Combined Transformations | 组合变换

📚 Combined Transformations | 组合变换

Combined transformations in A-Level Mathematics describe how you can apply two or more translations, reflections, stretches, or compressions to the graph of a function. Understanding them is essential for solving problems in pure mathematics, sketching graphs, and analysing functions in applied contexts. In Edexcel A-Level exams, straightforward questions often ask you to give the equation of a transformed graph or state the transformations in a specific order.

在A-Level数学中，组合变换描述如何对一个函数图像依次应用两个或更多个平移、反射、伸缩或压缩。理解组合变换对于解决纯数学问题、绘制函数图像以及在应用背景下分析函数都至关重要。在Edexcel A-Level考试中，基础题目通常要求你写出变换后的图像方程，或按指定顺序说明变换过程。

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1. The Four Basic Transformations | 四种基本变换

Before combining transformations, you must be completely confident with the four basic types. The table below summarises the effect of each transformation on the graph of \( y = f(x) \).

在组合变换之前，你必须对四种基本变换完全熟悉。下表总结了每种变换对 \( y = f(x) \) 图像的影响。

Transformation	Effect	中文解释
\( y = f(x) + a \)	Vertical translation by \( a \) units	向上（\( a>0 \)）或向下（\( a<0 \)）平移 \( a \) 个单位
\( y = f(x + a) \)	Horizontal translation by \( -a \) units	向左（\( a>0 \)）或向右（\( a<0 \)）平移 \( |a| \) 个单位
\( y = -f(x) \)	Reflection in the x-axis	关于 x 轴反射
\( y = f(-x) \)	Reflection in the y-axis	关于 y 轴反射
\( y = a f(x) \)	Vertical stretch by factor \( a \)	纵向伸展 \( a \) 倍；若 \( a<0 \) 则还包含 x 轴反射
\( y = f(ax) \)	Horizontal stretch by factor \( \frac{1}{a} \)	横向压缩或伸展 \( \frac{1}{a} \) 倍；若 \( a<0 \) 则还包含 y 轴反射

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2. Reading Combined Transformations from an Equation | 从方程识别组合变换

When you see an equation such as \( y = a f(bx + c) + d \), the key is to rearrange the horizontal part first. Factorise the coefficient of \( x \) inside the brackets so that the expression has the form \( f(b(x – h)) \). The transformed function can then be written as:

当你看到 \( y = a f(bx + c) + d \) 这样的方程时，关键是要先整理水平方向的部分。把括号内部 \( x \) 的系数提出来，使表达式变为 \( f(b(x – h)) \) 的形式。这样变换后的函数就可以写成：

\( y = a f(b(x – h)) + d \)

Here, \( a \) controls vertical stretch, \( d \) controls vertical translation, \( b \) controls horizontal stretch, and \( h \) controls horizontal translation. This form makes it easy to identify all transformations at once.

其中，\( a \) 控制纵向伸缩，\( d \) 控制纵向平移，\( b \) 控制横向伸缩，\( h \) 控制横向平移。这种形式可以让你一眼看出所有变换。

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3. The Golden Rule: Multiplication Before Addition | 黄金法则：先乘后加

For transformations on the same axis, the order matters. The golden rule is: when a stretch/reflection and a translation are both present, apply the stretch/reflection first, then the translation. This applies separately to the x-direction and the y-direction.

对于同一轴上的变换，顺序非常重要。黄金法则是：当伸缩/反射和平移同时出现时，先进行伸缩/反射，再进行平移。这一点对 x 方向和 y 方向分别适用。

For example, \( y = 2f(x) + 3 \) means: first stretch vertically by factor 2, then translate up 3 units. In contrast, \( y = 2(f(x) + 3) \) means: first translate up 3 units, then stretch vertically by factor 2. These produce different graphs.

例如，\( y = 2f(x) + 3 \) 表示：先纵向伸展 2 倍，再向上平移 3 个单位。而 \( y = 2(f(x) + 3) \) 表示：先向上平移 3 个单位，再纵向伸展 2 倍。这两种情况得到的图像是不同的。

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4. Horizontal Transformations: Factorise First | 水平变换：先提取因子

A very common mistake is to apply a horizontal translation before the horizontal stretch when the equation is not factorised. For example, consider \( y = f(2x – 6) \). Some students incorrectly describe this as ‘a horizontal stretch by \( \frac{1}{2} \), then a translation right by 6’. This is wrong.

一个非常常见的错误是：在没有提取因子时，就先进行水平平移再进行水平拉伸。例如，考虑 \( y = f(2x – 6) \)。有些学生错误地描述为“横向压缩 \( \frac{1}{2} \)，然后向右平移 6 个单位”。这是错误的。

The correct approach is to factorise: \( y = f(2(x – 3)) \). Therefore the correct sequence is: first a horizontal stretch by factor \( \frac{1}{2} \), then a translation right by 3 units.

正确的做法是先提取因子：\( y = f(2(x – 3)) \)。因此正确的变换顺序是：先横向压缩为原来的 \( \frac{1}{2} \)，再向右平移 3 个单位。

Always rewrite \( f(bx + c) \) as \( f(b(x – h)) \) before describing transformations.

在描述变换前，一定要把 \( f(bx + c) \) 改写成 \( f(b(x – h)) \)。

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5. Worked Example 1: Describing a Sequence | 例题1：描述变换顺序

Let \( f(x) = x^2 \). Describe a sequence of transformations that maps \( y = f(x) \) to \( y = 3f(x – 2) + 4 \).

设 \( f(x) = x^2 \)。请描述将 \( y = f(x) \) 变换到 \( y = 3f(x – 2) + 4 \) 的变换顺序。

Step 1: The horizontal shift is inside the function: \( f(x – 2) \), which is a translation right by 2 units.

第1步：水平平移在函数内部：\( f(x – 2) \) 表示向右平移 2 个单位。

Step 2: The coefficient 3 gives a vertical stretch by factor 3.

第2步：系数 3 表示纵向伸展 3 倍。

Step 3: The final +4 gives a translation up by 4 units.

第3步：最后的 +4 表示向上平移 4 个单位。

Because there is only one horizontal transformation, and the vertical stretch happens before the vertical translation, the full sequence is:

因为水平方向只有一个变换，且纵向伸展发生在纵向平移之前，所以完整的变换顺序是：

Translate right by 2 → vertical stretch by 3 → translate up by 4.

向右平移2个单位 → 纵向伸展3倍 → 向上平移4个单位。

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6. Worked Example 2: Using the Factorised Form | 例题2：利用提取因子后的形式

Describe a sequence of transformations that maps \( y = \sin x \) to \( y = 3 \sin(2x – \pi) + 1 \

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