📚 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 1k 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 1k 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(f(x) + 1) = 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(f(x) + 1) = 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 1a , then translate left by ba 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 1a 进行水平伸缩，再向左平移 ba 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(f(x) + b) , 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(f(x) + b) ，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(a(x - h)) before deciding the order. This is equivalent to y = f(ax - ah) . The first step is a horizontal stretch by factor 1a , 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(a(x - h)) 的形式再决定顺序。这等价于 y = f(ax - ah) 。第一步是按因子 1a 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(2(x - 3)) . Start with y = f(x) , stretch horizontally by factor 12 to get y = f(2x) , then translate right by 3 to get y = f(2(x - 3)) = f(2x - 6) .
例如：要得到 y = f(2x - 6) ，先改写为 y = f(2(x - 3)) 。从 y = f(x) 开始，先按因子 12 水平伸缩得到 y = f(2x) ，再向右平移 3 得到 y = f(2(x - 3)) = 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(b(x - h)) + 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(b(x - h)) + k 上的像是：
x' = xb + h , y' = a y + k
This is because the horizontal stretch changes x to xb , 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 xb ，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 12 , 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 12 ，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 12 gives y = f(2x) . Shifting right by 3 gives y = f(2(x - 3)) . Reflecting in the x-axis gives y = -f(2(x - 3)) . Finally shifting up by 4 gives y = -f(2(x - 3)) + 4 .
例如，从 y = f(x) 开始，施加水平伸缩因子 12 得到 y = f(2x) 。向右平移 3 得到 y = f(2(x - 3)) 。关于 x 轴反射得到 y = -f(2(x - 3)) 。最后向上平移 4 得到 y = -f(2(x - 3)) + 4 。
7. Worked Example 1: Quadratic | 例题1：二次函数
Let f(x) = x2 . 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) = x2 。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 = 3x2 . Shift down by 2: y = 3x2 - 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 = 3x2 。向下平移 2： y = 3x2 - 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 = x2 becomes (-1, -2) . Substitute x = -1 : y = 3(0)2 - 2 = -2 , correct.
用点法检验。 y = x2 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 + π) + 1 .
设 f(x) = sin x 。描述产生 y = 2 sin(3x + π) + 1 的变换。
Rewrite the inside: 3x + π = 3(x + π3) . So the horizontal sequence is: stretch horizontally by factor 13 , then shift left by π3 . The vertical sequence is: stretch vertically by factor 2, then shift up by 1.
改写内部：\( 3
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