📚 Combined Transformations of Functions | 函数变换的组合
In A-Level Mathematics, understanding how individual transformations affect a function is essential. When several transformations are applied at once, the result is a combined transformation. This lesson explains how to combine translations, reflections and stretches in the correct order, how to read the new equation, and how to interpret the resulting graph.
在A-Level数学中，理解单个变换如何影响一个函数至关重要。当多个变换同时应用时，结果就是组合变换。本课将讲解如何按正确顺序合并平移、反射和伸缩，如何写出新的方程，以及如何解读最终图像。
1. Individual Transformations | 单个变换回顾
Before combining transformations, you must be confident with the four basic types. Each transformation acts on either the x-coordinate or the y-coordinate of every point on the original graph y=f(x).
在组合变换之前，你必须熟练掌握四种基本变换类型。每种变换作用于原图像 y=f(x) 上每一点的 x 坐标或 y 坐标。
Translation: y=f(x)+a moves the graph vertically by a units; y=f(x-a) moves it horizontally by a units.
平移：y=f(x)+a 将图像垂直移动 a 个单位；y=f(x-a) 将其水平移动 a 个单位。
Reflection: y=-f(x) reflects in the x-axis; y=f(-x) reflects in the y-axis.
反射：y=-f(x) 关于 x 轴对称；y=f(-x) 关于 y 轴对称。
Vertical stretch: y=af(x) stretches by factor a parallel to the y-axis.
垂直伸缩：y=af(x) 沿 y 轴方向拉伸 a 倍。
Horizontal stretch: y=f(bx) compresses or stretches horizontally by factor 1b.
水平伸缩：y=f(bx) 沿 x 轴方向压缩或拉伸至 1b 倍。
2. Translations | 平移变换
A translation shifts every point of a graph by the same displacement. In combined transformations, translations must be identified and applied after stretches and reflections unless the equation is written in a specific form.
平移将图像的每个点移动相同的位移。在组合变换中，除非方程写成特定形式，否则必须在伸缩和反射之后处理平移。
y = f(x-a) + b
Here, the graph is first shifted right by a units, then up by b units. The order of these two translations does not matter because they act on different axes.
这里，图像先向右移动 a 个单位，再向上移动 b 个单位。这两个平移的顺序无关紧要，因为它们作用在不同的坐标轴上。
3. Reflections | 反射变换
Reflections flip the graph across an axis. A negative sign in front of the function y=-f(x) reflects across the x-axis. A negative sign inside the function y=f(-x) reflects across the y-axis.
反射将图像沿坐标轴翻转。函数前有负号 y=-f(x) 表示关于 x 轴对称；函数内有负号 y=f(-x) 表示关于 y 轴对称。
y=-f(x): all y-values change sign, so every point (x,y) becomes (x,-y).
y=-f(x)：所有 y 值变号，每一点 (x,y) 变为 (x,-y)。
y=f(-x): all x-values change sign, so every point (x,y) becomes (-x,y).
y=f(-x)：所有 x 值变号，每一点 (x,y) 变为 (-x,y)。
A combined transformation may include two reflections. For example, y=-f(-x) produces a rotation of 180° about the origin, which is equivalent to reflecting in both axes.
组合变换可能包含两个反射。例如，y=-f(-x) 产生绕原点旋转 180°，这等价于同时关于两个坐标轴反射。
4. Stretches | 伸缩变换
Vertical stretches multiply the output: y = a f(x). Horizontal stretches multiply the input: y = f(bx). The factor b is not always intuitive because the graph appears scaled by 1/b.
垂直伸缩乘以输出：y = a f(x)。水平伸缩乘以输入：y = f(bx)。因子 b 并不总是直观，因为图像的缩放倍数是 1/b。
Vertical stretch factor a: y = a f(x)
Horizontal stretch factor 1/b: y = f(bx)
When multiple stretches affect the same variable, they must be applied in order. For instance, in y = 3f(2x), the graph is first compressed horizontally by factor 12, then stretched vertically by factor 3.
当多个伸缩作用于同一变量时，必须按顺序应用。例如，在 y = 3f(2x) 中，图像先水平压缩至 12，再垂直拉伸 3 倍。
5. Order of Transformations | 变换顺序
The correct order is essential when a translation is combined with a stretch or reflection. The general equation y = a f(b(x-c)) + d shows the intended sequence clearly.
当平移与伸缩或反射结合时，正确的顺序至关重要。一般式 y = a f(b(x-c)) + d 清楚地展示了应有的顺序。
Start with the basic graph y=f(x).
从基本图像 y=f(x) 开始。
Apply the horizontal stretch/reflection: replace x by bx.
应用水平伸缩/反射：将 x 替换为 bx。
Apply the horizontal translation: replace bx by b(x-c). This shifts right by c.
应用水平平移：将 bx 替换为 b(x-c)。这表示向右平移 c 个单位。
Apply the vertical stretch/reflection: multiply the whole function by a.
应用垂直伸缩/反射：将整个函数乘以 a。
Apply the vertical translation: add d.
应用垂直平移：加上 d。
This order assumes you are transforming the graph from the basic form outward. If you are given a final equation, you must reverse-engineer these steps.
这个顺序假设你从基本形式出发逐步变换图像。如果给出最终方程，则必须逆向推导这些步骤。
6. Applying Combined Transformations to Graphs | 对图像进行组合变换
To transform a graph containing known points, apply each transformation to the coordinates of those points in the correct order. Work horizontally first, then vertically.
要变换含有已知点的图像，应按正确顺序将这些变换分别作用于点的坐标。先进行水平方向的处理，再进行垂直方向的处理。
Suppose a point (x, y) lies on y=f(x). For y = a f(b(x-c)) + d:
设点 (x, y) 在 y=f(x) 上。对于 y = a f(b(x-c)) + d：
Horizontal change: x → xb + c.
水平变化：x → xb + c。
Vertical change: y → a y + d.
垂直变化：y → a y + d。
(x, y) → ( xb + c, a y + d )
For example, the point (1,2) on y=f(x) becomes (1,6) on y=3f(x) because b=1, c=0, d=0 and y=3(2)=6.
例如，y=f(x) 上的点 (1,2) 在 y=3f(x) 上变为 (1,6)，因为 b=1, c=0, d=0，且 y=3(2)=6。
7. Finding the Equation from the Graph | 从图像求方程
Given a transformed graph, you can deduce the equation by identifying how the original asymptotes, intercepts and turning points have moved.
给定一个变换后的图像，你可以通过识别原始渐近线、截距和驻点如何移动来推断方程。
For a rational function y=1x, the vertical asymptote x=0 and horizontal asymptote y=0 are convenient markers. If the vertical asymptote becomes x=2, the graph must contain f(x-2). If the horizontal asymptote becomes y=5, the equation must contain +5.
对于有理函数 y=1x，垂直渐近线 x=0 和水平渐近线 y=0 是方便的标记。如果垂直渐近线变为 x=2，则图像必含有 f(x-2)。如果水平渐近线变为 y=5，则方程必含有 +5。
Combined transformations can be read from the positions of two distinct points or features. Always compare the transformed graph with the original, not with an unrelated horizontal line.
组合变换可以通过两个不同点或特征的位置来解读。始终将变换后的图像与原始图像比较，而不是与无关的水平线比较。
8. Worked Example 1 | 例题一
Let f(x)=x2. Find the equation of the graph obtained when f(x) is stretched vertically by factor 3, then translated left by 2 units.
设 f(x)=x2。求 f(x) 先垂直拉伸 3 倍，再向左平移 2 个单位所得图像的方程。
Step 1: vertical stretch gives g(x)=3f(x)=3x2.
第一步：垂直拉伸得 g(x)=3f(x)=3x2。
Step 2: translation left by 2 means replace x by x+2: h(x)=g(x+2)=3(x+2)2.
第二步：向左平移 2 个单位意味着将 x 替换为 x+2：h(x)=g(x+2)=3(x+2)2。
Final answer: y=3(x+2)2.
最终答案：y=3(x+2)2。
Remember that "left by 2" corresponds to x+2 inside the function, not x-2.
记住“向左平移 2 个单位”对应函数内部是 x+2，而不是 x-2。
9. Worked Example 2 | 例题二
The point (3, -1) lies on y=f(x). Find its image on y = -2f(4(x-1)).
点 (3, -1) 在 y=f(x) 上。求它在 y = -2f(4(x-1)) 上的像。
Horizontal transformations: replace x by 4(x-1). This means a horizontal compression by factor 14, then a translation right by 1. So the new x-coordinate is 34 + 1 = 1.75.
水平变换：将 x 替换为 4(x-1)。这表示水平压缩至 14，然后向右平移 1。因此新的 x 坐标是 34 + 1 = 1.75。
Vertical transformations: multiply y by -2. So the new y-coordinate is -2 × (-1) = 2.
垂直变换：将 y 乘以 -2。因此新的 y 坐标是 -2 × (-1) = 2。
The image point is (1.75, 2).
像点为 (1.75, 2)。
(3, -1) → ( 34+1, -2(-1) ) = (1.75, 2)
10. Common Mistakes and Tips | 常见错误与提示
Students often confuse the direction of horizontal translations or the factor of horizontal stretches. Use the inner bracket notation b(x-c) to avoid errors.
学生经常混淆水平平移的方向或水平伸缩的倍数。使用内括号形式 b(x-c) 可以避免错误。
Mistake: y=f(x+3) shifts right by 3. Correction: it shifts left by 3 because x+3 = x-(-3).
错误：y=f(x+3) 向右平移 3。纠正：它向左平移 3，因为 x+3 = x-(-3)。
Mistake: y=f(2x) stretches horizontally by 2. Correction: it compresses horizontally by factor 12.
错误：y=f(2x) 水平拉伸 2 倍。纠正：它水平压缩至 12。
Mistake: applying translations before stretches. Correction: always factor the coefficient of x before reading the translation.
错误：先做平移再做伸缩。纠正：先提取 x 的系数，再读取平移量。
When solving exam questions, rewrite the equation in the form a f(b(x-c)) + d first. This instantly reveals all four parameters a, b, c, d.
作答考试题时，先将方程写成 a f(b(x-c)) + d 的形式。这一步能立即揭示四个参数 a, b, c, d。
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