Transforming Functions | 函数变换

📚 Transforming Functions | 函数变换

Transformations of functions are one of the most frequently tested topics in Edexcel A-Level Mathematics. Understanding how graphs shift, reflect, and stretch is essential not only for pure mathematics questions but also for applications in mechanics and statistics. In this article, we will systematically explore every type of function transformation you need to master for your exams.

函数变换是 Edexcel A-Level 数学中考查频率最高的知识点之一。理解图像如何平移、反射和拉伸，不仅对纯数学问题至关重要，在力学和统计学中的应用也同样重要。在本文中，我们将系统地探讨考试中需要掌握的每一种函数变换类型。

1. What Are Transformations of Functions? | 什么是函数变换？

A transformation of a function is an operation that modifies the graph of a function in a systematic way. Given a base function y = f(x), we apply transformations to obtain new functions that are related to the original graph. These transformations can be classified into three main categories: translations (shifts), reflections, and stretches.

函数变换是对函数图像进行系统性修改的一种运算。给定一个基本函数 y = f(x)，我们通过对它施加变换来获得与原图像相关的新函数。这些变换可以分为三大类：平移（位移）、反射和拉伸。

For each transformation, we must understand three things: the algebraic rule that defines it, the geometric effect on the graph, and the impact on key features such as asymptotes, intercepts, and turning points.

对于每一种变换，我们需要理解三件事：定义变换的代数规则、对图像的几何影响，以及对渐近线、截距和转向点等关键特征的影响。

2. Vertical Translations: y = f(x) + a | 垂直平移：y = f(x) + a

The transformation y = f(x) + a translates the graph of y = f(x) vertically by a units. If a is positive, the graph moves upward; if a is negative, it moves downward. Every point on the original graph moves by the same vertical distance, so the shape of the graph is unchanged.

变换 y = f(x) + a 将 y = f(x) 的图像垂直平移 a 个单位。若 a 为正，图像向上移动；若 a 为负，图像向下移动。原图像上的每一个点都移动相同的垂直距离，因此图像的形状保持不变。

y = f(x) + a → Translation by vector (0, a)

For example, if f(x) = x², then f(x) + 3 = x² + 3 shifts the parabola up by 3 units. The vertex moves from (0, 0) to (0, 3), and the axis of symmetry remains x = 0.

例如，若 f(x) = x²，则 f(x) + 3 = x² + 3 将抛物线向上移动 3 个单位。顶点从 (0, 0) 移动到 (0, 3)，对称轴仍为 x = 0。

Key point: adding a constant outside the function affects the output (y-values) only, so the domain of the function is unchanged.

要点：在函数外部加常数只影响输出值（y 值），因此函数的定义域不变。

3. Horizontal Translations: y = f(x + a) | 水平平移：y = f(x + a)

The transformation y = f(x + a) translates the graph of y = f(x) horizontally. Crucially, a positive value of a shifts the graph to the LEFT by a units, and a negative value shifts it to the RIGHT. This is often a source of confusion, so let us examine why.

变换 y = f(x + a) 将 y = f(x) 的图像水平平移。关键的是，a 为正时图像向左移动 a 个单位，a 为负时图像向右移动。这常常是混淆的来源，让我们来分析一下原因。

y = f(x + a) → Translation by vector (−a, 0)

Consider f(x) = x² and the transformed function g(x) = f(x + 2) = (x + 2)². The vertex of the original parabola is at x = 0. For g(x), we need x + 2 = 0, which gives x = −2. The vertex has moved to (−2, 0) — two units to the left.

考虑 f(x) = x² 和变换后的函数 g(x) = f(x + 2) = (x + 2)²。原抛物线的顶点在 x = 0 处。对于 g(x)，我们需要 x + 2 = 0，即 x = −2。顶点已移动到 (−2, 0)——向左移动了两个单位。

A helpful way to remember this: solve the equation inside the parentheses for zero to find where a key point lands. Making the argument smaller requires moving left on the x-axis.

一个有用的记忆方法：令括号内的表达式等于零，解出关键点的新位置。要使自变量变小，就需要在 x 轴上向左移动。

4. Reflections: y = −f(x) and y = f(−x) | 反射：y = −f(x) 和 y = f(−x)

Reflections flip the graph across a line of symmetry. There are two reflections you must know for A-Level.

反射是将图像沿对称轴翻转。在 A-Level 中你需掌握两种反射。

y = −f(x) → Reflection in the x-axis

y = f(−x) → Reflection in the y-axis

The transformation y = −f(x) multiplies every output value by −1, so every point (x, y) maps to (x, −y). This flips the graph over the x-axis. For example, if f(x) = eˣ, then y = −eˣ reflects the exponential curve below the x-axis. The horizontal asymptote y = 0 remains unchanged, but all positive y-values become negative.

变换 y = −f(x) 将每个输出值乘以 −1，因此每个点 (x, y) 对应到 (x, −y)。这使图像绕 x 轴翻转。例如，若 f(x) = eˣ，则 y = −eˣ 将指数曲线反射到 x 轴下方。水平渐近线 y = 0 保持不变，但所有正的 y 值变为负值。

The transformation y = f(−x) replaces every x with −x, mapping each point (x, y) to (−x, y). This flips the graph over the y-axis. For instance, y = √(−x) is the reflection of y = √x across the y-axis; its domain changes from x ≥ 0 to x ≤ 0.

变换 y = f(−x) 将每个 x 替换为 −x，将每个点 (x, y) 映射到 (−x, y)。这使图像绕 y 轴翻转。例如，y = √(−x) 是 y = √x 关于 y 轴的反射；其定义域从 x ≥ 0 变为 x ≤ 0。

Note: an even function has the property f(−x) = f(x), so its graph is unchanged by a y-axis reflection. An odd function satisfies f(−x) = −f(x), meaning a reflection in the y-axis produces the same result as a reflection in the x-axis.

注意：偶函数具有 f(−x) = f(x) 的性质，因此其图像在 y 轴反射下不变。奇函数满足 f(−x) = −f(x)，意味着 y 轴反射与 x 轴反射产生相同的结果。

5. Vertical Stretches: y = af(x) | 垂直拉伸：y = af(x)

The transformation y = af(x), where a > 0, is a vertical stretch with scale factor a relative to the x-axis. Each point (x, y) maps to (x, ay). If a > 1, the graph becomes taller; if 0 < a < 1, the graph becomes shorter (a compression).

变换 y = af(x)（其中 a > 0）是相对于 x 轴、比例因子为 a 的垂直拉伸。每个点 (x, y) 映射到 (x, ay)。若 a > 1，图像变得更高；若 0 < a < 1，图像变得更矮（压缩）。

It is important to recognise that the x-intercepts never move under this transformation, because if f(x) = 0, then af(x) = 0 regardless of the value of a. However, the y-intercept is multiplied by a.

需要认识到，在这种变换下 x 截距永远不会移动，因为若 f(x) = 0，则无论 a 为何值，af(x) = 0。然而，y 截距会被乘以 a。

Consider f(x) = sin x and y = 2 sin x. The amplitude changes from 1 to 2, while the period remains 2π. The x-intercepts at multiples of π stay fixed, and the maximum and minimum values change from ±1 to ±2.

考虑 f(x) = sin x 和 y = 2 sin x。振幅从 1 变为 2，而周期保持 2π 不变。π 的整数倍处的 x 截距保持不变，最大值和最小值从 ±1 变为 ±2。

If a is negative, the vertical stretch is combined with a reflection in the x-axis. For example, y = −3f(x) stretches by factor 3 and reflects across the x-axis.

若 a 为负，则垂直拉伸与 x 轴反射相结合。例如，y = −3f(x) 先按因子 3 拉伸，再关于 x 轴反射。

6. Horizontal Stretches: y = f(ax) | 水平拉伸：y = f(ax)

The transformation y = f(ax), where a > 0, is a horizontal stretch with scale factor 1/a relative to the y-axis. Each point (x, y) maps to (x/a, y). If a > 1, the graph compresses horizontally; if 0 < a < 1, the graph stretches horizontally.

变换 y = f(ax)（其中 a > 0）是相对于 y 轴、比例因子为 1/a 的水平拉伸。每个点 (x, y) 映射到 (x/a, y)。若 a > 1，图像在水平方向压缩；若 0 < a < 1，图像在水平方向拉伸。

y = f(ax) → Horizontal stretch with scale factor 1/a

Note the reciprocal relationship: the scale factor is 1/a, not a. This is the most common error students make. Remember that the x-values are divided by a, so the graph shrinks when a > 1.

注意倒数关系：比例因子是 1/a 而非 a。这是学生最容易犯的错误。记住 x 值被除以 a，因此当 a > 1 时图像缩小。

For example, y = sin(2x) has period π, while y = sin x has period 2π. The graph of y = sin(2x) completes one full cycle in half the horizontal distance. Similarly, y = sin(x/2) has period 4π — the graph stretches horizontally.

例如，y = sin(2x) 的周期为 π，而 y = sin x 的周期为 2π。y = sin(2x) 的图像在水平距离一半内完成一个完整周期。类似地，y = sin(x/2) 的周期为 4π——图像在水平方向被拉伸。

The y-intercept remains unchanged under horizontal stretches, since x = 0 maps to x = 0. The x-intercepts move closer to or further from the y-axis depending on the value of a.

水平拉伸下 y 截距保持不变，因为 x = 0 映射到 x = 0。x 截距根据 a 的值而靠近或远离 y 轴。

7. Combined Transformations | 组合变换

In exam questions, you will often be asked to apply multiple transformations to a single function. When combining transformations, the order of operations matters significantly, and Edexcel specifically tests your understanding of this ordering.

在考试题目中，你常常需要将多种变换应用于同一个函数。组合变换时，运算顺序非常重要，Edexcel 会特别考查你对顺序的理解。

Here is the general rule of thumb: transformations that affect the coordinate outside the function (vertical changes) are applied in the natural order they appear; transformations inside the function (horizontal changes) work opposite to intuition.

这里有一个通用经验法则：影响函数外部坐标的变换（垂直变化）按出现顺序自然应用；函数内部的变换（水平变化）则与直觉相反。

Consider building the function y = 2f(x + 1) − 3 from y = f(x). The correct order is:

考虑从 y = f(x) 构建函数 y = 2f(x + 1) − 3。正确的顺序是：

Horizontal translation: y = f(x + 1) shifts the graph 1 unit to the left.

水平平移：y = f(x + 1) 将图像向左移动 1 个单位。
Vertical stretch: y = 2f(x + 1) stretches the graph vertically by factor 2.

垂直拉伸：y = 2f(x + 1) 将图像垂直拉伸因子 2。
Vertical translation: y = 2f(x + 1) − 3 shifts the graph down by 3 units.

垂直平移：y = 2f(x + 1) − 3 将图像向下移动 3 个单位。

A useful strategy is to track the transformation of a single key point, such as a vertex or a turning point, throughout each step. This helps verify your final graph is correct.

一个有用的策略是追踪单个关键点（如顶点或转向点）在每一步中的变换。这有助于验证最终图形是否正确。

8. Transformations of Key Features: Points, Asymptotes, Intercepts | 关键特征的变换：点、渐近线、截距

When transforming a function, every key feature must also be transformed. Let us examine how each type of feature is affected.

变换函数时，每个关键特征也必须被变换。让我们考察每类特征如何受到影响。

Turning points: A maximum or minimum point (x₀, y₀) on y = f(x) maps to:

转向点：y = f(x) 上的极大值或极小值点 (x₀, y₀) 映射为：

y = f(x) + a → (x₀, y₀ + a)

y = f(x + a) → (x₀ − a, y₀)

y = af(x) → (x₀, ay₀)

y = f(ax) → (x₀/a, y₀)

Asymptotes: For rational functions such as f(x) = 1/x, a vertical asymptote at x = c transforms under horizontal translations and stretches. Under y = f(x + a), it moves to x = c − a. Under y = f(ax), it moves to x = c/a. Horizontal asymptotes transform under vertical translations and stretches only.

渐近线：对于有理函数如 f(x) = 1/x，垂直渐近线 x = c 在水平平移和拉伸下变换。在 y = f(x + a) 下，它移动到 x = c − a。在 y = f(ax) 下，它移动到 x = c/a。水平渐近线仅在垂直平移和拉伸下变换。

For example, f(x) = 1/x has asymptotes x = 0 and y = 0. The function g(x) = 1/(x − 2) + 1 has vertical asymptote x = 2 and horizontal asymptote y = 1.

例如，f(x) = 1/x 有渐近线 x = 0 和 y = 0。函数 g(x) = 1/(x − 2) + 1 有垂直渐近线 x = 2 和水平渐近线 y = 1。

Intercepts: The x-intercepts are found by setting y = 0. Vertical translations and stretches change the y-intercept, while horizontal transformations do not. Reflection in the y-axis changes the sign of the x-coordinates of all x-intercepts.

截距：x 截距由令 y = 0 求得。垂直平移和拉伸改变 y 截距，而水平变换不改变。y 轴反射改变所有 x 截距的 x 坐标符号。

9. Invariant Points | 不变点

An invariant point is a point that does not change position under a transformation. Identifying invariant points can help you sketch transformed graphs accurately and check your work.

不变点是在变换下位置不发生改变的点。识别不变点可以帮助你准确绘制变换后的图像并检查你的解答。

Under y = f(x) + a, there are usually no invariant points unless a = 0.

在 y = f(x) + a 下，通常没有不变点，除非 a = 0。
Under y = −f(x), invariant points are where f(x) = 0 — the x-intercepts.

在 y = −f(x) 下，不变点是 f(x) = 0 的位置——即 x 截距。
Under y = f(−x), invariant points satisfy x = −x, so x = 0. The y-intercept is invariant.

在 y = f(−x) 下，不变点满足 x = −x，即 x = 0。y 截距是不变点。
Under y = af(x), invariant points occur where f(x) = af(x), i.e., f(x) = 0 (assuming a ≠ 1). The x-intercepts remain fixed.

在 y = af(x) 下，不变点出现在 f(x) = af(x) 处，即 f(x) = 0（假设 a ≠ 1）。x 截距保持固定。
Under y = f(ax), invariant points satisfy ax = x, so x = 0 (assuming a ≠ 1). The y-intercept remains fixed.

在 y = f(ax) 下，不变点满足 ax = x，即 x = 0（假设 a ≠ 1）。y 截距保持固定。

These invariant points serve as anchors on the graph and are excellent reference points when sketching.

这些不变点作为图像上的锚点，在绘制草图时是极好的参考点。

10. Summary Table | 总结表格

The following table summarises all the transformations you need to know for Edexcel A-Level Mathematics.

下表总结了 Edexcel A-Level 数学中你需要掌握的所有变换。

Transformation 变换	Rule 规则	Effect 效果	Key Point Mapping 关键点映射
Vertical translation 垂直平移	y = f(x) + a	Moves up by a (a > 0) 向上移动 a（a > 0）	(x, y) → (x, y + a)
Horizontal translation 水平平移	y = f(x + a)	Moves left by a (a > 0) 向左移动 a（a > 0）	(x, y) → (x − a, y)
Reflection in x-axis x 轴反射	y = −f(x)	Flips over x-axis 绕 x 轴翻转	(x, y) → (x, −y)
Reflection in y-axis y 轴反射	y = f(−x)	Flips over y-axis 绕 y 轴翻转	(x, y) → (−x, y)
Vertical stretch 垂直拉伸	y = af(x)	Stretch factor a parallel to y-axis 平行于 y 轴拉伸因子 a	(x, y) → (x, ay)
Horizontal stretch 水平拉伸	y = f(ax)	Stretch factor 1/a parallel to x-axis 平行于 x 轴拉伸因子 1/a	(x, y) → (x/a, y)

Memorise this table thoroughly. In particular, be very careful with the horizontal transformations y = f(x + a) and y = f(ax), because the direction and scale factor are opposite to what intuition suggests.

请彻底记住此表。特别是要非常小心水平变换 y = f(x + a) 和 y = f(ax)，因为其方向和比例因子与直觉相反。

11. Worked Example | 例题解析

Let us work through a typical exam-style question step by step.

让我们逐步解析一道典型的考试风格题目。

Question: The function f(x) = x² has a minimum point at (0, 0). The function g(x) = 3f(x − 2) + 1 is formed from f(x). Find the coordinates of the minimum point of g(x), and describe the sequence of transformations.

题目：函数 f(x) = x² 的最小值点为 (0, 0)。函数 g(x) = 3f(x − 2) + 1 由 f(x) 变换而来。求 g(x) 的最小值点坐标，并描述变换的顺序。

Solution: We track the point (0, 0) through each transformation:

解答：我们追踪点 (0, 0) 在每一步变换中的位置：

Step 1 — Horizontal translation: g₁(x) = f(x − 2). This shifts the graph right by 2 units. The point becomes (0 + 2, 0) = (2, 0).

第 1 步——水平平移：g₁(x) = f(x − 2)。图像向右移动 2 个单位。该点变为 (0 + 2, 0) = (2, 0)。
Step 2 — Vertical stretch: g₂(x) = 3f(x − 2). The point becomes (2, 3 × 0) = (2, 0). The y-coordinate is unchanged because 0 × 3 = 0.

第 2 步——垂直拉伸：g₂(x) = 3f(x − 2)。该点变为 (2, 3 × 0) = (2, 0)。因为 0 × 3 = 0，y 坐标不变。
Step 3 — Vertical translation: g(x) = 3f(x − 2) + 1. The point becomes (2, 0 + 1) = (2, 1).

第 3 步——垂直平移：g(x) = 3f(x − 2) + 1。该点变为 (2, 0 + 1) = (2, 1)。

Therefore, the minimum point of g(x) is (2, 1). The full expression is g(x) = 3(x − 2)² + 1, which is the vertex form of a parabola confirming our answer.

因此，g(x) 的最小值点为 (2, 1)。完整表达式为 g(x) = 3(x − 2)² + 1，这是抛物线的顶点式，证实了我们的答案。

g(x) = 3(x − 2)² + 1, Minimum point: (2, 1)

The order of transformations is: translate 2 units to the right, stretch vertically by factor 3, then translate 1 unit upward.

变换顺序为：向右平移 2 个单位，垂直拉伸因子 3，然后向上平移 1 个单位。

12. Exam Tips and Common Pitfalls | 考试技巧与常见错误

Finally, here are the most important tips and pitfalls to help you secure full marks on transformation questions.

最后，这里是一些最重要的技巧和易错点，帮助你在此类题目中拿到满分。

Horizontal translations move in the opposite direction. y = f(x + 2) moves LEFT, not RIGHT.

水平平移向相反方向移动。y = f(x + 2) 向左移动，而非向右。
Horizontal stretch factor is the reciprocal. y = f(2x) compresses by factor 1/2, not 2.

水平拉伸因子是倒数。y = f(2x) 压缩因子为 1/2，而非 2。
Apply transformations in the correct order. Horizontal changes inside the brackets are applied before or after vertical changes? Always track one point through each step to avoid mistakes.

按正确顺序应用变换。括号内的水平变化与垂直变化的先后顺序？务必追踪一个点经过每一步以避免错误。
Asymptotes transform too. Do not forget to transform asymptotes; this is a common source of lost marks in sketch questions.

渐近线也要变换。不要忘记变换渐近线；这是在作图题中失分的常见原因。
Use invariant points as checks. After sketching a transformed graph, verify that invariant points remain in the correct positions.

用不变点进行检验。绘制变换后的图像后，验证不变点是否保持在正确位置。
State full transformation descriptions. When describing a transformation, always specify the direction, the axis, and the scale factor. Marks are often awarded for each of these details.

完整描述变换。描述变换时，要明确指出方向、轴和比例因子。这些细节常常分点给分。

Mastering function transformations requires practice and careful attention to detail. Work through as many past paper questions as you can, paying special attention to the order of combined transformations. With consistent practice, these questions become some of the easiest marks on the paper.

掌握函数变换需要练习和对细节的仔细关注。尽可能多做历年真题，特别注意组合变换的顺序。坚持不懈地练习，这类题目就会成为试卷上最容易拿的分数。

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Transforming Functions | 函数变换