📚 A-Level Maths: Locating Roots and Graph Analysis | A-Level 数学：方程根的定位与图像分析
Solving equations exactly is not always possible. In A-Level Mathematics, you are often asked to locate a root approximately, to show that a root exists, or to analyse how the graph of a function relates to the solutions of an equation. This article covers the key techniques for locating roots and interpreting graphs.
在 A-Level 数学中，精确求解方程并不总是可行。你常常需要近似地定位根、证明根的存在，或分析函数图像与方程解之间的关系。本文涵盖求根定位与图像分析的关键技巧。
1. What Is a Root? | 什么是方程的根？
A root of an equation f(x) = 0 is a value x such that f(x) = 0. Geometrically, a root is an x-coordinate where the graph of y = f(x) crosses or touches the x-axis.
方程 f(x) = 0 的根是指使得 f(x) = 0 的 x 值。从几何上看，根就是函数 y = f(x) 的图像与 x 轴相交或相切处的 x 坐标。
For example, the equation x² − 4 = 0 has two real roots, x = −2 and x = 2, because the parabola y = x² − 4 meets the x-axis at those points.
例如，方程 x² − 4 = 0 有两个实根：x = −2 和 x = 2，因为抛物线 y = x² − 4 在这两点处与 x 轴相交。
Key idea: To solve an equation like f(x) = g(x), you can rearrange it as h(x) = f(x) − g(x) = 0. The roots of h correspond to the intersection points of y = f(x) and y = g(x).
关键思想：要解形如 f(x) = g(x) 的方程，可以将其改写为 h(x) = f(x) − g(x) = 0。h 的根对应于 y = f(x) 与 y = g(x) 图像的交点。
2. The Intermediate Value Theorem | 中间值定理
If f is continuous on the closed interval [a, b], and f(a) and f(b) have opposite signs, then there is at least one root in (a, b). This is the Intermediate Value Theorem applied to root location.
如果函数 f 在闭区间 [a, b] 上连续，且 f(a) 与 f(b) 异号，那么在开区间 (a, b) 内至少存在一个根。这就是中间值定理在求根定位中的应用。
Formal statement: If f is continuous on [a, b] and k lies between f(a) and f(b), then there exists c ∈ (a, b) such that f(c) = k.
严格叙述：若 f 在 [a, b] 上连续，且 k 介于 f(a) 与 f(b) 之间，则存在 c ∈ (a, b)，使得 f(c) = k。
For root location, we take k = 0. Therefore, a sign change of a continuous function across an interval guarantees a root. This is the basis of many numerical methods.
对于根的定位，我们取 k = 0。因此，连续函数在一个区间两端符号相反，就保证了该区间内至少有一个根。这是许多数值方法的基础。
f(a) × f(b) < 0 ⇒ at least one c ∈ (a, b) with f(c) = 0
3. Sign Change Method | 符号变化法
To locate a root between two integers, evaluate f at consecutive integer values. A sign change between f(n) and f(n + 1) indicates a root in (n, n + 1).
要在两个整数之间定位根，可在连续整数处计算 f 的值。如果 f(n) 与 f(n + 1) 符号相反，则说明根位于区间 (n, n + 1) 内。
Example: Show that f(x) = x³ − 2x − 5 has a root between 2 and 3.
示例：证明 f(x) = x³ − 2x − 5 在 2 与 3 之间存在一个根。
f(2) = 8 − 4 − 5 = −1 < 0, and f(3) = 27 − 6 − 5 = 16 > 0. Since f is a polynomial (hence continuous) and the sign changes, there is a root in (2, 3).
f(2) = 8 − 4 − 5 = −1 < 0，而 f(3) = 27 − 6 − 5 = 16 > 0。由于 f 是多项式（因此连续）且符号改变，故在 (2, 3) 内有一个根。
Important: A sign change guarantees a root, but the absence of a sign change does NOT prove that no root exists. For example, f(x) = x² has a root at x = 0, yet f(−1) and f(1) are both positive.
重要：符号变化能保证根存在，但没有符号变化并不能证明根不存在。例如，f(x) = x² 在 x = 0 处有根，但 f(−1) 和 f(1) 均为正。
4. Graphical Analysis: Sketching | 图像分析：草图绘制
Sketching y = f(x) helps you estimate the number of roots and their approximate locations. Pay attention to intercepts, turning points, asymptotes, and behaviour as x → ±∞.
绘制 y = f(x) 的草图有助于估计根的个数和大致位置。注意截距、转向点、渐近线以及 x → ±∞ 时的行为。
For rational functions, vertical asymptotes occur where the denominator is zero, and horizontal asymptotes are determined by the degrees of numerator and denominator. Roots of the function are where the numerator is zero (provided the denominator is not also zero).
对于有理函数，垂直渐近线出现在分母为零处，水平渐近线由分子分母的次数决定。函数的根出现在分子为零处（前提是分母不同时为零）。
Example: Sketch y = (x − 1)(x + 2) / (x − 3). The roots are x = 1 and x = −2, the vertical asymptote is x = 3, and the horizontal asymptote is y = 1 (since degrees are equal and leading coefficients are both 1).
示例：绘制 y = (x − 1)(x + 2) / (x − 3) 的草图。根为 x = 1 和 x = −2，垂直渐近线为 x = 3，水平渐近线为 y = 1（因为分子分母次数相同，最高次项系数均为 1）。
5. Repeated Roots and Tangency | 重根与相切
A repeated root occurs when f(x) = 0 and f′(x) = 0 at the same point. For example, f(x) = (x − 2)² has a double root at x = 2; the graph touches the x-axis but does not cross it.
重根出现在同一点处 f(x) = 0 且 f′(x) = 0。例如，f(x) = (x − 2)² 在 x = 2 处有二重根；图像与 x 轴相切但不穿过它。
If the multiplicity of the root is odd, the graph crosses the x-axis. If the multiplicity is even, the graph touches the x-axis and turns around.
如果根的重数为奇数，图像会穿过 x 轴；如果重数为偶数，图像与 x 轴相切并反向折回。
This is important when interpreting graphs: a point of tangency to the x-axis corresponds to a repeated root, not a simple crossing.
这在解释图像时很重要：与 x 轴相切的点对应重根，而不是简单的穿越。
Example: f(x) = x(x − 1)²(x + 2)³ has a simple root at x = 0, a double root at x = 1, and a triple root at x = −2. The graph crosses at x = 0 and x = −2, but touches at x = 1.
示例：f(x) = x(x − 1)²(x + 2)³ 在 x = 0 处有单根，在 x = 1 处有二重根，在 x = −2 处有三重根。图像在 x = 0 和 x = −2 处穿过，但在 x = 1 处相切。
6. The Bisection Method | 二分法
The bisection method uses repeated sign changes to narrow down an interval containing a root. Starting from an interval [a, b] with f(a)f(b) < 0, compute the midpoint m = (a + b) / 2.
二分法利用反复的符号变化来缩小包含根的区间。从满足 f(a)f(b) < 0 的区间 [a, b] 出发，计算中点 m = (a + b) / 2。
If f(m) = 0, then m is the root. If f(a) and f(m) have opposite signs, the root lies in [a, m]; otherwise it lies in [m, b]. Repeat until the interval is sufficiently small.
如果 f(m) = 0，则 m 就是根。如果 f(a) 与 f(m) 异号，根在 [a, m] 内；否则根在 [m, b] 内。重复此过程，直到区间足够小。
m = (a + b) / 2
Each iteration halves the interval length. After n iterations, the error is at most (b − a) / 2ⁿ. This method always converges for a continuous function with a sign change, but convergence is relatively slow.
每次迭代将区间长度减半。经过 n 次迭代后，误差不超过 (b − a) / 2ⁿ。对于连续且符号改变的函数，这种方法必定收敛，但收敛速度相对较慢。
7. Newton-Raphson Method | 牛顿-拉弗森法
The Newton-Raphson method is an iterative formula that produces successively better approximations to a root. Start with an initial guess x₀, then compute:
牛顿-拉弗森法是一种迭代公式，能够产生对根的逐次改进近似。从一个初始猜测 x₀ 出发，然后计算：
xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ)
Geometrically, xₙ₊₁ is the x-intercept of the tangent to y = f(x) at x = xₙ. Provided f′(xₙ) ≠ 0 and the initial guess is close enough, the method usually converges quickly.
从几何上看，xₙ₊₁ 是曲线 y = f(x) 在 x = xₙ 处的切线的 x 轴截距。只要 f′(xₙ) ≠ 0 且初始猜测足够接近，该方法通常收敛很快。
Example: For f(x) = x² − 5, the iteration is xₙ₊₁ = xₙ − (xₙ² − 5) / (2xₙ) = (xₙ + 5/xₙ) / 2. Starting from x₀ = 2 gives x₁ = 2.25, x₂ ≈ 2.2361, which is close to √5 ≈ 2.23607.
示例：对于 f(x) = x² − 5，迭代公式为 xₙ₊₁ = xₙ − (xₙ² − 5) / (2xₙ) = (xₙ + 5/xₙ) / 2。从 x₀ = 2 出发，得到 x₁ = 2.25，x₂ ≈ 2.2361，与 √5 ≈ 2.23607 非常接近。
Be careful: if f′(xₙ) is close to zero, the tangent is nearly horizontal and xₙ₊₁ may be far from the root. The method can fail if the initial guess is not near the root or if the function has a stationary point near the root.
注意：如果 f′(xₙ) 接近零，切线几乎水平，xₙ₊₁ 可能远离根。当初始猜测离根太远，或函数在根附近有驻点时，该方法可能失效。
8. Iterative Formulas xₙ₊₁ = g(xₙ) | 迭代公式 xₙ₊₁ = g(xₙ)
Many root-finding methods can be written in the form xₙ₊₁ = g(xₙ). A root α of f(x) = 0 can be rewritten as x = g(x), so that α is a fixed point of g.
许多求根方法可以写成 xₙ₊₁ = g(xₙ) 的形式。方程 f(x) = 0 的根 α 可以改写为 x = g(x)，这样 α 是 g 的不动点。
For example, the equation x³ − 2x − 5 = 0 can be rearranged as x = (2x + 5)^{1/3} or x = (x³ − 5)/2, among others. Each rearrangement gives a different iteration.
例如，方程 x³ − 2x − 5 = 0 可以改写为 x = (2x + 5)^{1/3} 或 x = (x³ − 5)/2 等多种形式。每一种改写对应不同的迭代。
The iteration converges to α if |g′(α)| < 1. If |g′(α)| > 1, the iteration diverges away from the root; if |g′(α)| = 1, convergence is uncertain.
当 |g′(α)| < 1 时，迭代收敛于 α；当 |g′(α)| > 1 时，迭代会远离根；当 |g′(α)| = 1 时，收敛性不确定。
Key skill: You may be asked to show that a given iteration converges to a root. Differentiate g and evaluate the derivative at the known approximate root.
关键技能：你可能会被要求证明某个迭代收敛于某个根。此时应对 g 求导，并在已知的近似根处计算导数值。
9. Transformations of Graphs and Their Roots | 图像变换与根的关系
Graph transformations change the location and number of roots. For example, y = f(x) + k shifts the graph vertically; if k is large enough, all roots may disappear.
图像变换会改变根的位置和数量。例如，y = f(x) + k 使图像垂直平移；如果 k 足够大，所有根都可能消失。
y = f(x − a) shifts the graph horizontally by a units; every root x = r becomes x = r + a. The number of roots is unchanged.
y = f(x − a) 使图像水平平移 a 个单位；每个根 x = r 变为 x = r + a，根的数目不变。
y = f(kx) compresses or stretches horizontally, changing roots by a factor of 1/k. Reflection in the y-axis, y = f(−x), changes each root r to −r.
y = f(kx) 水平压缩或拉伸，根变为原来的 1/k。关于 y 轴对称的 y = f(−x) 将每个根 r 变为 −r。
When solving f(x) = g(x), sketch both graphs. The x-coordinates of their intersection points are the roots. This graphical perspective often reveals how many solutions exist and whether they are positive or negative.
在解 f(x) = g(x) 时，可以同时画出两个图像。交点对应的 x 坐标就是根。这种图像视角常常能揭示方程有多少个解，以及解是正还是负。
10. Applications and Modelling | 应用与建模
Locating roots appears in optimisation problems, kinematics, and economic models. For example, finding the time when an object returns to its starting position requires solving a displacement equation.
根的定位常见于优化问题、运动学和经济学模型中。例如，求解物体回到起始位置的时间就需要解位移方程。
Example: The volume of a box made by cutting squares of side x from a 10 cm by 20 cm sheet is V(x) = x(10 − 2x)(20 − 2x). To find the value of x that gives V = 100, you need to solve 4x³ − 60x² + 200x − 100 = 0.
示例：从一张 10 cm × 20 cm 的纸板上裁去边长为 x 的正方形，所制盒子的体积为 V(x) = x(10 − 2x)(20 − 2x)。要求使 V = 100 的 x，就需要解方程 4x³ − 60x² + 200x − 100 = 0。
You can use a sign-change search to locate each root, then refine with Newton-Raphson or bisection. Always check that the root lies in the practical domain, here 0 < x < 5.
你可以使用符号变化搜索来定位每个根，然后用牛顿法或二分法进行细化。始终要检查根是否位于实际可行域内，此处为 0 < x < 5。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: Assuming a sign change is necessary for a root. A double root shows no sign change. Always consider multiplicity.
错误一：认为根的附近一定有符号变化。二重根并不会改变符号。务必考虑重数。
Mistake 2: Using Newton-Raphson when f′(xₙ) = 0. This causes division by zero in the formula.
错误二：在 f′(xₙ) = 0 时使用牛顿法。这会导致公式中出现除以零。
Mistake 3: Forgetting to check continuity before applying the sign-change rule. A discontinuity can produce a sign change without a root.
错误三：在应用符号变化规则之前忘记检查连续性。不连续也可能导致符号变化，但并没有根。
Exam tips: When asked to “show that a root lies between a and b”, simply evaluate f(a) and f(b) and state that f is continuous. When using an iteration, always write the formula clearly and show at least two iterations. Give answers to the specified number of decimal places.
考试技巧：当题目要求“证明根位于 a 与 b 之间”时，只需计算 f(a) 和 f(b) 并说明 f 连续。使用迭代时，一定要清楚写出迭代公式，并至少展示两次迭代。答案要按题目要求保留小数位数。
12. Conclusion | 总结
Locating roots is a fundamental skill in A-Level Mathematics. The sign-change rule and the Intermediate Value Theorem provide a rigorous way to prove existence, while the bisection method and Newton-Raphson method give practical ways to approximate roots accurately.
根的定位是 A-Level 数学中的基本技能。符号变化规则与中间值定理为证明根的存在提供了严谨的方法，而二分法和牛顿-拉弗森法则为精确近似根提供了实用的工具。
Graph analysis complements these techniques by giving you a visual understanding of where roots lie and how many there are. By combining algebraic, numerical and graphical approaches, you can solve a wide range of equations confidently.
图像分析则通过直观地展示根的位置与数量，为这些技巧提供了补充。将代数、数值与图像的方法结合起来，你就能自信地解决各种类型的方程问题。
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