📚 Scale Rule; Shift Rule; Factors tn and ekt | 尺度规则；平移规则；因子 tⁿ 与 eᵏᵗ
Understanding how to stretch, compress, and shift graphs is essential for modelling real-world data with functions. The scale rule and shift rule allow us to transform a parent function into the precise form we need, while the factors tn and ekt arise repeatedly in power-law growth and exponential growth. This article connects the geometric transformations with the algebraic forms of these two major families, providing clear explanations, side-by-side comparisons, and IB-style worked reasoning.
理解如何拉伸、压缩和平移图像，对于用函数对现实数据进行建模至关重要。尺度规则与平移规则使我们能够将父函数变换为所需的精确形式，而因子 tⁿ 和 eᵏᵗ 则反复出现在幂律增长和指数增长中。本文把几何变换与这两大函数族的代数形式联系起来，提供清晰的解释、并列比较和 IB 风格的推导演示。
1. Function Transformations Overview | 函数变换概览
When we alter a function's equation, we change its graph. The two fundamental types of modification are scaling – which stretches or compresses the graph – and shifting – which moves it horizontally or vertically. These actions are often combined to match data patterns, especially when dealing with power and exponential models.
当我们改写一个函数的方程时，其图像也随之改变。两种基本的修改类型是尺度变换——拉伸或压缩图像——以及平移——将图像水平或垂直移动。这些操作经常组合使用，以匹配数据模式，特别是在处理幂函数模型和指数模型时。
For any parent function f(x), the transformed function can be written as a · f(b(x − h)) + k. Here a and b control scaling, while h and k control shifting. The order in which we apply these changes matters, but the standard form above always represents a clear sequence: horizontal shift by h, horizontal scale by 1/b, vertical scale by a, and vertical shift by k.
对于任意父函数 f(x)，变换后的函数可写作 a · f(b(x − h)) + k。其中 a 与 b 控制尺度，而 h 与 k 控制平移。应用这些变化的顺次很重要，但上述标准形式始终代表明确的顺序：先水平平移 h，再水平缩放 1/b，然后垂直缩放 a，最后垂直平移 k。
2. The Scale Rule – Vertical and Horizontal Stretches | 尺度规则——垂直与水平伸缩
Vertical scaling multiplies all y-coordinates by a factor a. If |a| > 1, the graph stretches away from the x‑axis; if 0 < |a| < 1, it compresses towards the x‑axis. A negative a also reflects the graph across the x‑axis.
垂直尺度变换会将所有 y 坐标乘以因子 a。当 |a| > 1 时，图像远离 x 轴拉伸；当 0 < |a| < 1 时，它向 x 轴压缩。负的 a 还会使图像关于 x 轴反射。
Horizontal scaling works on the input: replacing x by bx multiplies the x‑coordinates by 1/b. A value |b| > 1 compresses the graph horizontally, while 0 < |b| < 1 stretches it. The scale rule is therefore: stretch factors are a (vertical) and 1/b (horizontal).
水平尺度变换作用在输入上：用 bx 替换 x，会将 x 坐标乘以 1/b。|b| > 1 时图像被水平压缩，0 < |b| < 1 时图像被水平拉伸。因此，尺度规则可概括为：垂直伸缩因子为 a，水平伸缩因子为 1/b。
al stretch by 3 and horizontal compression by factor 1/2. Understanding this dual nature is key to linking equation parameters to visual changes.
例如，y = 3 sin(2x) 具有垂直拉伸 3 倍和水平压缩 1/2 倍。理解这种双重特性是将方程参数与视觉变化联系起来的关键。
3. The Shift Rule – Translations | 平移规则——平移
Shifting a graph does not change its shape; it simply repositions it. A horizontal shift is given by (x − h): positive h moves the graph to the right, negative h moves it to the left. This often causes confusion, but remember that the expression (x − h) means the x‑coordinate that originally produced a certain feature now appears h units later.
平移图像不会改变其形状，只会重新定位。水平平移由 (x − h) 给出：正的 h 使图像右移，负的 h 使图像左移。这常常引起混淆，但请记住表达式 (x − h) 表示原本产生某个特征的 x 坐标现在晚出现 h 个单位。
Vertical shift is more intuitive: adding k moves the graph up, subtracting k moves it down. Together, (h, k) represent the translation vector from the parent function's origin or key point.
垂直平移更直观：加上 k 向上移动，减去 k 向下移动。(h, k) 一起表示从父函数原点或关键点出发的平移向量。
n possesses an asymptote, shifts also move the asymptote. For exponential functions, y = ex + k has a horizontal asymptote at y = k instead of y = 0.
当函数具有渐近线时，平移也会移动渐近线。对于指数函数，y = eˣ + k 的水平渐近线位于 y = k 而非 y = 0。
4. Applying the Rules to Exponential Factor ekt | 将规则应用于指数因子 ekt
The natural exponential parent function y = et can be transformed into y = A ek(t − h) ertically scales the curve: it is the initial value (when t = h) and also determines the y‑ th or decay. The shift h moves the curve horizontally, effectively making the growth start 'later' C vertically shifts the entire graph and often serves as a baseline or carrying capacity.
自然指数父函数 y = eᵗ 可以变换为 y = A eᵏ⁽ᵗ⁻ʰ⁾ + C。因子 A 对曲线进行垂直缩放：它是初始值（当 t = h 时），并且如果 h = 0，也决定了 y 轴截距。参数 k 缩放时间；更大的 k 产生更快的增长或衰减。平移 h 将曲线水平移动，实际上使增长“延迟”开始，而 C 垂直移动整个图像，常作为基线或环境承载力。
Many IB problems present the model P(t) = P₀ ekt and k is the continuous growth rate. If the model includes a time shift, say P(t) = 200 e0.03(t − 5), the population behaves as if the clock started 5 years earlier.
许多 IB 题目会给出模型 P(t) = P₀ eᵏᵗ，其中 t 以年为单位。这是 h = 0 且 C = 0 的特例。尺度因子 P₀ 设定初始种群，k 为连续增长率。若模型包含时间平移，例如 P(t) = 200 e⁰·⁰³⁽ᵗ⁻⁵⁾，则种群的表现就好像时间提前了 5 年开始一样。
The horizontal scale factor 1/k compresses or stretches the growth process. Compared to the parent et, a larger k makes the curve rise more steeply, as if time units are shrinking.
水平尺度因子 1/k 会压缩或拉伸增长过程。相较于父函数 eᵗ，较大的 k 使曲线上升得更陡，仿佛时间单位在缩短。
5. Applying the Rules to Power Factor tn | 将规则应用于幂因子 tⁿ
The basic power function y = tn can be transformed into y = A (t − h)n ctor A stretches the graph vertically and can be thought of as a proportionality constant. For t large, the term (t − h) behaves like t, but h shifts the apparent start of the power-law behaviour.
基本幂函数 y = tⁿ 可以变换为 y = A (t − h)ⁿ + k。指数 n 决定曲率，并常与物理定律相关（例如，动能中 n = 2，体积缩放中 n = 3）。尺度因子 A 垂直拉伸图像，可视为比例常数。当 t 很大时，(t − h) 表现得像 t，但 h 将幂律行为的表观起点进行了平移。
or example, y = 5/(t − 2) + 1 has vertical asymptote t = 2 and horizontal asymptote y = 1.
如果 n 为负值，该函数表示反比例关系且具有渐近线。此时加上 k 会移动水平渐近线，加上 h 会移动垂直渐近线。例如，y = 5/(t − 2) + 1 的垂直渐近线为 t = 2，水平渐近线为 y = 1。
Both integer and fractional powers follow the same transformation logic. The scale rule tells us that A acts as a vertical stretch and the exponent n governs the shape, while shifts reposition the power curve without altering its fundamental growth type.
整数幂和分数幂都遵循相同的变换逻辑。尺度规则告诉我们，A 起垂直拉伸的作用，指数 n 决定形状，而平移只重新定位幂函数曲线，并不改变其基本增长类型。
6. Combining Scale and Shift: The General Recipe | 尺度与平移的组合：通用方法
rovides a universal template. To sketch a transformed graph, first identify the parent function. Then apply shifts: move the key points or asymptotes horizontally by h and vertically by k. After that, apply scale factors: multiply x‑distances from the new centre by 1/b and y‑distances by a. Finally, reflect if any factor is negative.
组合形式 y = a · f(b(x − h)) + k 提供了一个通用模板。要画出变换后的图像，首先确定父函数。然后应用平移：将关键点或渐近线水平移动 h，垂直移动 k。接着应用尺度因子：将从新中心出发的 x 距离乘以 1/b，y 距离乘以 a。最后，若任一因子为负，则进行反射。
This order – shift first, then scale – mirrors the algebra inside the bracket. Always read (x − h) as 'the input is delayed by h'. Conversely, the vertical scale a and shift k act directly on the output.
这一顺序——先平移，再缩放——正好对应括号内的代数运算。始终将 (x − h) 理解为“输入延迟了 h”。相反，垂直尺度 a 和平移 k 直接作用在输出上。
For modelling, we often fit data to such transformed functions. The parameter values then carry physical meaning: A is an amplitude or initial value, k is a growth/decay rate, h is a time offset, and C is a baseline or equilibrium.
在建模中，我们经常将数据拟合成这类变换函数。此时参数具有物理意义：A 是振幅或初始值，k 是增长/衰减率，h 是时间偏移量，C 是基线或平衡值。
7. Comparing tn and ekt in Modelling | 建模中 tⁿ 与 ekt 的比较
Power functions of the form y = A tn to a power of time, such as diffusion or allometric scaling. Exponential functions y = A ekt model situations where the rate of change is proportional to the current value, such as population growth or radioactive decay. Both types often require a shift of axes to fit real data.
形如 y = A tⁿ 的幂函数用于模拟增长率与时间的幂成正比的现像，例如扩散或异速生长。指数函数 y = A eᵏᵗ 则模拟变化率与当前值成正比的情形，如种群增长或放射性衰变。这两类函数通常都需要平移坐标轴以拟合实际数据。
| Feature 特征 | Power factor tn 幂因子 | Exponential factor ekt 指数因子 |
|---|---|---|
| Growth type 增长类型 | Polynomial-like 多项式型 | Multiplicative 乘积型 |
| Relative growth rate 相对增长率 | Decreases for n>0 递减 | Constant 恒定 |
| Long-term domination 长期优势 | Outpaced by exponential 被指数超越 | Eventually overtakes any power 最终超越任何幂函数 |
| Linearisation method 线性化方法 | Log-log plot 双对数图 | Semi-log plot 半对数图 |
Understanding which factor dominates helps in long-term predictions. An IB question might ask why a logistic model is preferred over a pure exponential when a population has an upper limit; logistic models arise from transformations that shift the asymptote.
理解哪个因子起主导作用有助于进行长期预测。IB 试题可能会问，当种群存在上限时，为何逻辑模型优于纯指数模型；逻辑模型正是通过对渐近线进行平移变换而产生的。
8. Linearisation via Logarithmic Scales | 通过对数尺度实现线性化
ion of the scale rule appears when we take logarithms. For an exponential function y = A ekt kt. This is a linear relation in t, where k becomes the slope and ln A the intercept. The vertical scale of the original data is transformed into a logarithmic scale, and the exponential curve becomes a straight line.
当我们取对数时，尺度规则得到了一个强有力的应用。对于指数函数 y = A eᵏᵗ，取自然对数后得到 ln y = ln A + kt。这是关于 t 的线性关系，其中 k 成为斜率，ln A 成为截距。原始数据的垂直尺度被转化为对数尺度，指数曲线变成了一条直线。
Similarly, for a power function y = A tn t log t gives a straight line with slope n. Here both axes use a log scale, turning the power curve into a straight line. This linearisation is extremely useful when determining the exponent n from experimental data.
类似地，对于幂函数 y = A tⁿ，两边取对数得到 log y = log A + n log t。将 log y 对 log t 作图会得到一条斜率为 n 的直线。这里两个坐标轴都使用了对数尺度，将幂函数曲线转变为直线。这种线性化方法在根据实验数据确定指数 n 时极为有用。
e also interacts with linearisation: if the model contains a vertical shift, e.g., y = A ekt + C, a simple log transform does not yield a linear graph unless C is known or estimated. This is a common IB investigation point.
平移规则也与线性化产生交互：如果模型包含垂直平移，例如 y = A eᵏᵗ + C，简单对数变换并不能产生线性图像，除非已知或估算了 C。这是 IB 常见的探究考点。
9. Interpreting Parameters as Scale and Shift Factors | 将参数解读为尺度与平移因子
Every parameter in a transformed exponential or power model corresponds directly to a scale or shift operation. The table below summarises this mapping.
变换后的指数或幂模型中的每个参数都直接对应一个尺度或平移操作。下表总结了这一对应关系。
| Parameter 参数 | Transformation type 变换类型 | Effect on graph 对图像的影响 |
|---|---|---|
| A (in A ekt or A tn) | Vertical scale 垂直缩放 | Stretches or compresses vertically 垂直拉伸或压缩 |
| k (in ekt) | Horizontal scale 水平缩放 | Compresses time for |k| > 1 压缩时间 |
| n (in tn) | Shape / curvature 形状 / 曲率 | Determines concavity and growth degree 决定凹凸性和增长次数 |
| h (x − h) | Horizontal shift 水平平移 | Shifts left/right 左右移动 |
| C (+ C) | Vertical shift 垂直平移 | Shifts up/down, moves asymptote 上下移动，移动渐近线 |
When you read a model like N(t) = 150 e0.04(t − 10) + 30, immediately identify: initial value (at t=10) is 150+30 = 180, growth rate is 4% ation between algebra and graph is exactly what the scale and shift rules train you to do.
当你读到 N(t) = 150 e⁰·⁰⁴⁽ᵗ⁻¹⁰⁾ + 30 这样的模型时，应立即识别出：初始值（t=10 时）为 150+30 = 180，增长率为 4%，整个曲线向右平移了 10 年，向上平移了 30 个单位。这种在代数与图形之间的流畅转换，正是尺度规则和平移规则训练你掌握的能力。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
shifts. Remember: y = f(x − 3) moves the graph right by 3, not left. Thinking in terms of 'what x makes the bracket zero' helps: x = 3 restores the original f(0) point.
一个常见错误是弄反水平平移的方向。请记住：y = f(x − 3) 是将图像右移 3，不是左移。思考“让括号内为零的 x 值”会有所帮助：x = 3 时恢复原始的 f(0) 点。
because the graph is compressed. When modelling, students sometimes treat k as a horizontal shift rather than a scale factor; k stretches or compresses the time axis, it does not move it.
另一个陷阱是忘记水平缩放涉及 1/b。若看到 y = sin(2x)，周期减半，因为图像被压缩了。在建模时，学生有时将 k 视作水平平移而非尺度因子；k 拉伸或压缩时间轴，而不会移动它。
In logarithmic linearisation, a vertical shift C prevents a straight line unless subtracted first. Always check whether a model passes through the origin if expected, and be prepared to discuss the implications of shifts for an asymptote.
在对数线性化中，垂直平移 C 会阻止得到一条直线，除非先将其减去。务必检查模型是否按照预期通过原点，并准备好讨论平移对渐近线的影响。
For IB exams, practise writing transformations in the precise form a f(b(x − h)) + k, and extract the meaning of each parameter. When a question asks for a “scale factor” or “translation”, be explicit: “vertical stretch by factor 2” or “horizontal translation 4 units left”.
为准备 IB 考试，请多练习将变换写成 a f(b(x − h)) + k 的确切形式，并提取每个参数的含义。当题目要求写出“尺度因子”或“平移”时，请表述清楚：“垂直拉伸 2 倍”或“水平向左平移 4 个单位”。
11. Worked Example: Modelling with Shift and Scale | 例题演练：带平移与缩放的建模
The temperature T (in ℃) of a cooling liquid is modelled by T(t) = 25 + 60 e−0.2(t − 3), where t is minutes after noon. Describe the transformations that turn f(t) = et into this model, and find the room temperature and initial liquid temperature at noon.
某冷却液体的温度 T（℃）的模型为 T(t) = 25 + 60 e⁻⁰·²⁽ᵗ⁻³⁾，其中 t 是正午后的分钟数。请描述将 f(t) = eᵗ 变成此模型的变换，并求室温及正午时液体的初始温度。
First rewrite as T(t) = 60 e−0.2(t − 3) + 25. Compare with a f(b(t − h)) + k: a = 60, b = −0.2, h = 3, k = 25. The parent is et. Step 1: horizontal shift right by 3. Step 2: horizontal stretch by factor 1/|−0.2| = 5, and reflect across y‑axis (since b is negative, the decay becomes e−0.2t Step 4: vertical shift up by 25. The room temperature is the horizontal asymptote, T = 25℃. At noon t=0, T(0) = 25 + 60 e−0.2(−3) = 25 + 60 e0.6 ≈ 25
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