n 边形的对称轴数量：n

8. Rotational Symmetry | 旋转对称

A shape has rotational symmetry if it can be rotated about its centre by an angle less than 360° and still look exactly the same. The order of rotational symmetry is the number of times the shape matches its original position in one full 360° rotation. A shape with no rotational symmetry other than a full turn is said to have order 1.

如果一个图形绕其中心旋转一个小于 360° 的角度后仍与原来完全一样，则该图形具有旋转对称性。旋转对称的阶是图形在旋转 360° 的过程中与原来位置重合的次数。除旋转一整圈外没有其他旋转对称的图形称为 1 阶旋转对称。

Order of rotational symmetry = \(\frac{360°}{\text{smallest angle of rotation}}\)

旋转对称阶 = \(\frac{360°}{最小旋转角}\)

A rectangle has order 2: it matches at 180° and 360°.
矩形是 2 阶：在 180° 和 360° 时重合。
A regular hexagon has order 6: it matches at 60° intervals.
正六边形是 6 阶：每隔 60° 重合一次。
Every shape has at least order 1.
每个图形至少有 1 阶。

9. Invariant Points | 不动点

An invariant point is a point that maps to itself under a transformation. For a reflection, every point on the mirror line is invariant. For a rotation, the centre of rotation is invariant. For a translation with a non-zero vector, there are no invariant points. Recognising invariant points can help you identify a transformation in reverse-engineering problems.

不动点是在某种变换下映射到自身的点。对于反射，镜线上的每一点都是不动点；对于旋转，旋转中心是不动点；对于非零向量的平移，没有任何不动点。识别不动点有助于在逆向推导问题中判断变换类型。

Not all points inside a shape remain invariant; only those that lie on the mirror line or at the centre of rotation, as applicable. When a problem asks for invariant lines, these are lines that map to themselves as a whole, even if individual points move along them.

并非图形内所有点都是不动点；只有位于镜线上或旋转中心处的点才是不动点。当问题问及不动直线时，这些直线作为整体映射到自身，即使直线上的个别点发生了移动。

10. Mixed Practice and Common Exam Traps | 综合练习与常见陷阱

Exam questions on transformations frequently combine several concepts. You might be given a triangle and asked to apply two transformations and then describe the final image. Alternatively, you might see a diagram where the object and image are shown, and you must deduce the transformation.

考试的变换题经常综合多个概念。你可能会得到一个三角形，要求先施加两个变换再描述最终图像。或者，题目给出原图形和像的图形，要求你推断变换。

Trap 1: Confusing the order of combined transformations. Always apply the stated sequence step by step.
陷阱1：混淆组合变换的顺序。必须按所给顺序逐步应用。
Trap 2: Applying a rotation in the wrong direction. 90° clockwise is not the same as 90° anticlockwise.
陷阱2：旋转方向错误。顺时针 90° 与逆时针 90° 不同。
Trap 3: Forgetting to state the centre for rotation or enlargement. Half marks are lost easily without it.
陷阱3：遗漏旋转中心或缩放中心。缺少中心容易丢掉一半的分数。
Trap 4: Using a negative scale factor but not inverting through the centre.
陷阱4：使用负比例因子却没有绕中心反转。
Trap 5: Writing the translation vector incorrectly: \(\binom{x}{y}\) means right/left first, up/down second.
陷阱5：平移向量写反：\(\binom{x}{y}\) 是先水平后垂直。
Trap 6: Failing to check line symmetry for all orientations.
陷阱6：未检查所有方向的对称轴。

To master Algebra 8, practise with tracing paper for rotations and reflections, grid paper for enlargements, and always double-check your descriptions against the marks available. Work through as many past paper questions as you can, paying close attention to the command words: “rotate”, “reflect”, “translate”, “enlarge”, and “describe”.

要掌握”代数8″，请使用描图纸练习旋转和反射、用方格纸练习缩放，并始终根据分值核查你的描述。尽可能多做历年真题，特别注意题目命令词：”rotate（旋转）”、”reflect（反射）”、”translate（平移）”、”enlarge（缩放）”和”describe（描述）”。

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Algebra 8: Transformations and Symmetry | 代数8：变换与对称