Combined Transformations of Graphs | 图像的组合变换

📚 Combined Transformations of Graphs | 图像的组合变换

When a function is transformed by more than one operation, the result is a combination of transformations. Understanding how to combine translations, reflections and stretches in the correct order is essential for solving many A-Level exam questions.

当一个函数受到不止一种变换时，其结果就是变换的组合。理解如何以正确的顺序组合平移、反射和伸缩，对于解答许多A-Level考试题至关重要。

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

We assume you are already familiar with the basic transformations of a function \( y = f(x) \) — but we will write them using plain text notation. A vertical translation is written as \( y = f(x) + a \), which moves the graph up by \( a \) units. A horizontal translation is written as \( y = f(x + a) \), which moves the graph left by \( a \) units if \( a > 0 \).

我们假定你已经熟悉函数 \( y = f(x) \) 的基本变换——但这里我们会用纯文本符号表示它们。垂直平移写为 \( y = f(x) + a \)，它将图像向上移动 \( a \) 个单位。水平平移写为 \( y = f(x + a) \)，当 \( a > 0 \) 时图像向左移动 \( a \) 个单位。

A reflection in the x-axis is \( y = -f(x) \), and a reflection in the y-axis is \( y = f(-x) \). A vertical stretch by factor \( k \) is written as \( y = k f(x) \), while a horizontal stretch by factor \( \frac{1}{k} \) is written as \( y = f(kx) \). These are the building blocks of every combined transformation.

关于 x 轴的反射是 \( y = -f(x) \)，关于 y 轴的反射是 \( y = f(-x) \)。垂直伸缩因子 \( k \) 写为 \( y = k f(x) \)，而水平伸缩因子 \( \frac{1}{k} \) 写为 \( y = f(kx) \)。这些是每一种组合变换的基本构件。

2. Why Order Matters | 为什么顺序很重要

Transformations are not commutative in general. For example, \( y = 2f(x) + 1 \) means ‘stretch vertically by factor 2 first, then translate up by 1’. If you translate first, you would write \( 2\big(f(x) + 1\big) = 2f(x) + 2 \), which is a different graph.

一般来说，变换并不是可交换的。例如，\( y = 2f(x) + 1 \) 表示“先垂直伸缩 2 倍，再向上平移 1”。如果先平移再伸缩，你会得到 \( 2\big(f(x) + 1\big) = 2f(x) + 2 \)，那将是另一条不同的图像。

The same applies to horizontal transformations. For \( y = f(ax + b) \), you must first stretch horizontally by factor \( \frac{1}{a} \), then translate left by \( \frac{b}{a} \) units. The order inside the function is: stretch first, translate second.

水平变换也是如此。对于 \( y = f(ax + b) \)，你必须先按因子 \( \frac{1}{a} \) 进行水平伸缩，再向左平移 \( \frac{b}{a} \) 个单位。函数内部的顺序是：先伸缩，后平移。

3. Combining Vertical Transformations | 垂直变换的组合

Consider \( y = a f(x) + b \). The correct sequence is: first apply the vertical stretch or reflection \( a \), then apply the vertical translation \( b \). For instance, \( y = -3f(x) + 2 \) means ‘reflect in the x-axis, stretch vertically by factor 3, then move up by 2’.

考虑 \( y = a f(x) + b \)。正确的顺序是：先施加垂直伸缩或反射 \( a \)，再施加垂直平移 \( b \)。例如，\( y = -3f(x) + 2 \) 表示“关于 x 轴反射，垂直伸缩 3 倍，再向上移动 2”。

If the translation is applied before the stretch, the equation becomes \( y = a\big(f(x) + b\big) \), which is equivalent to \( y = a f(x) + ab \). Notice that the final vertical shift would then be \( ab \), not \( b \). This is why the order matters.

如果先平移再伸缩，方程会变成 \( y = a\big(f(x) + b\big) \)，相当于 \( y = a f(x) + ab \)。此时最终的垂直平移量是 \( ab \) 而不是 \( b \)。这就是顺序重要的原因。

4. Combining Horizontal Transformations | 水平变换的组合

For a combined horizontal transformation, write the function in the form \( y = f\big(a(x – h)\big) \) before deciding the order. This is equivalent to \( y = f(ax – ah) \). The first step is a horizontal stretch by factor \( \frac{1}{a} \), and the second step is a translation right by \( h \) units.

对于水平变换的组合，先将函数写成 \( y = f\big(a(x – h)\big) \) 的形式再决定顺序。这等价于 \( y = f(ax – ah) \)。第一步是按因子 \( \frac{1}{a} \) 进行水平伸缩，第二步是向右平移 \( h \) 个单位。

Example: to obtain \( y = f(2x – 6) \), rewrite it as \( y = f\big(2(x – 3)\big) \). Start with \( y = f(x) \), stretch horizontally by factor \( \frac{1}{2} \) to get \( y = f(2x) \), then translate right by 3 to get \( y = f\big(2(x – 3)\big) = f(2x – 6) \).

例如：要得到 \( y = f(2x – 6) \)，先改写为 \( y = f\big(2(x – 3)\big) \)。从 \( y = f(x) \) 开始，先按因子 \( \frac{1}{2} \) 水平伸缩得到 \( y = f(2x) \)，再向右平移 3 得到 \( y = f\big(2(x – 3)\big) = f(2x – 6) \)。

5. The Point Method | 点变换法

A reliable way to find the image of a point under a combined transformation is to work with coordinates. If a point \( (x, y) \) lies on \( y = f(x) \), then its image on \( y = a f\big(b(x – h)\big) + k \) is given by:

寻找一个点在组合变换下的像，一个可靠的方法就是坐标法。如果点 \( (x, y) \) 在 \( y = f(x) \) 上，那么它在 \( y = a f\big(b(x – h)\big) + k \) 上的像是：

\( x’ = \frac{x}{b} + h , \quad y’ = a y + k \)

This is because the horizontal stretch changes \( x \) to \( \frac{x}{b} \), then the horizontal shift adds \( h \). The vertical transformation multiplies \( y \) by \( a \), then adds \( k \).

这是因为水平伸缩把 \( x \) 变成 \( \frac{x}{b} \)，然后水平平移再加上 \( h \)。垂直变换先把 \( y \) 乘以 \( a \)，再加上 \( k \)。

6. Finding the Transformed Function Rule | 求变换后的函数表达式

Given a description such as ‘stretch horizontally by factor \( \frac{1}{2} \), shift right by 3, reflect in the x-axis and shift up by 4’, you can reverse the process. First decide whether the transformation is inside (horizontal) or outside (vertical), then apply them step by step to \( f(x) \).

当已知类似“水平伸缩因子 \( \frac{1}{2} \)，向右平移 3，关于 x 轴反射，向上平移 4”的描述时，你可以逆向推导。先判断变换是作用于内部（水平）还是外部（垂直），再逐步应用到 \( f(x) \) 上。

For example, starting with \( y = f(x) \), applying a horizontal stretch factor \( \frac{1}{2} \) gives \( y = f(2x) \). Shifting right by 3 gives \( y = f\big(2(x – 3)\big) \). Reflecting in the x-axis gives \( y = -f\big(2(x – 3)\big) \). Finally shifting up by 4 gives \( y = -f\big(2(x – 3)\big) + 4 \).

例如，从 \( y = f(x) \) 开始，施加水平伸缩因子 \( \frac{1}{2} \) 得到 \( y = f(2x) \)。向右平移 3 得到 \( y = f\big(2(x – 3)\big) \)。关于 x 轴反射得到 \( y = -f\big(2(x – 3)\big) \)。最后向上平移 4 得到 \( y = -f\big(2(x – 3)\big) + 4 \)。

7. Worked Example 1: Quadratic | 例题1：二次函数

Let \( f(x) = x^2 \). Find the equation of the graph obtained by stretching vertically by factor 3, shifting down by 2, and shifting left by 1.

设 \( f(x) = x^2 \)。求先垂直伸缩 3 倍、再向下平移 2、再向左平移 1 所得图像的方程。

Solution: Vertical stretch first: \( y = 3x^2 \). Shift down by 2: \( y = 3x^2 – 2 \). Shift left by 1: replace \( x \) with \( x + 1 \), giving \( y = 3(x + 1)^2 – 2 \).

解答：先垂直伸缩：\( y = 3x^2 \)。向下平移 2：\( y = 3x^2 – 2 \)。向左平移 1：把 \( x \) 替换为 \( x + 1 \)，得到 \( y = 3(x + 1)^2 – 2 \)。

Check using the point method. The vertex \( (0,0) \) of \( y = x^2 \) becomes \( (-1, -2) \). Substitute \( x = -1 \): \( y = 3(0)^2 – 2 = -2 \), correct.

用点法检验。\( y = x^2 \) 的顶点 \( (0,0) \) 变为 \( (-1, -2) \)。代 \( x = -1 \)：\( y = 3(0)^2 – 2 = -2 \)，正确。

8. Worked Example 2: Trigonometric | 例题2：三角函数

Let \( f(x) = \sin x \). Describe the transformations that produce \( y = 2 \sin(3x + \pi) + 1 \).

设 \( f(x) = \sin x \)。描述产生 \( y = 2 \sin(3x + \pi) + 1 \) 的变换。

Rewrite the inside: \( 3x + \pi = 3\left(x + \frac{\pi}{3}\right) \). So the horizontal sequence is: stretch horizontally by factor \( \frac{1}{3} \), then shift left by \( \frac{\pi}{3} \). The vertical sequence is: stretch vertically by factor 2, then shift up by 1.

改写内部：\( 3

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Combined Transformations of Graphs | 图像的组合变换