Transformations of Functions | 函数的变换

📚 Transformations of Functions | 函数的变换

Transformations allow us to modify the shape, position, and orientation of a graph without changing its fundamental character. Understanding how to translate, stretch, and reflect functions is essential for mastering IB Mathematics (both Analysis & Approaches and Applications & Interpretation), as it forms the basis for modelling real-world phenomena and analyzing complex functions. This article systematically covers all key transformation types, their algebraic and geometric impacts, and how to combine them correctly to build new graphs from parent functions.

变换使我们能够改变图形的形状、位置和方向，而不改变其根本特征。理解如何平移、伸缩和反射函数对于掌握IB数学（无论是分析与方法还是应用与解释）至关重要，因为它是建立真实世界模型和分析复杂函数的基础。本文系统地涵盖了所有关键的变换类型、它们的代数与几何影响，以及如何正确组合它们，从父函数构建新图形。

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1. What Are Transformations? | 什么是变换？

A transformation is an operation that moves or changes a graph in some way. When we apply a transformation to a function y = f(x), we alter its input, its output, or both, resulting in a new function whose graph is a modified version of the original. Transformations preserve the general shape of the graph, but they can shift it, stretch it, compress it, or flip it across an axis.

变换是一种以某种方式移动或改变图形的操作。当我们对一个函数 y = f(x) 应用变换时，我们改变了它的输入、输出或两者，从而得到一个图形是原始版本修改后的新函数。变换保留了图形的大致形状，但可以将其平移、拉伸、压缩或绕坐标轴翻转。

In IB Mathematics, transformations are usually studied in relation to elementary functions such as linear, quadratic, trigonometric, exponential, and logarithmic functions. Mastery of transformations enables you to sketch graphs quickly and solve equations involving shifted or stretched curves.

在IB数学中，变换通常与初等函数有关，如线性函数、二次函数、三角函数、指数函数和对数函数。掌握变换可以让你快速绘制图形，并解决涉及平移或拉伸曲线的方程。

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2. Vertical Translation | 垂直平移

The simplest transformation is a vertical shift. If we add a constant d to a function, the graph moves up by d units; if we subtract d, it moves down. Algebraically, this is represented as y = f(x) + d, where d is a real number.

最简单的变换是垂直平移。如果我们给一个函数加上常数d，图形就向上移动d个单位；如果减去d，则向下移动。代数上表示为 y = f(x) + d，其中d为实数。

For example, starting from the parent function f(x) = x², the graph of g(x) = x² + 3 is the parabola shifted 3 units upward. Every point (x, y) on the original graph becomes (x, y+3). This transformation only affects the y-coordinate.

例如，从父函数 f(x) = x² 开始，g(x) = x² + 3 的图形是向上平移3个单位的抛物线。原图形上的每个点 (x, y) 变为 (x, y+3)。这个变换只影响y坐标。

The vertical translation does not change the shape or x-intercepts in a simple way; x-intercepts will shift vertically, possibly changing their number. The vertical asymptote of a rational function is also unaffected because it depends on x, not y.

垂直平移不会以简单的方式改变形状或x截距；x截距会垂直移动，可能改变其个数。有理函数的垂直渐近线也不受影响，因为它依赖于x，而不是y。

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3. Horizontal Translation | 水平平移

Adding a constant to the input x causes a horizontal shift. The function y = f(x – c) shifts the graph to the right by c units if c > 0, and to the left if c < 0. This may seem counterintuitive because subtracting a positive c from x means that a larger x is needed to achieve the same output value, causing the graph to appear shifted to the right.

给输入x加上一个常数会引起水平平移。函数 y = f(x – c) 当 c > 0 时将图形向右移动c个单位，c < 0 时向左移动。这可能看起来反直觉，因为从x中减去一个正数c意味着需要更大的x才能得到相同的输出值，导致图形看起来向右平移。

For instance, if f(x) = √x, then y = √(x – 2) produces a graph that starts at (2,0) instead of (0,0), so the entire square root curve is shifted 2 units to the right. Points on the original graph (x, y) become (x+2, y).

例如，如果 f(x) = √x，那么 y = √(x – 2) 产生的图形从 (2,0) 开始，而不是 (0,0)，因此整个平方根曲线向右平移了2个单位。原图形上的点 (x, y) 变为 (x+2, y)。

Horizontal translations can change x-intercepts and vertical asymptotes, because those depend on x. For example, the vertical asymptote of y = 1/(x – 3) moves to x = 3.

水平平移可以改变x截距和垂直渐近线，因为这些依赖于x。例如，y = 1/(x – 3)的垂直渐近线移到了 x = 3。

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4. Vertical Stretch and Compression | 垂直伸缩与压缩

Multiplying the entire function by a constant factor a gives y = a f(x). If |a| > 1, the graph is stretched vertically away from the x-axis; if 0 < |a| < 1, it is compressed vertically towards the x-axis. If a is negative, there is also a reflection in the x-axis (covered later).

将整个函数乘以常数因子a得到 y = a f(x)。如果 |a| > 1，图形垂直拉伸远离x轴；如果 0 < |a| < 1，图形垂直压缩朝向x轴。如果a为负，还会同时发生关于x轴的反射（稍后讲解）。

Consider f(x) = sin x. The function y = 3 sin x has an amplitude of 3, so the graph is stretched vertically by a factor of 3. Points (x, y) become (x, 3y). The period remains 2π, but maximum and minimum values change.

考虑 f(x) = sin x。函数 y = 3 sin x 的振幅为3，因此图形垂直拉伸了3倍。点 (x, y) 变为 (x, 3y)。周期仍然是2π，但最大值和最小值改变了。

A vertical compression by factor ½ transforms f(x) = x² into y = ½ x², making the parabola wider. The vertex remains at the origin, but every y-value is halved.

垂直压缩因子½将 f(x) = x² 变换为 y = ½ x²，使抛物线变宽。顶点仍在原点，但每个y值减半。

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5. Horizontal Stretch and Compression | 水平伸缩与压缩

Modifying the input by a factor b inside the function gives y = f(bx). If |b| > 1, the graph is compressed horizontally towards the y-axis; if 0 < |b| < 1, it is stretched horizontally away from the y-axis. This is the opposite of what many students initially expect: a larger b value squashes the graph horizontally because the input x is multiplied, so to achieve a given output, the x-coordinate must be divided by b.

在函数内部通过因子b修改输入得到 y = f(bx)。如果 |b| > 1，图形水平压缩朝向y轴；如果 0 < |b| < 1，图形水平拉伸远离y轴。这与许多学生最初的预期相反：较大的b值会使图形水平压缩，因为输入x被乘以，为了达到给定的输出，x坐标必须除以b。

For example, f(x) = cos x has period 2π; the graph of y = cos(2x) has period π, so it is compressed horizontally by a factor of ½. In terms of points, (x, y) on the original becomes (x/2, y).

例如，f(x) = cos x 的周期为2π；y = cos(2x) 的图形的周期为π，因此它被水平压缩了因子½。就点而言，原图上的 (x, y) 变为 (x/2, y)。

A horizontal stretch by factor 2 is achieved by y = f(x/2). The graph of y = (x/2)² expands the parabola so that it is wider. All x-coordinates are doubled.

通过 y = f(x/2) 实现因子2的水平拉伸。y = (x/2)² 的图形扩张了抛物线，使其变宽。所有x坐标都翻倍。

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6. Reflections | 反射

Reflections flip the graph across a line. The two most common reflections are across the x-axis and the y-axis. Reflecting in the x-axis is given by y = -f(x); this changes the sign of all y-coordinates. Reflecting in the y-axis is given by y = f(-x), which changes the sign of all x-coordinates.

反射将图形沿一条直线翻转。最常见的两种反射是关于x轴和y轴的反射。关于x轴的反射由 y = -f(x) 给出；这改变了所有y坐标的符号。关于y轴的反射由 y = f(-x) 给出，改变了所有x坐标的符号。

If we start with f(x) = eˣ, then y = -eˣ reflects the exponential growth curve across the x-axis, turning it into a curve that decreases from 0 to negative infinity. The asymptote remains y = 0, but the graph lies below the x-axis.

如果我们从 f(x) = eˣ 开始，那么 y = -eˣ 将指数增长曲线关于x轴反射，使其变成一条从0下降到负无穷的曲线。渐近线仍然是 y = 0，但图形位于x轴下方。

Reflecting y = ln x across the y-axis yields y = ln(-x), which is defined only for x < 0 and is the mirror image of the logarithmic curve in the second quadrant.

将 y = ln x 关于y轴反射得到 y = ln(-x)，它仅在 x < 0 时有定义，是第二象限中对数曲线的镜像。

A double reflection across both axes results in y = -f(-x), which is equivalent to a 180° rotation about the origin. This can be seen as a point reflection.

关于两个轴的双重反射结果为 y = -f(-x)，相当于关于原点旋转180°。这可以看作点反射。

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7. Summary of Transformations | 变换总结表格

The following table summarises the primary transformations applied to a function y = f(x) and their effects on coordinates.

下表总结了应用于函数 y = f(x) 的主要变换及其对坐标的影响。

Transformation (变换)	Algebraic Form (代数形式)	Effect on (x, y) (对 (x,y) 的影响)
Vertical translation (垂直平移)	y = f(x) + d	(x, y+d)
Horizontal translation (水平平移)	y = f(x – c)	(x+c, y)
Vertical stretch/compression (垂直伸缩)	y = a f(x)	(x, a y)
Horizontal stretch/compression (水平伸缩)	y = f(bx)	(x/b, y)
Reflection in x-axis (关于x轴反射)	y = -f(x)	(x, -y)
Reflection in y-axis (关于y轴反射)	y = f(-x)	(-x, y)

Note that horizontal transformations act in the opposite direction inside the bracket: f(x – c) moves right, f(bx) compresses horizontally for |b|>1. This is a key concept to remember for exam questions.

注意，水平变换在括号内以相反方向作用：f(x – c) 向右移动，f(bx) 在 |b|>1 时水平压缩。这是考试题目中需要记住的关键概念。

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8. Combined Transformations | 复合变换

Most IB exam questions will ask you to apply multiple transformations to a single parent function. A general expression combining all transformations is y = a f(b(x – c)) + d. The parameters a, b, c, d control vertical stretch/compression/reflection, horizontal stretch/compression/reflection, horizontal shift, and vertical shift, respectively. However, the order in which you apply transformations matters.

大多数IB考试题目会要求你对一个父函数应用多个变换。包含所有变换的一般表达式为 y = a f(b(x – c)) + d。参数 a, b, c, d 分别控制垂直伸缩/压缩/反射、水平伸缩/压缩/反射、水平平移和垂直平移。然而，你应用变换的顺序很重要。

Consider transforming f(x) = x³ to g(x) = 2(x + 1)³ – 4. This involves a horizontal shift left by 1, a vertical stretch by factor 2, and a vertical shift down by 4. The correct sequence follows the “inside-out” approach: first deal with horizontal transformations inside the function argument, then vertical ones outside.

考虑将 f(x) = x³ 变换为 g(x) = 2(x + 1)³ – 4。这包括向左水平平移1个单位、垂直拉伸因子2和向下垂直平移4个单位。正确的顺序遵循“由内向外”的方法：首先处理函数参数内的水平变换，然后处理外部的垂直变换。

A step-by-step approach: (1) Start with y = x³. (2) Horizontal shift left 1: y = (x+1)³. (3) Vertical stretch by 2: y = 2(x+1)³. (4) Vertical shift down 4: y = 2(x+1)³ – 4. If you swap the vertical stretch and vertical shift, you get a different result because the stretch would affect the shift as well.

逐步方法：(1) 从 y = x³ 开始。(2) 向左水平平移1个单位：y = (x+1)³。(3) 垂直拉伸2倍：y = 2(x+1)³。(4) 向下垂直平移4个单位：y = 2(x+1)³ – 4。如果你交换垂直拉伸和垂直平移的顺序，你会得到不同的结果，因为拉伸也会影响平移。

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9. Order of Transformations | 变换顺序

The safest order when given a transformed function in the form y = a f(b(x – c)) + d is:

当给定形式为 y = a f(b(x – c)) + d 的变换函数时，最安全的顺序是：

• Horizontal stretch/compression and reflection (b)

  水平伸缩/压缩及反射 (b)

• Horizontal translation (c)

  水平平移 (c)

• Vertical stretch/compression and reflection (a)

  垂直伸缩/压缩及反射 (a)

• Vertical translation (d)

  垂直平移 (d)

This “horizontal first, then vertical” sequence works because horizontal transformations directly change the input before the function is evaluated, while vertical transformations act on the output. If you apply vertical transformations first, you might inadvertently alter the effect of later horizontal shifts.

这种“先水平后垂直”的顺序之所以有效，是因为水平变换在函数被求值之前直接改变了输入，而垂直变换作用于输出。如果你先应用垂直变换，你可能会无意中改变后来水平平移的效果。

For the function y = -2 f(3x – 6) + 5, begin by rewriting it as y = -2 f(3(x – 2)) + 5. Then follow the order: horizontal compression by factor 1/3 (from 3x), shift right 2, vertical stretch by 2 and reflection in x-axis (from -2), and finally shift up 5.

对于函数 y = -2 f(3x – 6) + 5，首先将其重写为 y = -2 f(3(x – 2)) + 5。然后按照顺序：水平压缩因子1/3（来自3x），向右平移2，垂直拉伸2倍并关于x轴反射（来自-2），最后向上平移5。

Misordering these steps is the most common error in IB transformation problems. Always factor out the b coefficient inside the bracket before applying translations.

这些步骤的顺序错误是IB变换问题中最常见的错误。在应用平移之前，一定要将括号内的b系数提取出来。

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10. Mapping Notation | 映射表示法

IB questions often use mapping notation to describe transformations. For a point (x, y) on the original graph, the image point under the transformation y = a f(b(x – c)) + d is given by:

IB题目经常使用映射表示法来描述变换。对于原图形上的一个点 (x, y)，在变换 y = a f(b(x – c)) + d 下的像点为：

(x, y) → (x/b + c, a y + d)

This compact form is extremely useful for quickly determining the new coordinates of specific points, such as intercepts or turning points, without having to re-sketch the whole function.

这种紧凑的形式对于快速确定特定点（如截距或转折点）的新坐标非常有用，而无需重新绘制整个函数。

For example, suppose the vertex (2, 5) of a parabola is transformed by y = 3 f(½(x + 4)) – 1. The mapping becomes: x’ = x/(½) + (-4)? Wait: b = ½, c = -4 (since it’s x+4 = x – (-4)). So x’ = x/b + c = x/(½) + (-4) = 2x – 4. y’ = 3y – 1. The new vertex is (2·2 – 4, 3·5 – 1) = (0, 14). This approach helps verify your sketched graph.

例如，假设抛物线顶点 (2, 5) 通过变换 y = 3 f(½(x + 4)) – 1。映射变为：x’ = x/(½) + (-4) = 2x – 4，y’ = 3y – 1。新顶点为 (2·2 – 4, 3·5 – 1) = (0, 14)。这种方法有助于验证你绘制的图形。

In exam papers, you may be asked to give the image of a point under a given transformation or to determine the transformation from a mapping rule. Always express transformations in the standard form to avoid sign errors.

在试卷中，你可能会被要求给出某点在给定变换下的像点，或者根据映射规则确定变换。始终将变换表达为标准形式以避免符号错误。

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11. Transformations of Points | 点的变换

When applying transformations to specific points rather than entire graphs, the same mapping rules apply. If a function undergoes multiple transformations, each coordinate of a point is transformed independently according to the horizontal and vertical orders.

当对特定点而不是整个图形应用变换时，同样的映射规则适用。如果一个函数经历了多个变换，点的每个坐标根据水平和垂直顺序独立变换。

Consider transposing a point through the transformation y = -2 f(¼ x – 1) + 3. Rewrite as y = -2 f(¼(x – 4)) + 3. A point (p, q) on the original graph maps to (4p + 4, -2q + 3). This is derived from: x’ = p/(¼) + 4 = 4p + 4, y’ = -2q + 3.

考虑通过变换 y = -2 f(¼ x – 1) + 3 的点。重写为 y = -2 f(¼(x – 4)) + 3。原图上的点 (p, q) 映射为 (4p + 4, -2q + 3)。这源于 x’ = p/(¼) + 4 = 4p + 4, y’ = -2q + 3。

This technique is also reverse-engineered: given a transformed point, you can find the original point by solving the equations. This is helpful when you know the image of a maximum or minimum and need to work backwards.

这种技术也可以反向推导：给定变换后的点，你可以通过解方程找到原始点。当你知道最大值或最小值的像点并需要反向推导时，这很有帮助。

Be careful with reflections: a reflection in the y-axis changes the sign of the x-coordinate, while a reflection in the x-axis changes the sign of the y-coordinate. Combined reflections can be tricky, so draw a quick sketch if in doubt.

注意反射：关于y轴的反射改变x坐标的符号，而关于x轴的反射改变y坐标的符号。复合反射可能会很棘手，如果有疑问，可以快速画一个草图。

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12. Exam Tips | 考试技巧

• Always rewrite the function in the form y = a f(b(x – c)) + d before identifying parameters. Factor out any coefficient of x inside the argument.

  在确定参数之前，始终将函数重写为 y = a f(b(x – c)) + d 的形式。提取参数内x的系数。

• Remember that horizontal transformations act counterintuitively: f(x + 2) shifts left, f(2x) compresses horizontally. A common trap is choosing the opposite direction in multiple-choice questions.

  记住水平变换的作用是反直觉的：f(x + 2) 向左平移，f(2x) 水平压缩。选择题中常见的陷阱是选择相反的方向。

• When describing transformations in words, follow the exact sequencing (horizontal then vertical) and use precise language: “stretch by factor 3 parallel to the y-axis”, “translate by vector (2, -5)”.

  当用文字描述变换时，遵循确切的顺序（先水平后垂直），并使用精确的语言：“平行于y轴拉伸因子3”、“按向量 (2, -5) 平移”。

• Check your work by selecting a key point (like a vertex or intercept) and applying the mapping rule to see if it lands on the expected location on the transformed graph.

  通过选择一个关键点（如顶点或截距）并应用映射规则检查你的工作，看它是否落在变换后图形的预期位置。

• Trigonometric functions have additional considerations: vertical stretch changes amplitude, horizontal stretch changes period. For y = a sin(bx + c) + d, the period is 2π/|b| and the phase shift is c/b to the left or right.

  三角函数有额外考虑：垂直拉伸改变振幅，水平拉伸改变周期。对于 y = a sin(bx + c) + d，周期为 2π/|b|，相移为 c/b 向左或向右。

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