📚 Year 9 WJEC Further Maths: Formula & Theorem Quick Reference Guide | 公式定理速查手册
This quick reference guide collects the essential formulas and theorems required for Year 9 WJEC Further Mathematics. Use it to revise algebra, geometry, trigonometry, vectors, matrices, sequences, probability and more – all presented with clear notation and paired explanations.
这本速查手册汇集了 Year 9 WJEC 进阶数学所需的基本公式与定理。无论是代数、几何、三角、向量、矩阵、数列还是概率，书中均以清晰的符号与中英对照的说明逐一呈现，方便你随时复习与查阅。
1. Algebraic Expressions and Operations | 代数表达式与运算
+ c) = ab + ac. Factorisation reverses this by taking out the highest common factor (HCF).
分配律用于展开括号：a(b + c) = ab + ac。因式分解则是逆过程，需提取最大公因数 (HCF)。
Key factorisation identities appear frequently – memorise them to save time in exams.
以下因式分解恒等式经常出现，记住它们能显著提高解题速度。
a² – b² = (a + b)(a – b)
平方差公式：a² – b² = (a + b)(a – b)
(a + b)² = a² + 2ab + b²
和的平方公式：(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
差的平方公式：(a – b)² = a² – 2ab + b²
When working with algebraic fractions, always look for a common denominator before adding or subtracting.
进行代数分式加减时，务必先找到公分母再运算。
2. Linear Equations and Straight Lines | 线性方程与直线
form of a linear equation is y = mx + c, where m is the gradient and c is the y-intercept.
直线方程的一般形式为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
Gradient between two points (x₁, y₁) and (x₂, y₂): m = (y₂ – y₁) / (x₂ – x₁).
两点间斜率公式：m = (y₂ – y₁) / (x₂ – x₁)。
Equation of a line given a point and gradient: y – y₁ = m(x – x₁).
已知一点和斜率求直线方程：y – y₁ = m(x – x₁)。
To solve simultaneous linear equations, you can use substitution or elimination. Graphically, the solution is the point of intersection.
解联立线性方程可使用代入法或消元法；从图形上看，解即为两条直线的交点坐标。
3. Quadratic Equations and the Quadratic Formula | 二次方程与求根公式
A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. Its graph is a parabola.
二次方程标准形式为 ax² + bx + c = 0（a ≠ 0），其图像是一条抛物线。
x = (-b ± √b² – 4ac) / (2a)
求根公式：x = (-b ± √b² – 4ac) / (2a)
The discriminant Δ = b² – 4ac determines the nature of the roots:
判别式 Δ = b² – 4ac 可判断根的性质：
Δ > 0 → two distinct real roots
Δ > 0 → 两个相异实根
Δ = 0 → one repeated real root
Δ = 0 → 一个重根（两个相等实根）
Δ < 0 → no real roots (complex roots)
Δ < 0 → 无实根（复数根）
The vertex of the parabola y = a(x – h)² + k is (h, k). Completing the square rewrites the quadratic in this form.
抛物线 y = a(x – h)² + k 的顶点坐标为 (h, k)。配方法可将一般式化为顶点式。
4. Indices and Surds | 指数与根式
The laws of indices are fundamental for simplifying algebraic expressions. They are summarised below.
指数法则是化简代数表达式的基石，下表列出核心法则。
| Rule | Formula |
|---|---|
| Product of powers | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Quotient of powers | aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ |
| Power of a product | (ab)ᵐ = aᵐ bᵐ |
| Zero exponent | a⁰ = 1 (a ≠ 0) |
| Negative exponent | a⁻ⁿ = 1 / aⁿ |
Surds are expressions containing a root symbol. The key rules are:
根式是含有根号的表达式，其主要运算法则为：
√ab = √a × √b
√ab = √a × √b
√a / b = √a / √b (b ≠ 0)
√a / b = √a / √b（b ≠ 0）
Rationalising a denominator like 1/√a involves multiplying numerator and denominator by √a.
将分母有理化，例如 1/√a，需分子分母同乘 √a。
5. Sequences and Series | 数列与级数
An arithmetic sequence has a common difference d. The nth term is given by aₙ = a₁ + (n – 1)d.
等差数列有公差 d，第 n 项为 aₙ = a₁ + (n – 1)d。
Sum of the first n terms of an arithmetic series: Sₙ = n/2 (2a₁ + (n – 1)d) or Sₙ = n/2 (a₁ + aₙ).
等差数列前 n 项和：Sₙ = n/2 (2a₁ + (n – 1)d) 或 Sₙ = n/2 (a₁ + aₙ)。
A geometric sequence has a common ratio r. The nth term is aₙ = a₁ rⁿ⁻¹.
等比数列有公比 r，第 n 项为 aₙ = a₁ rⁿ⁻¹。
Sum of the first n terms of a geometric series (r ≠ 1): Sₙ = a₁ (1 – rⁿ) / (1 – r).
等比数列前 n 项和（r ≠ 1）：Sₙ = a₁ (1 – rⁿ) / (1 – r)。
If |r| < 1, the infinite geometric series converges: S∞ = a₁ / (1 – r).
当 |r| < 1 时，无穷等比级数收敛：S∞ = a₁ / (1 – r)。
6. Geometry: Triangles, Pythagoras & Trigonometry | 几何：三角形、勾股定理与三角学
In a right-angled triangle, Pythagoras' theorem states: a² + b² = c², where c is the hypotenuse.
在直角三角形中，勾股定理：a² + b² = c²，其中 c 为斜边。
The three basic trigonometric ratios are:
三个基本三角比定义如下：
sin θ = opposite / hypotenuse
sin θ = 对边 / 斜边
cos θ = adjacent / hypotenuse
cos θ = 邻边 / 斜边
tan θ = opposite / adjacent
tan θ = 对边 / 邻边
Exact trigonometric values for common angles should be memorised:
常见特殊角的精确三角值需牢记：
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
For any triangle, the area can be found using: Area = ½ ab sin C.
任意三角形面积公式：面积 = ½ ab sin C。
The sine rule: a / sin A = b / sin B = c / sin C.
正弦定理：a / sin A = b / sin B = c / sin C。
The cosine rule: a² = b² + c² – 2bc cos A.
余弦定理：a² = b² + c² – 2bc cos A。
7. Circles: Area, Circumference and Theorems | 圆：面积、周长与定理
For a circle with radius r (diameter d = 2r):
对于半径为 r（直径 d = 2r）的圆：
Circumference C = 2πr = πd
周长 C = 2πr = πd
Area A = πr²
面积 A = πr²
Arc length and sector area when the central angle is θ degrees:
当圆心角为 θ 度时，弧长与扇形面积计算如下：
Arc length L = (θ / 360) × 2πr
弧长 L = (θ / 360) × 2πr
Sector area A = (θ / 360) × πr²
扇形面积 A = (θ / 360) × πr²
Important circle theorems include:
重要的圆定理包括：
The angle subtended by a diameter at the circumference is 90°.
直径所对的圆周角为 90°。
Angles in the same segment are equal.
同弧上的圆周角相等。
The angle at the centre is twice the angle at the circumference (same arc).
圆心角等于同一弧上圆周角的两倍。
Opposite angles of a cyclic quadrilateral sum to 180°.
圆内接四边形对角互补（和为 180°）。
8. Vectors and Matrices | 向量与矩阵
A vector is written as (x y) or xi + yj. The magnitude (length) of a vector v = (x y) is |v| = √x² + y².
向量可记作 (x y) 或 xi + yj。向量 v = (x y) 的模长为 |v| = √x² + y²。
Vector operations: addition a + b = (x₁+x₂, y₁+y₂), subtraction a – b, and scalar multiplication k a = (kx, ky).
向量运算：加法 a + b = (x₁+x₂, y₁+y₂)，减法 a – b，以及数乘 k a = (kx, ky)。
action are performed element-wise. Matrix multiplication requires rows of the first to multiply columns of the second.
对于 2×2 矩阵，加减法是对应元素相加减；矩阵乘法要求前一矩阵的行乘以后一矩阵的列。
Determinant of matrix A = [[a, b], [c, d]]: det A = ad – bc.
矩阵 A = [[a, b], [c, d]] 的行列式：det A = ad – bc。
If det A ≠ 0, the inverse is A⁻¹ = (1 / det A) [[d, -b], [-c, a]].
若 det A ≠ 0，则逆矩阵 A⁻¹ = (1 / det A) [[d, -b], [-c, a]]。
9. Probability and Statistics | 概率与统计
number of favourable outcomes / total number of outcomes, provided all outcomes are equally likely.
事件 A 的概率为 P(A) = 有利结果数 / 总结果数（当所有结果等可能时）。
For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
若事件 A 与 B 互斥，则 P(A 或 B) = P(A) + P(B)。
For independent events, P(A and B) = P(A) × P(B). The probability of A given B is P(A|B) = 0 for mutually exclusive events.
若事件相互独立，则 P(A 且 B) = P(A) × P(B)。条件概率 P(A|B) 在互斥事件中为 0。
Statistical measures for a data set include:
数据集的统计量包括：
Mean (average) = sum of all data values / number of values
平均数 = 所有数据值之和 / 数据个数
Median = middle value when data are ordered
中位数 = 排序后中间位置的值
Mode = most frequent value
众数 = 出现次数最多的值
Range = maximum – minimum
极差 = 最大值 – 最小值
10. Transformations of Functions and Graphs | 函数图像变换
Given a function y = f(x), common graph transformations can be applied:
对于函数 y = f(x)，常见的图像变换如下：
y = f(x) + k translates the graph vertically by k units (up if k > 0, down if k < 0).
y = f(x) + k 使图像垂直平移 k 个单位（k > 0 向上，k < 0 向下）。
y = f(x + h) translates the graph horizontally by –h units (left if h > 0, right if h < 0).
y = f(x + h) 使图像水平平移 –h 个单位（h > 0 向左，h < 0 向右）。
y = –f(x) reflects the graph in the x-axis; y = f(–x) reflects it in the y-axis.
y = –
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