Transforming Functions | 函数变换

📚 Transforming Functions | 函数变换

Function transformations allow us to modify the graph of a given function f(x) in a systematic way. By applying translations, stretches, and reflections, we can sketch new graphs quickly and understand how altering an equation changes the shape and position of its curve. In Edexcel A‑Level Mathematics, you are expected to master both single transformations and combinations, always being careful about the order in which they are applied.

函数变换让我们能够系统地修改给定函数 f(x) 的图像。通过平移、伸缩和反射，我们可以快速绘制新图形，并理解方程的改动如何影响曲线的形状和位置。在爱德思 A‑Level 数学中，你需要掌握单一变换和复合变换，并始终注意它们应用的顺序。

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

A function transformation takes the graph of y = f(x) and produces a new graph by altering the equation in one or more ways. The main types are translations (shifts), stretches (expansions and compressions), and reflections (flips). Any transformed function can be written in the general form y = a f(b (x + c)) + d, where the parameters a, b, c and d each control a specific change.

函数变换将 y = f(x) 的图像通过以一种或多种方式修改方程来生成新图像。主要类型包括平移（移动）、伸缩（拉长和压缩）和反射（翻转）。任何变换后的函数都可以写成通式 y = a f(b (x + c)) + d，其中参数 a、b、c、d 分别控制一种特定的变化。

Understanding these transformations saves time when sketching graphs and is essential for solving equations and inequalities graphically. It also deepens your insight into the relationship between an algebraic formula and its geometric representation.

理解这些变换可以节省绘制图形的时间，并对利用图像解方程和不等式至关重要。它还能加深你对代数表达式与其几何表示之间关系的洞察。

Throughout this topic, always start with a clear picture of the parent function f(x) and then apply changes step by step. We will explore each parameter in detail.

在这个主题中，始终先清晰想象出原函数 f(x) 的图像，然后逐步施加变化。我们将详细探讨每一个参数。

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

Adding a constant a to f(x) moves the entire graph vertically. If a > 0, the graph shifts upwards by a units. If a < 0, the graph shifts downwards by |a| units. The shape of the graph remains unchanged.

将常数 a 加到 f(x) 上会使整个图像垂直移动。如果 a > 0，图像向上移动 a 个单位；如果 a < 0，图像向下移动 |a| 个单位。图形的形状保持不变。

For example, if f(x) = x², then y = x² + 3 is the parabola shifted up by 3, and y = x² − 2 is the same parabola shifted down by 2. This transformation directly affects the y‑coordinates: every point (x, y) becomes (x, y + a).

例如，若 f(x) = x²，则 y = x² + 3 是抛物线向上平移 3 个单位，y = x² − 2 是同一抛物线向下平移 2 个单位。这种变换直接影响 y 坐标：每个点 (x, y) 变为 (x, y + a)。

This is one of the easiest transformations to recognise and apply. Always remember that the constant is outside the function, so it only influences the output.

这是最容易识别和应用的变换之一。始终记住常数在函数外部，所以它只影响输出值。

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

Replacing x with (x + a) inside the function shifts the graph horizontally. The direction is opposite to the sign of a: y = f(x + a) moves the graph a units to the left, while y = f(x − a) moves it a units to the right. This feels counter‑intuitive at first, but it is because the new x must compensate to produce the same y as before.

在函数内部用 (x + a) 替换 x 会使图像水平移动。移动方向与 a 的符号相反：y = f(x + a) 将图像向左移动 a 个单位，而 y = f(x − a) 将图像向右移动 a 个单位。这起初感觉与直觉相反，因为新的 x 必须补偿才能得到和之前相同的 y。

For instance, if f(x) = x², then y = (x + 2)² is the graph of y = x² shifted 2 units left; y = (x − 3)² is shifted 3 units right. You can check by noting that the vertex (0,0) moves to (−2,0) and (3,0) respectively.

例如，若 f(x) = x²，则 y = (x + 2)² 是 y = x² 向左平移 2 个单位的图像；y = (x − 3)² 是向右平移 3 个单位。你可以通过观察顶点 (0,0) 分别移至 (−2,0) 和 (3,0) 来验证。

When dealing with more complicated functions, always set the bracket to zero to find the new position of a key point. This transformation is the source of many errors, so practise writing the bracket carefully.

当处理更复杂的函数时，总是令括号内的式子为零来找到关键点的新位置。这是出错最多的变换，所以请仔细书写括号中的表达式。

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4. Vertical Stretches and Compressions: y = a f(x) | 垂直拉伸与压缩：y = a f(x)

Multiplying the whole function f(x) by a constant a causes a vertical stretch or compression. If |a| > 1, the graph stretches away from the x‑axis; if 0 < |a| < 1, the graph is compressed towards the x‑axis. When a is negative, there is also a reflection in the x‑axis, which we will treat separately.

将整个函数 f(x) 乘以常数 a 会产生垂直拉伸或压缩。若 |a| > 1，图像沿远离 x 轴的方向拉伸；若 0 < |a| < 1，图像向 x 轴压缩。当 a 为负时，还包含关于 x 轴的反射，我们稍后单独讨论。

Think of the transformation as scaling all y‑values by factor a. A point (x, y) on the original graph moves to (x, a y). For example, y = 3 sin x triples the amplitude of the sine wave, while y = ½ sin x halves it.

可以将这种变换视为将所有 y 值按因子 a 缩放。原图像上的点 (x, y) 移动到 (x, a y)。例如，y = 3 sin x 将正弦波的振幅扩大为原来的三倍，而 y = ½ sin x 则将其缩小一半。

This transformation does not change the x‑intercepts, as they occur where the function value is zero and multiplying zero by any a still gives zero. It is crucial to note that vertical scaling happens after any vertical translation when both are present, unless brackets group them differently.

这种变换不会改变 x 轴截距，因为截距处函数值为零，零乘以任何 a 仍为零。需要注意的是，当同时存在垂直平移时，垂直缩放发生在平移之后，除非括号有不同的分组。

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5. Horizontal Stretches and Compressions: y = f(bx) | 水平拉伸与压缩：y = f(bx)

When x is multiplied by a constant b inside the function, the graph undergoes a horizontal stretch or compression. If |b| > 1, the graph is compressed horizontally towards the y‑axis by factor 1/|b|. If 0 < |b| < 1, the graph is stretched horizontally away from the y‑axis by factor 1/|b|. Again, a negative b also induces a reflection in the y‑axis.

当函数内部的 x 乘以常数 b 时，图像会发生水平拉伸或压缩。如果 |b| > 1，图像以因子 1/|b| 水平向 y 轴压缩；如果 0 < |b| < 1，图像以因子 1/|b| 水平远离 y 轴拉伸。同样，负的 b 还会引起关于 y 轴的反射。

This is the most confusing stretch because the scale factor is the reciprocal of b. For y = f(2x), every x‑value gets halved to produce the same y, so the graph looks ‘squeezed’ by factor 1/2. For y = f(x/3), which is y = f( (1/3)x ), the graph is stretched by factor 3.

这是最容易混淆的伸缩，因为缩放因子是 b 的倒数。对于 y = f(2x)，每个 x 值减半才能得到相同的 y，所以图像看起来被“挤压”了一个因子 1/2。对于 y = f(x/3)，即 y = f( (1/3)x )，图像被拉伸为原来的 3 倍。

In Edexcel exams, you might see notation like y = f(2x) or y = f(x/2). Always re‑write the inside as (b x) to identify b clearly. The x‑coordinates of key points are divided by b, while y‑coordinates stay the same.

在爱德思考试中，你可能会看到 y = f(2x) 或 y = f(x/2) 这样的写法。务必将其内部重写为 (b x) 的形式以清楚地识别出 b。关键点的 x 坐标除以 b，而 y 坐标保持不变。

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6. Reflections: y = −f(x) and y = f(−x) | 反射：y = −f(x) 与 y = f(−x)

Reflections flip the graph over a line. y = −f(x) reflects the graph in the x‑axis: every y‑coordinate changes sign, while x‑coordinates remain the same. The point (x, y) becomes (x, −y).

反射使图像围绕一条直线翻转。y = −f(x) 将图像关于 x 轴反射：每个 y 坐标改变符号，而 x 坐标保持不变。点 (x, y) 变为 (x, −y)。

Similarly, y = f(−x) reflects the graph in the y‑axis. The sign of x is flipped, so each point (x, y) becomes (−x, y). This transformation is often used with even and odd functions to test symmetry.

类似地，y = f(−x) 将图像关于 y 轴反射。x 的符号被翻转，因此每个点 (x, y) 变为 (−x, y)。这种变换常与偶函数和奇函数一起用来检验对称性。

Examples: y = −x² is an upside‑down parabola, while y = √(−x) is the reflection of the square‑root graph across the y‑axis, appearing only for x ≤ 0. Combining reflections with stretches is common: y = −2 f(x) is a vertical stretch by 2 and then a reflection in the x‑axis.

例如：y = −x² 是一个倒置的抛物线，而 y = √(−x) 是平方根图像关于 y 轴的反射，仅出现在 x ≤ 0 的区域。将反射与伸缩结合也很常见：y = −2 f(x) 表示先垂直拉伸 2 倍，再关于 x 轴反射。

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7. Combined Transformations and the Order | 复合变换及其顺序

When a function involves multiple transformations, the order in which you apply them matters, especially for horizontal changes. Write the transformed function in the form y = a f(b (x + c)) + d. The recommended sequence is:

当一个函数包含多个变换时，应用的顺序很重要，尤其是水平方向的变化。将变换后的函数写成 y = a f(b (x + c)) + d 的形式。推荐的顺序是：

• Horizontal translation by −c (move left if c > 0, right if c < 0)
• Horizontal stretch/compression by factor 1/b (if b ≠ 1)
• Vertical stretch/compression by factor a (including reflection if a < 0)
• Vertical translation by d

• 水平平移 −c（若 c > 0 则左移，若 c < 0 则右移）
• 水平拉伸/压缩（因子 1/b，若 b ≠ 1）
• 垂直拉伸/压缩（因子 a，若 a < 0 还包含反射）
• 垂直平移 d

This order follows ‘inside the bracket first, then outside’. For horizontal transformations, always handle the translation before the stretch when the argument is written as b(x + c). If the expression inside is not factorised, e.g., f(2x + 4), rewrite it as f(2(x + 2)) to see the translation clearly.

这个顺序遵循“先括号内，后括号外”的原则。对于水平变换，当自变量写成 b(x + c) 形式时，总是先处理平移再处理伸缩。如果括号内的表达式未因式分解，例如 f(2x + 4)，应重写为 f(2(x + 2)) 以看清平移量。

Consider y = 3 f(2(x − 1)) + 5. Start with the parent f(x). Shift right by 1, then compress horizontally by 1/2, then stretch vertically by 3, and finally shift up by 5. Alternatively, some steps can be swapped if careful, but the horizontal sequence must be translation then stretch.

考虑 y = 3 f(2(x − 1)) + 5。从原函数 f(x) 开始，先右移 1 个单位，然后水平压缩到 1/2，接着垂直拉伸 3 倍，最后上移 5 个单位。有时若注意得当，中间几步可以交换，但水平变换的顺序必须是先平移后伸缩。

A common mistake is to apply the horizontal stretch before the translation, which gives a wrong shift. Always check by testing a point on the original graph.

一个常见错误是先水平伸缩再平移，这会导致错误的位移。务必通过原图上的一个测试点来验证。

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8. Transforming Key Graphs: Quadratics and Trigonometry | 关键图像变换：二次函数与三角函数

Applying transformations to familiar graphs helps build intuition. For a quadratic f(x) = x², y = (x − 2)² + 3 shifts the vertex from (0,0) to (2,3) with the same shape. The axis of symmetry and intercepts change accordingly.

对熟悉的图像进行变换有助于建立直觉。对于二次函数 f(x) = x²，y = (x − 2)² + 3 将顶点从 (0,0) 移至 (2,3)，形状不变。对称轴和截距也相应改变。

For trigonometric functions, y = 2 sin(3x − 60°) is a good challenge. Rewrite as y = 2 sin(3(x − 20°)). The parent graph of sin x is shifted right by 20°, then compressed horizontally by factor 1/3 (so period becomes 360°/3 = 120°), and finally stretched vertically by 2. The amplitude becomes 2.

对于三角函数，y = 2 sin(3x − 60°) 是一道很好的练习题。将其重写为 y = 2 sin(3(x − 20°))。原 sin x 图像先右移 20°，然后水平压缩为原来的 1/3（因此周期变为 360°/3 = 120°），最后垂直拉伸 2 倍，振幅变为 2。

In exponential and logarithmic graphs, transformations can change asymptotes. For y = eˣ + 2, the horizontal asymptote moves from y = 0 to y = 2. For y = ln(x + 4), the vertical asymptote shifts from x = 0 to x = −4.

在指数函数和对数函数图像中，变换会改变渐近线。对于 y = eˣ + 2，水平渐近线从 y = 0 移到 y = 2；对于 y = ln(x + 4)，垂直渐近线从 x = 0 移到 x = −4。

Work through these step by step, plotting a few key points each time. This method builds the confidence needed to tackle exam questions on transformation of functions.

逐步练习这些步骤，每次描出几个关键点。这种方法能建立解答函数变换考题所需的信心。

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9. Identifying Transformations from Equations | 从方程识别变换

Given two functions, you may be asked to describe the sequence of transformations that map one onto the other. For instance, to map f(x) onto g(x) = 3 f(½ x − 1) − 4, first rewrite g(x) = 3 f(½(x − 2)) − 4. The transformations are:

给定两个函数，你可能需要描述将一个映射到另一个的一系列变换。例如，将 f(x) 映射到 g(x) = 3 f(½ x − 1) − 4，首先重写 g(x) = 3 f(½(x − 2)) − 4。其变换为：

• Translation right by 2
• Horizontal stretch by factor 2 (since b = ½)
• Vertical stretch by factor 3
• Translation down by 4

• 向右平移 2
• 水平拉伸为原来的 2 倍（因为 b = ½）
• 垂直拉伸 3 倍
• 向下平移 4

Always remember to factorise the x‑term to reveal the true horizontal translation. The expression inside the function must be in the form b(x + c) for the horizontal shift to be −c.

始终记得对含 x 的项因式分解以揭示真正的水平平移量。函数内部表达式必须化为 b(x + c) 的形式，水平位移才是 −c。

In some exam questions, you are asked ‘Describe fully the sequence of transformations.’ This demands a clear, logical order. State the translation, stretch and reflection clearly, and mention the factor and direction. Do not forget to mention whether the transformation is ‘in the x‑axis’ or ‘in the y‑axis’ for reflections.

在一些考题中，会要求“完整描述变换的顺序”。这需要清晰、有逻辑的顺序。清楚说明平移、伸缩和反射，并提及因子和方向。反射时不要忘记说明是“关于 x 轴”还是“关于 y 轴”。

Practice with pairs of diagrams; the more you link algebraic changes to geometric effects, the easier this becomes.

通过图形配对进行练习；越是将代数变化与几何效应联系起来，这就会变得越简单。

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10. Key Points and Common Pitfalls | 要点与常见误区

Transformations are highly predictable, but a few traps catch many students. Here are the essential reminders:

变换的规律性很强，但有几个陷阱常让学生出错。以下是一些重要提醒：

• Inside the function: the horizontal translation acts opposite to the sign. y = f(x + 3) moves left.
• Inside the function: the horizontal stretch has a reciprocal effect. y = f(2x) compresses by ½.
• Always factorise x‑terms: f(3x + 6) → f(3(x + 2)). Shift left by 2, then compress.
• Order for horizontal: translation before stretch for f(b(x + c)).
• Vertical transformations are applied last unless brackets create a different order.
• Reflections are just special stretches where the factor is −1.

• 函数内部：水平平移的符号作用相反。y = f(x + 3) 向左移动。
• 函数内部：水平伸缩具有倒数效应。y = f(2x) 压缩为原来的 ½。
• 务必因式分解含 x 的项：f(3x + 6) → f(3(x + 2))。向左平移 2，然后压缩。
• 水平方向的顺序：对于 f(b(x + c))，先平移后伸缩。
• 垂直变换最后进行，除非括号造成不同的顺序。
• 反射只是因子为 −1 的特殊伸缩。

To avoid mistakes, always test with a known point. For example, if the parent function has a point (p, q), what does it become after the transformations? If your result matches the equation’s description, you are on the right track.

为避免错误，总是用一个已知点来测试。例如，如果原函数有一个点 (p, q)，经过变换后它变成什么？若你的结果与方程的描述相符，就说明你走在了正确的道路上。

With consistent practice, transforming functions becomes a mechanical process that earns you reliable marks in the A‑Level exam.

通过持续练习，函数变换将成为一个机械化的过程，让你在 A‑Level 考试中稳健地得分。

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