📚 The Power Function | 幂函数
t and n is a real number. Understanding how the shape, derivative, and integral of a power function depend on the exponent n is essential for analysing curves, solving equations, and modelling real-world phenomena in physics, economics, and biology. In this revision guide, we will explore the graphical behaviour of y = xⁿ for different types of n, derive the power rule for differentiation and integration, and apply these ideas to problems typical of the Edexcel specification.
幂函数是A-Level数学中最基本的构件之一。它形如 f(x) = kxⁿ，其中 k 为常数，n 为实数。理解幂函数的图像形状、导数和积分如何依赖于指数 n，对于分析曲线、求解方程以及建立物理、经济学和生物学中的真实模型至关重要。在这份复习指南中，我们将探究 y = xⁿ 在不同类型的 n 下的图形行为，推导微分的幂法则和积分的逆幂法则，并将这些思想应用于 Edexcel 考纲中的典型问题。
1. What is a Power Function? | 什么是幂函数？
xⁿ, where k is a non-zero constant called the coefficient, and n is a real number called the exponent. The domain and range of a power function depend heavily on the value of n. For example, when n is a positive integer, the function is defined for all real x. When n is a fraction like ½, the function represents a root, so x must be non-negative for real outputs. If n is negative, x cannot be zero because division by zero is undefined.
幂函数定义为 f(x) = kxⁿ，其中 k 是非零常数（称为系数），n 是实数（称为指数）。幂函数的定义域和值域在很大程度上取决于 n 的值。例如，当 n 为正整数时，该函数对所有实数 x 都有定义。当 n 为 ½ 这样的分数时，该函数表示一个根式，因此 x 必须为非负数才能得到实数输出。如果 n 为负数，则 x 不能为零，因为除以零是未定义的。
Power functions appear in direct variation (y ∝ xⁿ) and form the basis of polynomial and root functions. Recognising a power function helps us apply universal differentiation and integration rules, avoiding case-by-case memorisation.
幂函数出现在正变关系中（y ∝ xⁿ），构成了多项式和根式函数的基础。识别出幂函数有助于我们应用统一的微分和积分法则，避免逐个记忆。
2. Graphs of y = xⁿ for Positive Integer n | 正整指数幂函数的图像
the graph is a parabola that touches the origin and is symmetric about the y-axis. For n = 3, the graph is a cubic that passes through the origin and is symmetric about the origin.
当 n 为正整数时，y = xⁿ 是单项多项式。当 n = 1 时，图像为通过原点、斜率为 1 的直线。当 n = 2 时，图像为抛物线，在原点处与 x 轴相切，且关于 y 轴对称。当 n = 3 时，图像为三次曲线，通过原点且关于原点对称。
In general, for even n, the graph is U-shaped and lies entirely in the first and second quadrants. For odd n, the graph passes from the third quadrant to the first quadrant, with rotational symmetry of order 2 about the origin. As n increases, the graph becomes flatter near the origin and steeper for |x| > 1.
一般而言，当 n 为偶数时，图像呈 U 形，完全位于第一和第二象限。当 n 为奇数时，图像从第三象限延伸至第一象限，关于原点具有二阶旋转对称性。随着 n 增大，图像在原点附近变得更平坦，而在 |x| > 1 的区域变得更陡峭。
3. Graphs of y = xⁿ for Negative n | 负指数幂函数的图像
symptote at the y-axis. For example, y = x⁻¹ is the rectangular hyperbola with branches in the first and third quadrants. For y = x⁻², the graph lies entirely above the x-axis and has the y-axis as a vertical asymptote and the x-axis as a horizontal asymptote.
当 n 为负整数，例如 n = -m 且 m > 0，函数变为 y = x⁻ᵐ = 1/xᵐ。这类函数在 x = 0 处无定义，因此图像在 y 轴处有一条垂直渐近线。例如，y = x⁻¹ 是等轴双曲线，分支位于第一和第三象限。对于 y = x⁻²，图像完全位于 x 轴上方，y 轴为垂直渐近线，x 轴为水平渐近线。
he origin. As x → 0⁺, y → +∞ for even m and diverges with the same sign as x for odd m. As |x| → ∞, y → 0. These features are crucial for sketching reciprocal and rational functions.
负指数函数的特征是在原点不连接的两段曲线。当 x → 0⁺ 时，对于偶数 m，y → +∞；对于奇数 m，y 的符号与 x 同号且绝对值趋向无穷。当 |x| → ∞ 时，y → 0。这些特征对于绘制倒数和有理函数的图像至关重要。
4. Graphs of y = x1/k and Rational Powers | 分数指数幂函数的图像
When the exponent is a unit fraction 1/k, y = x1/k = ᵏ√x ph lies in the first quadrant only. For odd k (e.g., cube root, fifth root), the function is defined for all real x and displays the characteristic ‘lazy S’ shape passing through the origin.
当指数为单位分数 1/k 时，y = x1/k = ᵏ√x。对于偶数 k（如平方根、四次方根），定义域为 x ≥ 0，图像仅位于第一象限。对于奇数 k（如立方根、五次方根），函数对所有实数 x 都有定义，并呈现出通过原点的‘拉长 S’形特征。
Rational powers like y = xp/q can be treated as (x1/q)ᵖ or (xᵖ)1/q. The graph’s shape depends on whether p and q are even or odd. For example, y = x2/3 is the cube root of x², defined for all real x, symmetric about the y-axis, and has a cusp at the origin where the gradient is infinite. Understanding these subtleties is vital for accurate sketch-work in exams.
有理指数如 y = xp/q 可视为 (x1/q)ᵖ 或 (xᵖ)1/q。图像的形状取决于 p 和 q 的奇偶性。例如，y = x2/3 是 x² 的立方根，对所有实数 x 都有定义，关于 y 轴对称，并在原点处有一个尖点，此处梯度为无穷大。理解这些细微之处对于考试中准确绘制图像至关重要。
5. Key Features and Symmetry | 关键特征与对称性
jor role: for even n, f(-x) = f(x), so the graph is symmetric about the y-axis (even function). For odd n, f(-x) = -f(x), so the graph is symmetric about the origin (odd function).
所有纯幂函数 y = kxⁿ 都经过点 (1, k)，因为 1ⁿ = 1。如果 n > 0，图像还经过原点 (0,0)。对称性起着重要作用：当 n 为偶数时，f(-x) = f(x)，图像关于 y 轴对称（偶函数）。当 n 为奇数时，f(-x) = -f(x)，图像关于原点对称（奇函数）。
The sign of the coefficient k controls vertical reflection and stretch. A negative k reflects the graph in the x-axis. The absolute value |k| changes the steepness but does not alter the basic shape or symmetry. When analysing unfamiliar power functions, identifying the sign and parity of n is the first step in sketching.
系数 k 的符号控制垂直反射和拉伸。负的 k 将图像关于 x 轴反射。|k| 的绝对值改变陡峭程度，但不改变基本形状或对称性。在分析不熟悉的幂函数时，确定 n 的符号和奇偶性是绘制图像的第一步。
6. Derivative of a Power Function: The Power Rule | 幂函数的导数：幂法则
For f(x) = kxⁿ, the derivative is given by f' ncluding integers, fractions, and negatives, provided the function is differentiable at that point.
对于 f(x) = kxⁿ，其导数为 f'(x) = knxⁿ⁻¹。这被称为幂法则，是最常用的微分公式之一。该法则对任意实数常数 n 都成立，包括整数、分数和负数，只要函数在该点可导。
x⁵, then dy/dx = 3 × 5 × x⁴ = 15x⁴. If y = 2x⁻³, dy/dx = 2 × (-3) × x⁻⁴ = -6x⁻⁴. If y = 4x½, dy/dx = 4 × (½) × x-½ = 2/√x. The rule makes differentiating roots and reciprocals much faster than using first principles.
例如，若 y = 3x⁵，则 dy/dx = 3 × 5 × x⁴ = 15x⁴。若 y = 2x⁻³，则 dy/dx = 2 × (-3) × x⁻⁴ = -6x⁻⁴。若 y = 4x½，则 dy/dx = 4 × (½) × x-½ = 2/√x。该法则使得根式和倒数函数的微分比使用第一原理快得多。
Power Rule: d/dx [kxⁿ] = knxⁿ⁻¹
幂法则：d/dx [kxⁿ] = knxⁿ⁻¹
7. Proving the Power Rule for Integer Exponents | 整数指数幂法则的证明
sing the binomial theorem or by induction. Using the limit definition of the derivative: f' by h yields nxⁿ⁻¹ + terms containing h. As h → 0, all terms with h vanish, leaving nxⁿ⁻¹.
正整数指数的幂法则可通过二项式定理或数学归纳法证明。利用导数的极限定义：f'(x) = lim(h→0) [(x+h)ⁿ - xⁿ]/h。用二项式定理展开 (x+h)ⁿ 得到 xⁿ + nxⁿ⁻¹h + [n(n-1)/2]xⁿ⁻²h² + ... + hⁿ。减去 xⁿ 并除以 h，得到 nxⁿ⁻¹ 加上含有 h 的项。当 h → 0 时，所有含 h 的项消失，留下 nxⁿ⁻¹。
the same rule holds: d/dx (x⁻ᵐ) = -mx⁻ᵐ⁻¹. This unified proof reinforces the internal consistency of calculus.
对于负整数指数，将 x⁻ᵐ 写为 1/xᵐ，应用商法则或利用链式法则重写后求导。结果证实相同的法则成立：d/dx (x⁻ᵐ) = -mx⁻ᵐ⁻¹。这一统一的证明强化了微积分的内在一致性。
8. Extending the Power Rule to Rational Exponents | 有理指数幂法则的扩展
the power rule for rational exponents p/q, we can use implicit differentiation. Let y = xp/q h sides with respect to x: qyᵍ⁻¹ (dy/dx) = p xᵖ⁻¹. Solving for dy/dx gives dy/dx = (p/q) xp/q - 1. This elegant derivation confirms that the power rule works for all rational n.
为证明有理指数 p/q 的幂法则，我们可以使用隐函数微分法。令 y = xp/q，然后将两边同时取 q 次方：yᵍ = xᵖ。两边对 x 求导：qyᵍ⁻¹ (dy/dx) = p xᵖ⁻¹。解出 dy/dx 得到 dy/dx = (p/q) xp/q - 1。这个巧妙的推导证实了幂法则对所有有理数 n 都成立。
For irrational exponents, the result still holds but the proof requires logarithms and limits beyond the scope of A-Level. In practice, you can confidently apply the power rule for any real exponent n when given a power function in the form kxⁿ.
对于无理指数，该结果同样成立，但证明需要超出A-Level范围的对数和极限知识。在实践中，当给定形如 kxⁿ 的幂函数时，你可以放心地对任何实数指数 n 应用幂法则。
9. Integration of Power Functions | 幂函数的积分
k × [xⁿ⁺¹/(n+1)] + C, where C is the constant of integration. This is often remembered as 'add one to the power and divide by the new power' is critical because division by zero would occur; the special case n = -1 integrates to ln|x| + C.
微分的逆过程给出不定积分。对于 f(x) = kxⁿ，且 n ≠ -1，∫ kxⁿ dx = k × [xⁿ⁺¹/(n+1)] + C，其中 C 是积分常数。这通常被记忆为“指数加一，再除以新指数”。条件 n ≠ -1 至关重要，因为此时会除以零；当 n = -1 时的特殊情况积分为 ln|x| + C。
For example, ∫ 5x³ dx = 5 × (x⁴/4) + C = (5/4)x⁴ + C. ∫ 3/√x dx = ∫ 3x-½ dx = 3 × (x½/(½)) + C = 6√x + C. Mastery of this rule allows rapid evaluation of definite integrals and areas under curves.
例如，∫ 5x³ dx = 5 × (x⁴/4) + C = (5/4)x⁴ + C。∫ 3/√x dx = ∫ 3x-½ dx = 3 × (x½/(½)) + C = 6√x + C。熟练掌握这一法则可以快速计算定积分和曲线下的面积。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1
10. Reversing the Power Rule | 幂法则的逆运算
s ask you to integrate a derivative back to the original function, or to find f(x) given f'(x) and an initial condition. For a power function derivative f' neral antiderivative is f(x) = (A/(n+1))xⁿ⁺¹ + C. Using a point on the curve determines C.
由于微分和积分互为逆运算，许多考题要求你将一个导数积分回原函数，或在给定 f'(x) 和初始条件的情况下求 f(x)。对于幂函数的导数 f'(x) = Axⁿ，其一般原函数为 f(x) = (A/(n+1))xⁿ⁺¹ + C。利用曲线上的一点即可确定常数 C。
This technique is particularly important in kinematics, where acceleration is a power function, and we integrate to find velocity and displacement. Similarly, in economics, marginal cost functions are often integrated to obtain total cost functions.
这一技巧在运动学中尤为重要，此时加速度是幂函数，我们通过积分求得速度和位移。同样，在经济学中，边际成本函数经常被积分以得到总成本函数。
11. Applications: Modelling with Power Functions | 应用：幂函数建模
e area of a circle A = πr² is a power function with n = 2. The period of a pendulum T = 2π√L/g can be rewritten as T = kL½ e square laws, F = kx⁻². In biology, metabolic rate often scales with mass to the power ¾.
幂函数可以对许多现实关系进行建模。例如，圆的面积 A = πr² 是一个指数为 2 的幂函数。单摆的周期 T = 2π√L/g 可改写成 T = kL½，即一个幂函数。在物理学中，万有引力和电场力遵循平方反比定律，F = kx⁻²。在生物学中，代谢速率通常与质量的 ¾ 次幂成正比。
inear relationship between ln y and ln x. Determining n from experimental data is a key skill assessed in the applied sections of A-Level Mathematics.
在双对数图上呈直线的数据表明其背后有幂律关系。这是因为对 y = kxⁿ 取对数得到 ln y = ln k + n ln x，这是 ln y 与 ln x 之间的线性关系。从实验数据中确定指数 n 是A-Level数学应用部分考察的一项关键技能。
12. Combined Functions and Transformations | 组合函数与变换
a power function. The chain rule must be used: differentiate the outer power, then multiply by the derivative of the inner expression. Transformations such as f(x) = a(x + b)ⁿ + c shift and stretch the basic power graph.
幂函数通常是更复杂表达式的组成部分。例如，f(x) = (2x² + 3)⁵ 是一个复合函数，其外层函数是幂函数。必须使用链式法则：先对外层幂函数求导，再乘以内层表达式的导数。诸如 f(x) = a(x + b)ⁿ + c 的变换会平移和拉伸基本的幂函数图像。
Understanding the effect of transformations: replacing x by (x + b) shifts the graph left by b units; adding c shifts it up; multiplying by a stretches vertically. Combinations of these allow you to write down derivatives and integrals quickly using the reverse chain rule or substitution methods, solidifying your command of power functions across the syllabus.
理解变换的效果：用 (x + b) 替换 x 会使图像向左平移 b 个单位；加上 c 会使图像向上平移；乘以 a 则会垂直拉伸。这些组合使得你可以利用逆链式法则或代换法快速写出导数和积分，巩固你在大纲范围内对幂函数的掌握。
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