📚 Mathematics Applications and Interpretation HL: Compression Question Analysis | 数学应用与解释HL：压缩变换题型解析
In the IB Mathematics: Applications and Interpretation HL course, understanding function transformations is essential for modelling and analysis. Among these transformations, compressions—both vertical and horizontal—are frequently examined in Paper 1 and Paper 2 questions. This article provides a comprehensive breakdown of compression-type questions, covering concepts, strategies, and common pitfalls to help you master this topic.
在IB数学：应用与解释HL课程中，理解函数变换对于建模和分析至关重要。在这些变换中，压缩——无论是垂直方向还是水平方向——经常在试卷一和试卷二的考题中出现。本文将对压缩题型进行全面剖析，涵盖概念、解题策略和常见误区，助你彻底掌握这一主题。
1. Introduction to Compression Transformations | 压缩变换概述
A compression transformation shrinks the graph of a function either towards the x-axis (vertical compression) or towards the y-axis (horizontal compression). Mathematically, a vertical compression occurs when the output values are multiplied by a factor a with 0 < a < 1: y = a f(x). A horizontal compression happens when the input variable x is multiplied by a factor b with b > 1: y = f(bx). Understanding these definitions is the first step in tackling compression questions.
压缩变换将函数图像向x轴（垂直压缩）或y轴（水平压缩）收缩。数学上，当输出值乘以因子a（0 < a < 1）时，发生垂直压缩：y = a f(x)。当输入变量x乘以因子b（b > 1）时，发生水平压缩：y = f(bx)。理解这些定义是解决压缩题型的第一步。
Compressions differ from stretches: a vertical stretch uses a > 1, while a vertical compression uses 0 < a < 1; a horizontal stretch uses 0 < b < 1, and a horizontal compression uses b > 1. The factor is often the reciprocal for horizontal transformations, so pay close attention.
压缩与拉伸不同：垂直拉伸使用a > 1，而垂直压缩使用0 < a < 1；水平拉伸使用0 < b < 1，水平压缩使用b > 1。水平变换的因子通常是倒数关系，因此需特别注意。
2. Vertical Compression: y = a f(x), 0 < a < 1 | 垂直压缩：y = a f(x), 0 < a < 1
When a function is transformed to y = a f(x) with 0 < a < 1, all y-coordinates are multiplied by a. This compresses the graph vertically towards the x-axis by a factor of a. For example, if f(x) = x², then g(x) = 0.5 x² is a vertical compression by factor ½. The vertex remains at the origin, but the parabola becomes wider.
当函数变换为 y = a f(x) 且 0 < a < 1 时，所有y坐标都乘以a。这将图像以因子a垂直向x轴压缩。例如，若 f(x) = x²，则 g(x) = 0.5 x² 是以因子½进行的垂直压缩。顶点仍在原点，但抛物线变得更宽。
In exam questions, you might be asked to identify the value of a given the coordinates of a point after transformation. Remember: new y = a × old y. So a = new y / old y, provided old y ≠ 0.
在考试题中，你可能会被要求根据变换后点的坐标确定a的值。记住：新y = a × 旧y。因此 a = 新y / 旧y，前提是旧y ≠ 0。
3. Horizontal Compression: y = f(bx), b > 1 | 水平压缩：y = f(bx), b > 1
A horizontal compression occurs when we replace x with bx where b > 1. The graph is compressed towards the y-axis by a factor of 1/b. For instance, if f(x) = sin x, then g(x) = sin(2x) compresses the period from 2π to π. Points on the graph have their x-coordinates divided by b.
当用 bx（b > 1）替换 x 时，发生水平压缩。图像以因子 1/b 向y轴压缩。例如，若 f(x) = sin x，则 g(x) = sin(2x) 将周期从 2π 压缩到 π。图像上点的x坐标除以 b。
To find b from two corresponding points, use b = old x / new x. This inverse relationship is a common source of error; many students incorrectly think b is the multiplier directly. Always test with a simple point: if (2, 3) maps to (1, 3) under y = f(bx), then b = 2 / 1 = 2.
要从两个对应点求 b，使用 b = 旧x / 新x。这种倒数关系是常见的错误来源；许多学生误以为 b 就是直接乘的倍数。总是用一个简单点来检验：如果点 (2, 3) 在 y = f(bx) 下映射到 (1, 3)，那么 b = 2 / 1 = 2。
4. Distinguishing Compression vs. Stretch | 区分压缩与拉伸
IB exam questions often test your ability to distinguish between compression and stretch by describing the transformation in words. For y = a f(x): a > 1 is a vertical stretch, a = 1 is no change, and 0 < a < 1 is a vertical compression. For y = f(bx): b > 1 is a horizontal compression, b = 1 is no change, and 0 < b < 1 is a horizontal stretch.
IB考题经常通过文字描述来测试你区分压缩与拉伸的能力。对于 y = a f(x)：a > 1 是垂直拉伸，a = 1 无变化，0 < a < 1 是垂直压缩。对于 y = f(bx)：b > 1 是水平压缩，b = 1 无变化，0 < b < 1 是水平拉伸。
A handy mnemonic: ‘Vertical is straightforward (a multiplies y); Horizontal is opposite (b divides x).’ Keep this in mind to avoid mistakes.
一个顺口溜：“垂直直接（a 乘 y），水平相反（b 除 x）。”牢记此点可避免错误。
5. Combining Compressions with Other Transformations | 与其他变换的组合
Compressions often appear alongside translations, reflections, and stretches. The order of transformations matters. When applying multiple transformations to a function, follow the order: horizontal transformations first (inside the brackets), then vertical transformations. For horizontal compressions combined with shifts, factor out the coefficient of x. Example: y = f(bx + c) should be written as y = f(b(x + c/b)) to identify the horizontal shift −c/b and the compression factor b.
压缩经常与平移、反射和拉伸同时出现。变换的顺序很重要。对一个函数应用多个变换时，遵循顺序：先进行水平变换（括号内），再进行垂直变换。对于水平压缩与平移的组合，要提取x的系数。例如：y = f(bx + c) 应写作 y = f(b(x + c/b))，以识别水平平移量 −c/b 和压缩因子 b。
Exam question: ‘Describe fully the sequence of transformations that maps f(x) to g(x) = 3 f(2x + 1) − 4.’ Start with y = f(x). Horizontal compression b=2, then shift left ½ unit, then vertical stretch by 3, then vertical shift down 4. Note the order: f(2x+1) = f(2(x+0.5)) so horizontal compression first, then shift left 0.5.
考题示例：“完整描述将 f(x) 映射到 g(x) = 3 f(2x + 1) − 4 的变换序列。”从 y = f(x) 开始。先水平压缩 b=2，然后左移 ½ 单位，然后垂直拉伸 3 倍，再垂直下移 4。注意顺序：f(2x+1) = f(2(x+0.5))，所以先压缩再平移。
6. Effects on Key Features: Intercepts, Asymptotes | 对关键特征的影响：截距、渐近线
Compression transformations alter key features of graphs. Vertical compression scales the y-intercept and local max/min values by a. Horizontal compression scales the x-intercepts by 1/b, and periodic features compress. For rational functions, vertical asymptotes may remain unchanged unless numerator affected; horizontal asymptotes are scaled by a in vertical compression. For horizontal compression, vertical asymptotes x = c become x = c/b, shifting towards the y-axis.
压缩变换会改变图像的关键特征。垂直压缩将y截距和局部极值按a缩放。水平压缩将x截距按 1/b 缩放，周期性特征也被压缩。对于有理函数，垂直渐近线一般不变（除非分子受影响）；水平渐近线在垂直压缩时按a缩放。对于水平压缩，垂直渐近线 x = c 变为 x = c/b，向y轴靠近。
When solving questions, always recalculate intercepts and asymptotes after applying transformations. Do not assume they stay the same.
解题时，应用变换后一定要重新计算截距和渐近线。不要假设它们保持不变。
7. Real-life Application: Modelling with Compression | 实际应用：建模中的压缩
In the AI HL syllabus, functions are used to model real-world phenomena. Compressions can represent scaling in time or amplitude. For example, a sound wave with higher frequency is a horizontally compressed sine wave. A damped oscillation exhibits vertical compression over time. Understanding compression helps interpret model parameters.
在AI HL大纲中，函数被用于对现实世界现象建模。压缩可以表示时间或幅度的缩放。例如，频率更高的声波是水平压缩的正弦波。阻尼振荡随时间呈现垂直压缩。理解压缩有助于解释模型参数。
Example: The height of a bungee jumper might be modelled as h(t) = 20 e⁻⁰·¹ᵗ cos(3t) + 30. The cos(3t) term is a horizontal compression of cos(t) with b=3, reducing the period. The exponential factor causes a gradual vertical compression over time.
示例：蹦极者的高度可建模为 h(t) = 20 e⁻⁰·¹ᵗ cos(3t) + 30。cos(3t) 一项是 cos(t) 的水平压缩（b=3），缩短了周期。指数因子随时间逐渐产生垂直压缩效应。
8. Exam-style Question Type 1: Graph Recognition | 题型一：图像识别
You may be given a transformed function and asked to choose the correct graph among options. Look for vertical compression if the amplitude of a periodic function is reduced, or horizontal compression if the period is shorter. For quadratics, a smaller leading coefficient (0 < a < 1) yields a wider parabola, indicating vertical compression.
你可能会得到一个变换后的函数，要求从选项中选出正确的图像。如果周期函数的振幅减小，寻找垂直压缩；如果周期变短，寻找水平压缩。对于二次函数，二次项系数较小（0 < a < 1）则抛物线更宽，表明垂直压缩。
Strategy: Identify the parent function, then apply in your mind the transformations step by step. Check key points like intercepts and turning points against the given graphs.
策略：识别母函数，然后在脑海中逐步应用变换。将截距和转折点等关键点与给定图像进行核对。
9. Exam-style Question Type 2: Equation Determination | 题型二：方程确定
Given a graph and its parent function, find the values of parameters a and b for compressions. Use coordinates of corresponding points. For vertical compression y = a f(x), pick a point (x, y) on the new graph, compute f(x) for that x on the parent, then a = y / f(x). For horizontal compression y = f(bx), use a point: if (p, q) is on the new graph, then f(bp) = q. Often you can find b by comparing x-coordinates for the same y-value.
给定图像及其母函数，求压缩参数 a 和 b 的值。利用对应点的坐标。对于垂直压缩 y = a f(x)，在新图像上取一点 (x, y)，计算母函数在 x 处的 f(x)，则 a
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