📚 Mastering Algebra and Functions for IB and CCEA Mathematics | IB 与 CCEA 数学代数与函数考点精讲
Algebra and functions form the bedrock of any advanced mathematics curriculum. Whether you are tackling the IB Analysis & Approaches or the CCEA GCE Mathematics specification, fluency in manipulating algebraic expressions and interpreting functional relationships is essential. This guide distils the key concepts, techniques and problem‑solving strategies that will help you build confidence and accuracy.
代数和函数是任何高等数学课程的基石。无论你学习的是 IB 分析与方法还是 CCEA GCE 数学，熟练地处理代数式并解读函数关系都至关重要。本指南提炼了最核心的概念、技巧和解题策略，帮助你建立信心、提高准确性。
1. Polynomial Expressions and Operations | 多项式表达式与运算
A polynomial in one variable x is an expression aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where n is a non‑negative integer and the coefficients aᵢ are real numbers. The degree, leading coefficient and constant term reveal its behaviour.
关于单个变量 x 的多项式是形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ 的表达式，其中 n 为非负整数，系数 aᵢ 为实数。次数、首项系数和常数项决定了它的基本特征。
Adding and subtracting polynomials relies on collecting like terms. Multiplying polynomials uses the distributive law, often organised with a grid or by careful expansion of brackets.
多项式的加减法依赖于合并同类项。多项式乘法则运用分配律，常常借助网格法或仔细地展开括号来完成。
Division of a polynomial by a linear factor can be performed via long division or synthetic division. The Remainder Theorem states that when P(x) is divided by (x − c), the remainder is P(c). The Factor Theorem then tells us (x − c) is a factor if and only if P(c) = 0.
多项式除以线性因式可用长除法或综合除法进行。余式定理指出，当 P(x) 除以 (x − c) 时，余数为 P(c)。因式定理则告诉我们，(x − c) 是 P(x) 的因式当且仅当 P(c) = 0。
2. Factorisation Techniques | 因式分解技巧
Factorising higher‑degree polynomials often begins with recognising common factors, grouping terms, or spotting standard patterns such as difference of squares a² − b² = (a − b)(a + b) and sum/difference of cubes.
高次多项式的因式分解通常始于提取公因式、分组分解，或者识别标准的模式，例如平方差 a² − b² = (a − b)(a + b) 以及立方和/立方差。
For quadratics x² + bx + c, we look for two numbers that multiply to c and add to b. When the leading coefficient is not 1, the ‘ac’ method or trial and error with brackets is used.
对于二次式 x² + bx + c，我们需要找到两个数，乘积为 c 且和为 b。当首项系数不是 1 时，则可采用 “ac” 方法或试探括号内的一次项。
Repeated application of the Factor Theorem combined with polynomial division allows complete factorisation into linear and irreducible quadratic factors. This skill is essential for solving polynomial equations and sketching graphs.
反复应用因式定理并结合多项式除法，可将多项式完全分解为线性因式和不可约二次因式的乘积。这一技能对求解多项式方程和绘制函数图像都不可或缺。
3. Quadratic Functions and Their Graphs | 二次函数及其图像
A quadratic function has the general form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a parabola that opens upwards if a > 0 and downwards if a < 0.
二次函数的一般形式为 f(x) = ax² + bx + c，其中 a ≠ 0。它的图像是一条抛物线，当 a > 0 时开口向上，当 a < 0 时开口向下。
The y‑intercept is (0, c). The x‑intercepts, or roots, are found by solving f(x) = 0. The axis of symmetry is the vertical line x = −b/(2a), and the vertex lies on this line.
y 轴截距为 (0, c)。x 轴截距（即根或零点）可通过求解 f(x) = 0 得到。对称轴是垂直直线 x = −b/(2a)，顶点就位于该直线上。
The discriminant Δ = b² − 4ac determines the nature of the roots: two distinct real roots for Δ > 0, one repeated real root for Δ = 0, and no real roots for Δ < 0 (two complex conjugates).
判别式 Δ = b² − 4ac 决定了根的性质：Δ > 0 时有两个相异实根，Δ = 0 时有一个重根，Δ < 0 时没有实根（两个共轭复数根）。
4. Completing the Square and Vertex Form | 配方法与顶点式
Completing the square transforms the standard quadratic into the vertex form f(x) = a(x − h)² + k, where (h, k) is the turning point of the parabola.
配方法将标准二次式转化为顶点式 f(x) = a(x − h)² + k，其中 (h, k) 为抛物线的顶点。
The process for x² + bx is to add and subtract (b/2)². When a ≠ 1, first factor a from the first two terms, then complete the square inside the bracket, remembering to balance the constant term.
对 x² + bx 形式的配方步骤是加上并减去 (b/2)²。当 a ≠ 1 时，从首两项中提取因子 a，然后在括号内配方，记得调整常数项以保持恒等。
Vertex form instantly reveals the maximum or minimum value of a quadratic function and the coordinates for the optimum. It is also the gateway to understanding function transformations.
顶点式可以立刻呈现二次函数的最大值或最小值以及达到最值时的坐标。它也是理解函数变换的入门钥匙。
5. Solving Quadratic Equations | 解二次方程
Quadratic equations ax² + bx + c = 0 can be solved by factorising, completing the square, or applying the quadratic formula: x = [−b ± √(b² − 4ac)] / (2a).
二次方程 ax² + bx + c = 0 可通过因式分解、配方法或使用求根公式 x = [−b ± √(b² − 4ac)] / (2a) 来求解。
When solving by factorising, set the equation to zero, factorise the left‑hand side, and then apply the zero‑product property: if pq = 0 then p = 0 or q = 0.
采用因式分解法时，先将方程设为零，再将左边因式分解，然后应用零乘积性质：若 pq = 0，则 p = 0 或 q = 0。
Equations that are not initially quadratic can sometimes be reduced via substitution, e.g. an equation involving x⁴, x² and a constant can be turned into a quadratic in t = x².
有些方程最初并非二次方程，但可以通过代换转化，例如包含 x⁴、x² 和常数的方程可令 t = x² 化归为二次方程。
Always check for hidden restrictions, such as denominators or even‑index radicals, and verify solutions in the original equation.
务必检查隐藏的限制条件，例如分母或偶次根式，并将所得解代入原方程进行检验。
6. Inequalities and Sign Diagrams | 不等式与符号图
To solve a quadratic inequality such as ax² + bx + c > 0, first find the real roots (if any). Next construct a sign diagram by testing intervals between the roots.
求解二次不等式如 ax² + bx + c > 0 时，首先求出实根（如果有的话），然后通过在根之间的区间上取测试点来构建符号图。
The sign of a polynomial changes only at roots of odd multiplicity. Roots of even multiplicity simply touch the x‑axis without crossing, leaving the sign unchanged.
多项式仅在奇数重根处改变符号。偶数重根只与 x 轴相切而不穿越，符号保持不变。
For rational inequalities like (x−a)/(x−b) ≤ 0, identify values that make the numerator or denominator zero, place them on a number line, and test each interval. Remember to exclude values that make the denominator zero.
对于有理式不等式，如 (x−a)/(x−b) ≤ 0，找出使分子或分母为零的值，将它们标在数轴上，并对每个区间进行测试。切记要剔除使分母为零的值。
7. Functions: Domain and Range | 函数的定义域与值域
A function f: X → Y assigns exactly one output in Y to each input in X. The domain is the set of all permissible inputs; the range is the set of all actual outputs.
函数 f: X → Y 将 X 中的每个输入对应到 Y 中唯一的一个输出。定义域是所有允许的输入值的集合；值域是所有实际输出值的集合。
For algebraic functions, domain restrictions arise from denominators (can’t be zero) and even roots (radicand must be ≥ 0). Logarithmic functions require strictly positive arguments.
对于代数函数，定义域的限制通常来自分母（不能为零）和偶次根式（被开方数必须 ≥ 0）。对数函数要求真数严格为正。
Range is often found by considering the behaviour of the function, including asymptotes, turning points, and end behaviour. Graphical analysis is a powerful tool.
值域通常通过考察函数的行为来确定，包括渐近线、极值点和末端趋势。图像分析是强有力的工具。
8. Composite and Inverse Functions | 复合函数与反函数
The composite function f(g(x)), written f ∘ g, means applying g first and then f. The domain of f ∘ g is the set of x in the domain of g such that g(x) is in the domain of f.
复合函数 f(g(x))，记作 f ∘ g，表示先作用 g 再作用 f。f ∘ g 的定义域是 g 的定义域中那些使得 g(x) 落在 f 的定义域内的 x 的集合。
An inverse function f⁻¹ undoes the action of f: if f(x) = y then f⁻¹(y) = x. For f⁻¹ to exist, f must be one‑one (pass the horizontal line test).
反函数 f⁻¹ 可以撤销 f 的操作：若 f(x) = y，则 f⁻¹(y) = x。反函数存在的条件是 f 必须是一对一的（通过水平线检验）。
To find an inverse algebraically, write y = f(x), swap x and y, and then solve for y. The domain of f⁻¹ equals the range of f, and vice versa.
用代数方法求反函数时，先写出 y = f(x)，交换 x 和 y，然后解出 y。f⁻¹ 的定义域等于 f 的值域，反之亦然。
9. Transformations of Functions | 函数变换
Transformations allow us to sketch related functions from a known base graph. The most common are translations, stretches, and reflections.
利用函数变换，我们可以从已知的基础图像绘制出相关函数的图像。最常见的变化包括平移、伸缩和反射。
For a constant k > 0:
- y = f(x) + k shifts the graph up by k (vertical translation).
- y = f(x + k) shifts the graph left by k (horizontal translation).
- y = a f(x) stretches vertically by factor |a|; if a is negative it also reflects in the x‑axis.
- y = f(bx) compresses horizontally by factor 1/|b|; if b is negative it also reflects in the y‑axis.
对于常数 k > 0：
- y = f(x) + k 将图像向上平移 k 个单位（垂直平移）。
- y = f(x + k) 将图像向左平移 k 个单位（水平平移）。
- y = a f(x) 将图像垂直伸缩至原来的 |a| 倍；若 a 为负还同时关于 x 轴反射。
- y = f(bx) 将图像水平压缩至原来的 1/|b|；若 b 为负还同时关于 y 轴反射。
When combining transformations, apply horizontal changes (inside the bracket) before vertical ones (outside). The order matters.
当组合多个变换时，先处理括号内的水平变换，再处理括号外的垂直变换。顺序非常重要。
10. Exponential and Logarithmic Functions | 指数函数与对数函数
Exponential functions of the form f(x) = a·bˣ (with b > 0, b ≠ 1) model growth and decay. The natural exponential function eˣ has a unique property: its derivative is itself.
形如 f(x) = a·bˣ（b > 0 且 b ≠ 1）的指数函数可用来描述增长与衰减。自然指数函数 eˣ 具有独特的性质：它的导数等于它自身。
Logarithms are the inverses of exponentials: logₐ y = x if and only if aˣ = y. Key rules include:
- logₐ (MN) = logₐ M + logₐ N
- logₐ (M/N) = logₐ M − logₐ N
- logₐ (Mʳ) = r·logₐ M
- Change of base: logₐ b = log꜀ b / log꜀ a
对数是指数的逆运算：logₐ y = x 当且仅当 aˣ = y。关键的运算法则有：
- logₐ (MN) = logₐ M + logₐ N
- logₐ (M/N) = logₐ M − logₐ N
- logₐ (Mʳ) = r·logₐ M
- 换底公式：logₐ b = log꜀ b / log꜀ a
Equations involving exponentials and logarithms are solved by taking logs of both sides or rewriting in exponential form. Always check the domain of logarithmic expressions (arguments > 0).
解指数方程或对数方程时，通常对两边取对数或改写为指数形式。务必检查对数表达式的定义域（真数 > 0）。
11. Rational Functions and Asymptotes | 有理函数与渐近线
A rational function is a ratio of two polynomials, f(x) = P(x)/Q(x). Its domain excludes values that make Q(x) = 0. Vertical asymptotes occur at those x‑values where the reduced form still has a zero denominator.
有理函数是两个多项式之比，f(x) = P(x)/Q(x)。其定义域不包括使 Q(x) = 0 的值。在化简后分母仍为零的 x 值处会出现垂直渐近线。
Horizontal asymptotes are determined by comparing the degrees of P and Q. If degree(P) < degree(Q), y = 0 is the asymptote. If degrees are equal, y = (leading coefficient of P)/(leading coefficient of Q). If degree(P) > degree(Q), there is no horizontal asymptote but possibly an oblique one.
水平渐近线通过比较 P 和 Q 的次数来确定。若 P 的次数 < Q 的次数，y = 0 为渐近线。若次数相等，y = (P 的首项系数)/(Q 的首项系数)。若 P 的次数 > Q 的次数，则没有水平渐近线，但可能存在斜渐近线。
To sketch rational functions, find intercepts, asymptotes, and test behaviour in each interval. Long division helps to identify the end‑behaviour function for oblique asymptotes.
描绘有理函数图像时，需要找到截距和渐近线，并在每个区间测试符号。长除法有助于确定斜渐近线所对应的末端行为函数。
12. Systems of Equations and Intersections | 方程组与交点
Solving simultaneous equations algebraically – by substitution or elimination – gives the intersection points of their graphs. For a linear–quadratic system ax + by = c and y = dx² + ex + f, substituting the linear expression into the quadratic yields a quadratic in one variable.
通过代入法或消元法解联立方程组，等同于求对应图像的交点。对于一次−二次方程构成的方程组，将一次式代入二次式可得到一个单变量的二次方程。
Graphically, intersections correspond to the real solutions of f(x) = g(x). Rearranging to f(x) − g(x) = 0 reframes the problem as finding roots of a new function. Understanding this connection is fundamental in calculus and applied contexts.
从图形上看，两个函数图像的交点对应于方程 f(x) = g(x) 的实数解。将其改写为 f(x) − g(x) = 0 即可将问题重新表述为求一个新函数的零点。理解这种联系是微积分和应用问题的基础。
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