Combined Transformations of Functions | 函数变换的组合

📚 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 \(\frac{1}{b}\).

水平伸缩：\(y=f(bx)\) 沿 \(x\) 轴方向压缩或拉伸至 \(\frac{1}{b}\) 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 \(\frac{1}{2}\), 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 \(\frac{1}{2}\)，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: \(x \to \frac{x}{b} + c\).

水平变化：\(x \to \frac{x}{b} + c\)。
Vertical change: \(y \to a y + d\).

垂直变化：\(y \to a y + d\)。

\((x, y) \to \left( \frac{x}{b} + c, \; a y + d \right)\)

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 \(y=\frac{1}{x}\), 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 \(y=\frac{1}{x}\)，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 \(f(x)=x^2\). Find the equation of the graph obtained when \(f(x)\) is stretched vertically by factor 3, then translated left by 2 units.

设 \(f(x)=x^2\)。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 \(g(x)=3f(x)=3x^2\).

第一步：垂直拉伸得 \(g(x)=3f(x)=3x^2\)。

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 \(\frac{1}{4}\), then a translation right by 1. So the new \(x\)-coordinate is \(\frac{3}{4} + 1 = 1.75\).

水平变换：将 \(x\) 替换为 \(4(x-1)\)。这表示水平压缩至 \(\frac{1}{4}\)，然后向右平移 1。因此新的 \(x\) 坐标是 \(\frac{3}{4} + 1 = 1.75\)。

Vertical transformations: multiply \(y\) by \(-2\). So the new \(y\)-coordinate is \(-2 \times (-1) = 2\).

垂直变换：将 \(y\) 乘以 \(-2\)。因此新的 \(y\) 坐标是 \(-2 \times (-1) = 2\)。

The image point is \((1.75, 2)\).

像点为 \((1.75, 2)\)。

\((3, -1) \to \left( \frac{3}{4}+1, \; -2(-1) \right) = (1.75, 2)\)

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 \(\frac{1}{2}\).

错误：\(y=f(2x)\) 水平拉伸 2 倍。纠正：它水平压缩至 \(\frac{1}{2}\)。
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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Combined Transformations of Functions | 函数变换的组合