KS3 WJEC Additional Mathematics: Formula & Theorem Quick Reference | KS3 WJEC 进阶数学：公式定理速查手册

📚 KS3 WJEC Additional Mathematics: Formula & Theorem Quick Reference | KS3 WJEC 进阶数学：公式定理速查手册

This quick-reference guide summarises the essential formulae and theorems required for the WJEC Level 2 Certificate in Additional Mathematics (often studied in KS3/KS4). It covers algebra, functions, coordinate geometry, trigonometry, introductory calculus, vectors, and sequences. Use it to check your memory before exams and to strengthen your problem-solving routine.

本速查手册汇总了 WJEC Level 2 Additional Mathematics（通常在 KS3/KS4 阶段学习）所必需的关键公式与定理，涵盖代数、函数、坐标几何、三角学、微积分入门、向量与数列。可在考前用来检验记忆并强化解题套路。

1. Algebraic Foundations | 代数基础

The laws of indices and surds form the backbone of algebraic manipulation. For any non‑zero real numbers a, b and rational exponents m, n:

指数律与根式运算是代数运算的基础。对于任意非零实数 a、b 及有理指数 m、n：

• aᵐ × aⁿ = aᵐ⁺ⁿ — aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ — aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ — (aᵐ)ⁿ = aᵐⁿ
• a⁻ⁿ = 1/aⁿ — a⁻ⁿ = 1/aⁿ
• a^(1/n) = √[n]a — a^(1/n) = n次根号a
• (ab)ⁿ = aⁿ bⁿ — (ab)ⁿ = aⁿ bⁿ

When simplifying surds, use √(ab) = √a √b and rationalise denominators, e.g. 1/(√a + b) → multiply numerator and denominator by (√a − b).

化简根式时使用 √(ab) = √a √b，并对分母进行有理化，例如 1/(√a + b) → 分子分母同乘 (√a − b)。

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2. Quadratic Expressions and Equations | 二次式与二次方程

A quadratic equation ax² + bx + c = 0 (a ≠ 0) can be solved by factorising, completing the square, or using the quadratic formula. The discriminant Δ = b² − 4ac determines the nature of the roots.

二次方程 ax² + bx + c = 0 (a ≠ 0) 可通过因式分解、配方法或求根公式求解。判别式 Δ = b² − 4ac 决定根的性质。

• Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a)
• 求根公式： x = [−b ± √(b² − 4ac)] / (2a)
• Discriminant: Δ > 0 → two distinct real roots; Δ = 0 → one repeated real root; Δ < 0 → no real roots.
• 判别式： Δ > 0 → 两个不等实根；Δ = 0 → 一个重根；Δ < 0 → 无实根。
• Completing the square: x² + bx = (x + b/2)² − (b/2)²
• 配方法： x² + bx = (x + b/2)² − (b/2)²

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3. Inequalities | 不等式

Solving linear and quadratic inequalities requires careful attention when multiplying or dividing by a negative number – the inequality sign reverses.

解一次和二次不等式时，若乘以或除以负数，不等号方向必须反转。

• Linear: ax + b > c ⇒ ax > c − b; if a < 0, reverse sign when dividing.
• 一次不等式：ax + b > c ⇒ ax > c − b；若 a < 0，除以 a 时不等号反向。
• Quadratic: solve ax² + bx + c > 0 by finding critical values and testing intervals or using a sketch graph. The solution is typically “x < α or x > β”.
• 二次不等式：解 ax² + bx + c > 0 先求临界值，再测试区间或画示意图。解通常为 “x < α 或 x > β”。

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4. Functions and Graphs | 函数与图像

Understanding function notation f(x), domain, range, and transformations is key. The main transformations on y = f(x) are:

理解函数记号 f(x)、定义域、值域与图像变换至关重要。y = f(x) 的主要变换如下：

• f(x) + a: vertical translation by a — 上移 a
• f(x + a): horizontal translation by −a — 左移 a
• a f(x): vertical stretch by factor a — 纵向拉伸 a 倍
• f(ax): horizontal stretch by factor 1/a — 横向压缩至 1/a
• −f(x): reflection in x‑axis — 关于 x 轴对称
• f(−x): reflection in y‑axis — 关于 y 轴对称

The parabola y = a(x − h)² + k has vertex (h, k) and axis of symmetry x = h.

二次函数 y = a(x − h)² + k 的顶点为 (h, k)，对称轴为 x = h。

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5. Indices, Exponentials and Logarithms | 指数、指数函数与对数

For a > 0, a ≠ 1, the exponential function aˣ and its inverse logₐ(x) satisfy the following identities:

对于 a > 0, a ≠ 1，指数函数 aˣ 与其反函数 logₐ(x) 满足以下恒等式：

• logₐ(xy) = logₐx + logₐy
• logₐ(x/y) = logₐx − logₐy
• logₐ(xⁿ) = n logₐx
• logₐa = 1, logₐ1 = 0
• Change of base: logₐb = log_c b / log_c a
• 换底公式：logₐb = log_c b / log_c a

The natural exponential y = eˣ and natural logarithm ln x = logₑx are central; ln(eˣ) = x and e^(ln x) = x.

自然指数 y = eˣ 与自然对数 ln x = logₑx 为核心；ln(eˣ) = x, e^(ln x) = x。

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6. Coordinate Geometry and Straight Lines | 坐标几何与直线

Key formulae for points and lines in the plane:

平面上点与直线的核心公式：

• Midpoint of (x₁, y₁) and (x₂, y₂): ((x₁+x₂)/2, (y₁+y₂)/2)
• 中点坐标：((x₁+x₂)/2, (y₁+y₂)/2)
• Gradient m = (y₂ − y₁)/(x₂ − x₁)
• 斜率 m = (y₂ − y₁)/(x₂ − x₁)
• Equation of a line: y − y₁ = m (x − x₁) → y = mx + c
• 直线方程：y − y₁ = m (x − x₁) → y = mx + c
• Parallel lines: m₁ = m₂. Perpendicular lines: m₁ × m₂ = −1.
• 平行：m₁ = m₂；垂直：m₁ × m₂ = −1。

The distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].

两点间距离：√[(x₂ − x₁)² + (y₂ − y₁)²]。

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7. Trigonometry | 三角学

Basic trigonometric ratios in a right‑angled triangle and exact values for standard angles:

直角三角形中的基本三角比及特殊角的精确值：

• sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
• sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边

Exact values table | 精确值表

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	½	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	½	√3
90°	1	0	undefined

Pythagorean identity: sin²θ + cos²θ = 1. Also, tan θ = sin θ / cos θ.

毕达哥拉斯恒等式：sin²θ + cos²θ = 1。此外 tan θ = sin θ / cos θ。

The sine rule: a/sin A = b/sin B = c/sin C. The cosine rule: a² = b² + c² − 2bc cos A. Area of triangle = ½ ab sin C.

正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A。三角形面积 = ½ ab sin C。

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8. Introductory Calculus – Differentiation | 微积分入门——微分

The derivative of f(x) is f'(x) = lim_{h→0} [f(x+h) − f(x)]/h. The power rule is the primary tool: if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹.

导数的定义为 f'(x) = lim_{h→0} [f(x+h) − f(x)]/h。幂函数法则为基本工具：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。

• Constant multiple: d/dx [c·f(x)] = c·f'(x)
• 常数倍法则：d/dx [c·f(x)] = c·f'(x)
• Sum/difference: d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
• 和/差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

The second derivative f”(x) determines concavity. At a stationary point f'(x) = 0, examine f”(x): if f”(x) > 0 → minimum; if f”(x) < 0 → maximum.

二阶导数 f”(x) 确定凹凸性。在驻点 f'(x) = 0 处检查 f”(x)：若 f”(x) > 0 为极小值点；若 f”(x) < 0 为极大值点。

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9. Introductory Calculus – Integration | 微积分入门——积分

Integration is the reverse of differentiation. The indefinite integral of xⁿ is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1.

积分是微分的逆运算。xⁿ 的不定积分为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。

• ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
• ∫ k·f(x) dx = k ∫ f(x) dx
• Definite integral ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x).
• 定积分 ∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F'(x) = f(x)。

The area under a curve y = f(x) between x = a and x = b is given by the definite integral. Area between curves: top – bottom integrated.

曲线 y = f(x) 下方、x = a 到 x = b 之间的面积由定积分给出。曲线间的面积：上方减下方积分。

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10. Vectors | 向量

A vector has both magnitude and direction. In two dimensions, a vector can be written as (x, y) or xi + yj.

向量既有大小又有方向。在二维中，向量可记为 (x, y) 或 xi + yj。

• Addition: (a, b) + (c, d) = (a+c, b+d)
• Scalar multiplication: k (a, b) = (ka, kb)
• Magnitude |v| = √(x² + y²)
• Unit vector: v / |v|
• Position vector of A relative to O: OA⃗; AB⃗ = OB⃗ − OA⃗
• A 相对于 O 的位置向量：OA⃗；AB⃗ = OB⃗ − OA⃗

Collinear points have position vectors that are scalar multiples of each other.

共线点的位置向量互为标量倍数。

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11. Sequences and Series | 数列与级数

Arithmetic sequences: n‑th term uₙ = a + (n−1)d, sum of first n terms Sₙ = n/2 [2a + (n−1)d] or n/2 (a + l). l is the last term.

等差数列：第 n 项 uₙ = a + (n−1)d，前 n 项和 Sₙ = n/2 [2a + (n−1)d] 或 n/2 (a + l)，其中 l 为末项。

Geometric sequences: uₙ = a rⁿ⁻¹, sum of first n terms Sₙ = a (1 − rⁿ)/(1 − r) for r ≠ 1. For an infinite convergent geometric series (|r| < 1), sum to infinity S∞ = a/(1−r).

等比数列：uₙ = a rⁿ⁻¹，前 n 项和 Sₙ = a (1 − rⁿ)/(1 − r)，r ≠ 1。对于无穷收敛等比级数 (|r| < 1)，无穷和 S∞ = a/(1−r)。

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12. Key Theorems and Reminders | 核心定理与备忘

Keep these theorems in mind for geometric and algebraic problem solving:

牢记以下定理，用于几何与代数解题：

• Circle theorems: angle at centre is twice angle at circumference; angle in a semicircle is 90°; tangents from a point are equal in length.
• 圆定理：圆心角是圆周角的两倍；直径所对圆周角为 90°；圆外一点到圆的两条切线长相等。
• Binomial expansion: (a + b)ⁿ = Σ [n choose k] aⁿ⁻ᵏ bᵏ, where [n choose k] = n!/(k!(n−k)!).
• 二项式展开：(a + b)ⁿ = Σ [n choose k] aⁿ⁻ᵏ bᵏ，其中 [n choose k] = n!/(k!(n−k)!)。
• Completing the square for quadratics and circles: x² + y² + 2gx + 2fy + c = 0 → centre (−g, −f), radius √(g² + f² − c).
• 配方法求圆方程：x² + y² + 2gx + 2fy + c = 0 → 圆心 (−g, −f)，半径 √(g² + f² − c)。

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