Combining Transformations | 复合变换

📚 Combining Transformations | 复合变换

In A-Level Mathematics, a ‘combined transformation’ means applying two or more graph transformations to a base function y = f(x). Edexcel questions often ask you to describe fully the transformations that map one curve onto another, or to find the equation after a given sequence. Success depends on knowing the standard toolkit, rewriting the function correctly, and keeping the order of transformations clear.

在 A-Level 数学中，“复合变换”是指对原函数 y = f(x) 连续施加两个或更多图像变换。Edexcel 试题常要求你完整描述将一条曲线映射到另一条曲线的变换，或根据给定顺序求出变换后的方程。成功的关键在于掌握标准变换工具、正确改写函数，并清楚区分变换顺序。

1. What Are Combined Transformations? | 什么是复合变换？

A combined transformation is any sequence of translations, stretches or reflections that changes y = f(x) into a new graph. For example, y = 2f(x + 3) – 5 means taking f(x), shifting it left 3, stretching it vertically by factor 2, and then moving it down 5. The word ‘combined’ simply emphasises that more than one transformation is needed.

复合变换是指将平移、伸缩或反射按一定顺序连续施加，使 y = f(x) 变为新的图像。例如，y = 2f(x + 3) – 5 表示将 f(x) 先向左平移 3 个单位，再沿 y 轴方向拉伸 2 倍，最后向下平移 5 个单位。“复合”一词强调需要用到不止一个变换。

2. Transformation Toolkit | 基本变换工具箱

Before combining transformations, you must know the standard single transformations and their effect on y = f(x). The table below summarises the key cases used in Edexcel exam questions.

在组合变换之前，你必须掌握标准单一变换及其对 y = f(x) 的影响。下表总结了 Edexcel 考试中的关键情形。

Remember: horizontal transformations act inside the bracket and usually have the ‘opposite’ effect on x. Vertical transformations act outside the bracket and follow the same sign as the constant.

记住：水平变换作用在括号内部，对 x 的影响通常是“反向”的；垂直变换作用在括号外部，其方向与常数的正负一致。

3. The General Case y = a f(b(x + c)) + d | 一般形式 y = a f(b(x + c)) + d

Most combined transformation questions can be handled by writing the target function in the form shown below. The constants a, b, c and d control different directions: a controls vertical stretch, b controls horizontal stretch, c controls horizontal translation, and d controls vertical translation.

大多数复合变换题都可以通过把目标函数写成下面的一般形式来处理。常数 a、b、c、d 控制不同方向：a 控制垂直伸缩，b 控制水平伸缩，c 控制水平平移，d 控制垂直平移。

y = a f(b(x + c)) + d

This form is important because b(x + c) shows the horizontal translation clearly. If the inside of the bracket is written as bx + bc, you should first factorise it to avoid mistakes in the translation amount.

这个形式很重要，因为 b(x + c) 能清楚地展示水平平移。如果括号内写成 bx + bc，你应当先提取公因式，以免平移量出错。

4. Horizontal Order: Translate or Stretch First? | 水平变换顺序：先平移还是先伸缩？

Horizontal transformations are the most frequent source of order errors. Suppose you need to map y = f(x) to y = f(bx + c). First rewrite the inside as b(x + c/b). There are two valid orders, but they have different translation amounts.

水平变换最容易出现顺序错误。假设要把 y = f(x) 变为 y = f(bx + c)，首先将括号内改写为 b(x + c/b)。有两种正确顺序，但它们的平移量不同。

y = f(bx + c) = f(b(x + c/b))

Option A: translate first by vector (-c, 0), then stretch horizontally by scale factor 1/b.

方案 A：先按向量 (-c, 0) 平移，再以 1/b 为尺度因子做水平伸缩。

Option B: stretch horizontally by scale factor 1/b first, then translate by vector (-c/b, 0).

方案 B：先以 1/b 为尺度因子做水平伸缩，再按向量 (-c/b, 0) 平移。

Both orders produce the same final graph, but the translation vector changes. That is why you must state the order clearly and use the correct translation amount for that order.

两种顺序得到相同的最终图像，但平移向量不同。因此你必须清楚说明所采用的顺序，并使用与该顺序对应的平移量。

5. Vertical Order: Stretch or Translate First? | 垂直变换顺序：先伸缩还是先平移？

For vertical transformations, the standard and most convenient order is to stretch first and then translate. Starting with y = f(x), the mapping to y = a f(x) + d is usually described as a vertical stretch scale factor a followed by a translation by vector (0, d).

对于垂直变换，标准且

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