📚 Algebra and Functions: Key Topic Review for IB & Edexcel Mathematics | 代数和函数：IB 与 Edexcel 数学考点精讲
Algebra and functions form the backbone of both IB Diploma Programme Mathematics (Analysis & Approaches and Applications & Interpretation) and Edexcel A Level Mathematics. A solid command of polynomial manipulation, equation solving, function properties, and transformations is essential for tackling advanced topics such as calculus, trigonometry, and statistics. This guide consolidates the key concepts, common pitfalls, and strategic approaches required by these curricula, ensuring you build a robust foundation whether you are preparing for IB Paper 1/2 or Edexcel Pure Mathematics papers.
代数和函数是 IB 文凭课程数学（分析与方法、应用与解释）以及 Edexcel A Level 数学的核心支柱。扎实掌握多项式运算、方程求解、函数性质与图像变换，是攻克微积分、三角学、统计等进阶内容的基础。本指南整合了两个课程体系共同要求的核心概念、常见易错点和解题策略，无论你备考 IB 试卷还是 Edexcel Pure 试卷，都能帮助你构建牢固的知识框架。
1. Polynomials and Factorisation | 多项式与因式分解
A polynomial in x is an expression of the form anxⁿ + an-1xⁿ⁻¹ + … + a₁x + a₀, where n is a non‑negative integer. Key factorisation methods include taking out the greatest common factor, grouping terms, and applying the difference of two squares: a² – b² = (a – b)(a + b).
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。核心因式分解方法包括提取最大公因式、分组分解以及运用平方差公式：a² – b² = (a – b)(a + b)。
The Factor Theorem is crucial for cubic or higher‑degree polynomials: if f(p) = 0, then (x – p) is a factor. Once one factor is found, polynomial division (long division or synthetic division) reduces the degree, allowing further factorisation.
因式定理对三次及更高次多项式至关重要：若 f(p) = 0，则 (x – p) 是一个因式。找到一个因式后，通过多项式除法（长除法或综合除法）降次，可实现进一步分解。
Both IB and Edexcel may ask students to factorise fully and hence solve polynomial equations. Remember that a repeated factor indicates a repeated root, which affects the shape of the graph.
IB 和 Edexcel 考试都可能要求学生彻底因式分解并据此求解多项式方程。注意重因式意味着重根，这会影响函数图像的形状。
2. Quadratic Functions and Equations | 二次函数与方程
The standard form is ax² + bx + c = 0, with solutions given by the quadratic formula: x = [–b ± √(b² – 4ac)] / 2a. Completing the square rewrites the quadratic as a(x + p)² + q, revealing the vertex (–p, q) and axis of symmetry x = –p.
标准形式为 ax² + bx + c = 0，其解由二次公式给出：x = [–b ± √(b² – 4ac)] / 2a。配方法将二次式改写为 a(x + p)² + q，由此可确定顶点 (–p, q) 与对称轴 x = –p。
The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots (two complex conjugates in further maths). Sum and product of roots α + β = –b/a and αβ = c/a are useful for forming equations and symmetry arguments.
判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个重实根；Δ < 0 无实根（进阶数学中为一对共轭复根）。根的和与积 α + β = –b/a，αβ = c/a 常用于构造方程和对称性推理。
Quadratic inequalities such as ax² + bx + c > 0 are solved by sketching the parabola and identifying intervals where the curve lies above or below the x‑axis. Always pay attention to whether the inequality is strict or inclusive.
二次不等式如 ax² + bx + c > 0 的解法是绘制抛物线草图，找出曲线在 x 轴上方或下方的区间。务必注意不等号是否带等号。
3. Exponentials and Logarithms | 指数与对数
Exponential functions are of the form f(x) = aˣ with a > 0, a ≠ 1. The natural base e ≈ 2.718 is fundamental. Logarithms are the inverses of exponentials: if y = aˣ, then x = logₐ y. The natural logarithm ln x is logₑ x.
指数函数形式为 f(x) = aˣ，其中 a > 0, a ≠ 1。自然底数 e ≈ 2.718 是常用底数。对数是指数的逆运算：若 y = aˣ，则 x = logₐ y。自然对数 ln x 即 logₑ x。
Laws of logarithms are essential: logₐ(MN) = logₐ M + logₐ N; logₐ(M/N) = logₐ M – logₐ N; logₐ Mᵏ = k logₐ M. The change‑of‑base formula logₐ b = log b / log a often appears when solving equations.
对数运算法则是核心：logₐ(MN) = logₐ M + logₐ N；logₐ(M/N) = logₐ M – logₐ N；logₐ Mᵏ = k logₐ M。换底公式 logₐ b = log b / log a 常在解方程时出现。
In both IB and Edexcel, you must be able to solve exponential equations by taking logarithms of both sides, and to model growth/decay with functions like N(t) = N₀eᵏᵗ. Pay attention to domain restrictions: logₐ x is only defined for x > 0.
在 IB 和 Edexcel 考试中，你必须能够通过对两边取对数来求解指数方程，并能用 N(t) = N₀eᵏᵗ 等函数建立增长或衰减模型。注意定义域限制：logₐ x 仅在 x > 0 时有定义。
4. Functions and Mappings | 函数与映射
A function is a mapping that assigns exactly one output to each input. It is defined by an expression, a domain (the set of allowed inputs), and a codomain. The set of all actual outputs is the range.
函数是一种映射，它为每个输入值分配唯一一个输出值。函数由表达式、定义域（允许的输入值集合）和陪域共同定义。所有实际输出值构成的集合称为值域。
Notation varies slightly between syllabi, but f: x ↦ 2x + 1, x ∈ ℝ is standard. One‑to‑one, many‑to‑one, and onto functions are examined especially in IB Analysis & Approaches, while Edexcel focuses more on one‑to‑one and many‑to‑one.
记号在不同大纲中略有差异，但 f: x ↦ 2x + 1, x ∈ ℝ 是标准写法。一对一、多对一和映上函数在 IB 分析与方法中考察较深，而 Edexcel 更侧重一对一和多对一函数。
A function is only invertible if it is one‑to‑one. The vertical line test checks if a graph represents a function; the horizontal line test checks if the function is one‑to‑one.
只有一对一函数才可逆。垂直线检验可判断一个图像是否表示函数；水平线检验可判断函数是否为一对一。
5. Domain and Range | 定义域与值域
The domain is usually stated explicitly or determined by the formula. Common restrictions: denominators cannot be zero, arguments of even roots must be ≥ 0, and log arguments must be > 0. For piecewise functions, check the conditions for each piece.
定义域通常直接给出或通过表达式确定。常见限制有：分母不能为零，偶次根号下的被开方数需 ≥ 0，对数真数必须 > 0。对于分段函数，需检查每一段的定义条件。
To find the range, either sketch the graph or consider the extreme values of the function. For quadratics in completed‑square form, the range is determined by the vertex. For rational functions, behaviour near asymptotes is key.
求值域时，可通过绘制图像或分析函数的极值。配成完全平方形式的二次函数，可由顶点确定值域。有理函数则需关注渐近线附近的行为。
IB often asks to find the largest possible domain of a function, while Edexcel may combine domain restrictions with inverse functions or composite functions. Always express domain and range in set notation or interval notation as required.
IB 经常要求找出函数的最大可能定义域，而 Edexcel 可能将定义域限制与反函数或复合函数结合考查。务必按题目要求使用集合记号或区间记号表示定义域和值域。
6. Composite Functions | 复合函数
The composite function f ∘ g (or fg) is defined by (f ∘ g)(x) = f(g(x)). The order matters: f(g(x)) is generally not the same as g(f(x)). The domain of f ∘ g is the set of x in the domain of g such that g(x) lies in the domain of f.
复合函数 f ∘ g（或 fg）定义为 (f ∘ g)(x) = f(g(x))。顺序至关重要：f(g(x)) 通常不等于 g(f(x))。f ∘ g 的定义域是 g 的定义域中使得 g(x) 落在 f 定义域内的 x 集合。
Both boards require you to form composite functions algebraically and specify their domains. A common mistake is to evaluate the range of the inner function but forget to restrict the outer function’s domain accordingly.
两个考试局都要求通过代数方式构造复合函数并指出其定义域。常见错误是只考虑了内层函数的值域，却忘记相应地限制外层函数的定义域。
You may also be given a composite function and asked to deduce the form of one of the component functions, often by substitution or comparison of coefficients.
题目也可能给出一个复合函数，要求反推其中某一组成函数的形式，通常可通过代换或比较系数来解决。
7. Inverse Functions | 反函数
If f is a one‑to‑one function with domain A and range B, its inverse f⁻¹ has domain B and range A, and satisfies f⁻¹(f(x)) = x for x ∈ A. Graphically, y = f⁻¹(x) is the reflection of y = f(x) in the line y = x.
若 f 是一对一函数，定义域为 A、值域为 B，则其反函数 f⁻¹ 的定义域为 B、值域为 A，且满足 f⁻¹(f(x)) = x，x ∈ A。图像上，y = f⁻¹(x) 与 y = f(x) 关于直线 y = x 对称。
To find an inverse: write y = f(x), swap x and y, then solve for y. If the original function is not one‑to‑one, you must restrict the domain to make it invertible – for example, restricting y = x² to x ≥ 0.
求反函数的步骤：写出 y = f(x)，交换 x 和 y，然后解出 y。若原函数不是一一对应的，必须限制定义域使其可逆——例如将 y = x² 限制在 x ≥ 0。
IB and Edexcel both test the relationship between the domain of f and the range of f⁻¹, and vice versa. Expect to work with logarithmic and exponential inverses, as well as trigonometric inverses in later topics.
IB 和 Edexcel 都会考查 f 的定义域与 f⁻¹ 的值域之间的关系，反之亦然。后续章节还将涉及对数与指数互为反函数，以及三角反函数等内容。
8. Transformations of Functions | 函数变换
Transformations allow you to sketch related graphs quickly. The main types are translations, stretches, and reflections. Given y = f(x), the transformed graph can be described by combinations of these.
函数变换使你能够快速绘制相关函数的图像。主要类型有平移、伸缩和对称。已知 y = f(x)，变换后的图像可由这些类型的组合描述。
| Transformation (变换) | Effect on y = f(x) | Description |
|---|---|---|
| f(x + a) | Translation left by a | 向左平移 a 个单位 |
| f(x) + a | Translation up by a | 向上平移 a 个单位 |
| f(–x) | Reflection in y‑axis | 关于 y 轴对称 |
| –f(x) | Reflection in x‑axis | 关于 x 轴对称 |
| af(x) (a > 1) | Vertical stretch by factor a | 垂直方向拉伸 a 倍 |
| f(ax) (a > 1) | Horizontal compression by factor 1/a | 水平方向压缩 1/a 倍 |
Order of transformations is critical: applying horizontal shifts before stretches yields a different result than the reverse. Both syllabi frequently test combined transformations where you must identify the sequence or describe the resulting graph.
变换的顺序至关重要：先水平平移再水平伸缩与先伸缩后平移的结果不同。两个大纲都经常考查复合变换，要求你识别变换顺序或描述最终图像。
9. Rational Functions and Partial Fractions | 有理函数与部分分式
A rational function is a ratio of two polynomials, e.g., f(x) = (px + q)/(rx + s). Key features include vertical asymptotes (where denominator = 0), horizontal or oblique asymptotes, and intercepts. Curve sketching often relies on algebraic division to separate the polynomial part.
有理函数是两个多项式的比，例如 f(x) = (px + q)/(rx + s)。关键特征包括垂直渐近线（分母为零处）、水平或斜渐近线以及截距。绘制曲线时常通过代数除法分离出多项式部分。
Partial fractions decompose a complex rational expression into a sum of simpler fractions. Types include linear factors (A/(ax+b)), repeated linear factors, and irreducible quadratic factors. This technique is essential for integration and series expansion in both IB and Edexcel.
部分分式将复杂的有理式分解为若干简单分式的和。类型包括线性因式 (A/(ax+b))、重复线性因式和不可约二次因式。这一技巧在 IB 和 Edexcel 的积分与级数展开中都是必不可少的。
For example, (3x+5)/(x²+x–2) can be split into A/(x+2) + B/(x–1). Solve for A and B by equating coefficients or substituting convenient x values. Always check for improper fractions first – perform division if the numerator’s degree is ≥ denominator’s degree.
例如，(3x+5)/(x²+x–2) 可拆分为 A/(x+2) + B/(x–1)。通过比较系数或代入便捷 x 值求出 A 和 B。务必先检查是否为假分式——若分子次数 ≥ 分母次数，需先进行除法运算。
10. Modulus Functions and Inequalities | 绝对值函数与不等式
The modulus function |x| is defined as |x| = x for x ≥ 0, and |x| = –x for x < 0. Its graph is V‑shaped. Functions like |f(x)| reflect the negative parts of y = f(x) above the x‑axis, while f(|x|) reflects the right‑hand side to the left.
绝对值函数 |x| 定义为：当 x ≥ 0 时 |x| = x；当 x < 0 时 |x| = –x。其图像呈 V 形。|f(x)| 会将 y = f(x) 的负值部分翻折到 x 轴上方，而 f(|x|) 则将右侧图像对称复制到左侧。
Modulus equations and inequalities such as |ax + b| = cx + d or |x – 3| > 2 require considering separate branches. The key is to square both sides (if both sides are non‑negative) or graph the functions to see intersection points.
形如 |ax + b| = cx + d 或 |x – 3| > 2 的绝对值方程与不等式需要分情况讨论。关键技巧是两边平方（前提是两边非负）或通过图像观察交点。
In IB, modulus inequalities often link to the definition |x| < a ⇔ –a < x < a. Edexcel may embed them within function transformations. Always express solutions as intervals clearly marking inclusive and exclusive boundaries.
在 IB 中，绝对值不等式常与定义 |x| < a ⇔ –a < x < a 相关联。Edexcel 则可能将绝对值融入函数变换中考查。最后务必用区间清晰表示解集，标出开闭区间。
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