📚 A-Level WJEC Maths: Algebra and Functions Key Points | A-Level WJEC 数学：代数和函数 考点精讲
Algebra and Functions form the backbone of the WJEC A-Level Mathematics specification. This article walks you through the essential skills, from simplifying expressions and solving quadratics to understanding composite functions and graph transformations. Each key topic is broken down with clear explanations, helping you revise efficiently and avoid common exam pitfalls.
代数和函数是 WJEC A-Level 数学大纲的核心基础。本文带你梳理从化简表达式、解二次方程，到理解复合函数与图像变换的重要技能。每个关键主题都有清晰解析，助你高效复习，避开常见考试陷阱。
1. Simplifying Algebraic Expressions | 化简代数表达式
To simplify an expression, collect like terms and use the laws of indices where necessary. Always watch out for hidden negative signs and ensure brackets are expanded correctly before simplifying.
化简代数表达式时，需要合并同类项，并在必要时使用指数法则。要特别注意隐藏的负号，确保先去括号再合并。
For example, 3x² + 2x − 5 − x² + 4x + 1 simplifies to 2x² + 6x − 4 by grouping x² terms, x terms and constants separately.
例如，3x² + 2x − 5 − x² + 4x + 1 通过分别合并 x² 项、x 项和常数项，化简为 2x² + 6x − 4。
When simplifying with indices, remember am × an = am+n and (am)n = amn. Apply these before combining terms.
涉及指数化简时，记住 am × an = am+n 以及 (am)n = amn。先处理指数再合并同类项。
2. Expanding and Factorising | 展开与因式分解
Expanding brackets means multiplying each term inside the bracket by the factor outside. For double brackets, use the FOIL method (First, Outer, Inner, Last) to ensure all terms are accounted for.
展开括号意味着将括号外的因子与括号内每一项相乘。对于两个括号相乘，使用 FOIL 方法（首、外、内、尾）确保不遗漏任何项。
Factorising is the reverse process. Always begin by looking for a common factor. For quadratics of the form ax² + bx + c, find two numbers that multiply to ac and add to b, then factor by grouping.
因式分解是展开的逆过程。始终先寻找公因式。对于 ax² + bx + c 形式的二次式，找到两个乘积为 ac 且和为 b 的数，然后通过分组进行分解。
Special forms like difference of two squares: a² − b² = (a − b)(a + b) appear frequently in WJEC papers and can save time when applied correctly.
平方差公式 a² − b² = (a − b)(a + b) 在 WJEC 试卷中频繁出现，正确运用可节省大量时间。
3. Quadratic Functions | 二次函数
A quadratic function is written as f(x) = ax² + bx + c, where a ≠ 0. Its graph is a parabola. If a > 0 the parabola opens upwards; if a < 0 it opens downwards. The vertex represents the minimum or maximum point.
二次函数写作 f(x) = ax² + bx + c，其中 a ≠ 0。其图像为抛物线。a > 0 时开口向上，a < 0 时开口向下，顶点即是最小值或最大值点。
Solving quadratic equations by factorising, using the quadratic formula, or completing the square are all examinable. The discriminant Δ = b² − 4ac determines the nature of roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
用因式分解法、求根公式法或配方法解二次方程都是考点。判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
Always check the discriminant when sketching graphs or showing that a quadratic has no real solutions.
在画图或证明二次方程无实数解时，一定要检查判别式。
4. Completing the Square | 配方
Completing the square transforms ax² + bx + c into the form a(x + p)² + q, where p and q are constants. This reveals the vertex coordinates (−p, q) directly and is essential for deriving the quadratic formula.
配方法将 ax² + bx + c 变形为 a(x + p)² + q 的形式，直接展示顶点坐标 (−p, q)，也是推导求根公式的基础。
For y = x² + 6x + 5, halve the coefficient of x (6/2 = 3), square it to get 9, then write (x + 3)² − 9 + 5 = (x + 3)² − 4. The vertex is (−3, −4).
以 y = x² + 6x + 5 为例，取 x 系数的一半 6/2=3，平方得 9，于是写作 (x + 3)² − 9 + 5 = (x + 3)² − 4，顶点为 (−3, −4)。
WJEC often asks you to show that a quadratic is always positive or find minimum values by completing the square.
WJEC 常要求用配方法证明某二次式恒为正，或求其最小值。
5. Polynomials and the Factor Theorem | 多项式与因式定理
A polynomial in x is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀. The factor theorem states that if f(a) = 0, then (x − a) is a factor of f(x). This is invaluable for factorising cubics and higher-degree polynomials.
多项式是具有 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ 形式的表达式。因式定理指出，若 f(a) = 0，则 (x − a) 是 f(x) 的因式。这对于分解三次及更高次多项式至关重要。
To factorise x³ − 4x² + x + 6, test small integer values like x = 2: 2³ − 4·2² + 2 + 6 = 8 − 16 + 2 + 6 = 0, so (x − 2) is a factor. Then use long division or synthetic division to find the quadratic factor.
分解 x³ − 4x² + x + 6 时，试 x = 2：2³ − 4·2² + 2 + 6 = 0，所以 (x − 2) 是因式，然后用长除法或综合除法求出二次因式。
Remainder theorem: when f(x) is divided by (x − a), the remainder is f(a). This is often tested in WJEC C1 and C2 papers.
余式定理：当 f(x) 除以 (x − a) 时，余数为 f(a)。这在 WJEC C1 与 C2 试卷中经常出现。
6. Rational Expressions | 有理表达式
Rational expressions are fractions with polynomials in the numerator and denominator. Simplifying them involves factorising both parts and cancelling common factors. Always state restrictions on x where the denominator equals zero.
有理表达式是分子与分母均为多项式的分式。化简时需要对分子和分母分别因式分解，再约去公因式。务必标出使分母为零的 x 限制值。
Adding and subtracting rational expressions requires a common denominator. Multiply numerator and denominator appropriately, then simplify the resulting expression.
有理表达式的加减运算需要通分。适当乘以分子分母，然后化简所得表达式。
When solving equations involving rational expressions, multiply through by the common denominator to clear fractions, but always check for extraneous solutions that make the denominator zero.
在解含有理表达式的方程时，两边乘以公分母去分母，但一定要检验会令原分母为零的增根。
7. Indices and Surds | 指数与根式
The laws of indices are fundamental: am × an = am+n, am ÷ an = am−n, (am)n = amn, and a0 = 1. Negative and fractional indices represent reciprocals and roots: a−n = 1/an, a1/n = ⁿ√a.
指数法则是基础：am × an = am+n，am ÷ an = am−n，(am)n = amn，以及 a0 = 1。负指数和分数指数分别表示倒数与根式：a−n = 1/an，a1/n = ⁿ√a。
Surds are irrational numbers like √2 or √3. Simplify surds by extracting square factors: √48 = √(16 × 3) = 4√3. Rationalising the denominator means removing surds from the bottom of a fraction, e.g. 1/(2 + √3) × (2 − √3)/(2 − √3).
根式如 √2 或 √3 是无理数。化简根式需提取平方因子：√48 = √(16 × 3) = 4√3。分母有理化即消去分母中的根号，例如 1/(2 + √3) 乘以 (2 − √3)/(2 − √3)。
WJEC exam questions frequently combine surds with expanding brackets, so practice expressions like (√5 + 2)(√5 − 3).
WJEC 考题常将根式与展开括号结合，因此要熟练掌握 (√5 + 2)(√5 − 3) 这类运算。
8. Logarithms | 对数
Logarithms are the inverse of exponentiation. If ax = N, then logₐN = x. You must know the laws: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, and logₐ(xⁿ) = n logₐx.
对数是指数运算的逆运算。若 ax = N，则 logₐN = x。必须掌握对数法则：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，logₐ(xⁿ) = n logₐx。
The change-of-base formula logba = logca / logcb is useful when evaluating logarithms with an unusual base. The natural logarithm ln x uses base e, where e ≈ 2.718.
换底公式 logba = logca / logcb 在计算非常见底数的对数时很有用。自然对数 ln x 以 e 为底，e ≈ 2.718。
Equations involving exponentials are often solved by taking logs on both sides. For instance, 3x = 7 leads to x ln 3 = ln 7, so x = ln 7 / ln 3.
含指数的方程常通过两边取对数求解。例如 3x = 7 两边取自然对数得 x ln 3 = ln 7，故 x = ln 7 / ln 3。
9. Functions: Domain and Range | 函数：定义域与值域
A function f maps each input x to exactly one output f(x). The domain is the set of all possible input values; the range is the set of all possible output values. In WJEC, you must be able to write domains and ranges using set notation or interval notation.
函数 f 将每个输入 x 映射到唯一输出 f(x)。定义域是所有可能输入值的集合；值域是所有可能输出值的集合。在 WJEC 中，你必须能用集合或区间符号表示定义域和值域。
For f(x) = √(x − 2), the expression under the root must be non‑negative, so x − 2 ≥ 0, giving domain x ≥ 2. The range is f(x) ≥ 0 because square roots are always non‑negative.
对于 f(x) = √(x − 2)，根号内的式子必须非负，所以 x − 2 ≥ 0，定义域为 x ≥ 2。值域为 f(x) ≥ 0，因为算术平方根始终非负。
Restrictions such as denominators not equal to zero or arguments of logarithms being positive must always be considered when finding the domain.
分母不为零、对数的真数为正等限制在求定义域时始终需要考虑。
10. Composite and Inverse Functions | 复合函数与反函数
A composite function fg(x) means applying g first, then applying f to the result: fg(x) = f(g(x)). The order matters: fg is generally different from gf. Always check the domain of the inside function and the overall composite.
复合函数 fg(x) 表示先执行 g，再对结果执行 f：fg(x) = f(g(x))。顺序很重要，fg 通常与 gf 不同。务必检查内层函数及整体复合的定义域。
An inverse function f⁻¹ undoes the effect of f. To find it, write y = f(x), swap x and y, then solve for y. The domain of f⁻¹ is the range of f. The graphs of f and f⁻¹ are reflections in the line y = x.
反函数 f⁻¹ 会抵消 f 的效果。求反函数时，令 y = f(x)，交换 x 和 y，然后解出 y。f⁻¹ 的定义域是 f 的值域。f 与 f⁻¹ 的图像关于直线 y = x 对称。
For f(x) = 2x + 3, write y = 2x + 3, swap to get x = 2y + 3, solve: y = (x − 3)/2, so f⁻¹(x) = (x − 3)/2.
以 f(x) = 2x + 3 为例，令 y = 2x + 3，交换得 x = 2y + 3，解出 y = (x − 3)/2，故 f⁻¹(x) = (x − 3)/2。
11. Transformations of Graphs | 函数图像的变换
Graph transformations allow you to sketch new functions based on a known shape. The four basic transformations are translation, stretch, reflection, and combination. For a function y = f(x):
图像变换使你能够根据已知函数形状绘制新函数图像。四种基本变换是平移、伸缩、对称及组合变换。对于函数 y = f(x)：
- f(x) + a → vertical translation by a
- f(x) + a → 竖直平移 a
- f(x + a) → horizontal translation by −a (left if a positive)
- f(x + a) → 水平平移 −a（a 为正时向左）
- a f(x) → vertical stretch by factor a
- a f(x) → 竖直方向缩放 a 倍
- f(ax) → horizontal stretch by factor 1/a
- f(ax) → 水平方向缩放 1/a 倍
- −f(x) → reflection in the x‑axis; f(−x) → reflection in the y‑axis.
- −f(x) → 关于 x 轴对称；f(−x) → 关于 y 轴对称。
Always apply horizontal transformations inside the brackets and vertical transformations outside when combining multiple changes. The order matters: stretches/reflections before translations.
进行多重变换时，水平变换（括号内）与竖直变换（括号外）分开处理。顺序至关重要：先做缩放与对称，再平移。
12. Modelling with Functions | 函数建模
Functions are used to model real-life situations such as projectile motion, population growth, or financial interest. You may be asked to construct a function from given data, analyse its behaviour, or find maximum or minimum values using differentiation or completing the square.
函数可用于对现实生活情境建模，如抛体运动、人口增长或利息计算。考试可能要求你根据给定数据构造函数，分析其性态，或用求导或配方求最大值或最小值。
Pay close attention to the practical constraints: domains might be limited to positive numbers, whole numbers or specific time intervals. Interpret your answers in context, stating units clearly.
要密切留意实际限制：定义域可能局限于正数、整数或特定时间段。在情境中解释答案，并清晰写出单位。
Common models include linear, quadratic, cubic, exponential and trigonometric functions. Knowing the shape and properties of each helps you decide which model fits the data best.
常见模型包括线性、二次、三次、指数和三角函数。熟悉每种函数的形状和性质，有助于你判断哪种模型与数据最匹配。
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