📚 Year 8 WJEC Further Mathematics Formula & Theorem Quick Reference | 八年级 WJEC 进阶数学公式定理速查手册
This revision guide pulls together the essential formulas, theorems, and algebraic rules you will meet in Year 8 WJEC Further Mathematics. It is designed to be a quick reference when you are practising past-paper questions, completing homework, or preparing for end-of-topic assessments. Each section presents a key idea in English, immediately followed by the same explanation in Chinese, so you can strengthen your understanding in both languages.
这本复习手册汇集了你在八年级 WJEC 进阶数学中将遇到的基本公式、定理和代数法则。它旨在成为你在练习历年真题、完成家庭作业或准备单元测试时的速查工具。每个部分先用英文阐述关键概念，紧接着用中文复述，帮助你在双语环境下巩固理解。
1. Algebraic Expansion and Factorisation | 代数展开与因式分解
The distributive law is used to remove brackets. For any numbers a, b, and c: a(b + c) = ab + ac. The same rule works when there is a minus sign: a(b – c) = ab – ac.
分配律用于去括号。对于任意数字 a、b 和 c：a(b + c) = ab + ac。括号前为减号时同样适用：a(b – c) = ab – ac。
To expand a product of two binomials, such as (x + p)(x + q), apply the FOIL method. The result is x² + (p + q)x + pq. Pay special attention to signs: (x – p)(x + q) becomes x² + ( – p + q)x – pq.
展开两个二项式的乘积，例如 (x + p)(x + q)，可以使用 FOIL 法则。结果为 x² + (p + q)x + pq。需特别注意符号：(x – p)(x + q) 展开后得到 x² + ( – p + q)x – pq。
Factorising is the reverse process. For a quadratic x² + bx + c, we look for two numbers that multiply to c and add to b. If those numbers are p and q, then x² + bx + c = (x + p)(x + q).
因式分解是展开的逆运算。对于二次式 x² + bx + c，我们要找到两个数，它们的乘积为 c、和为 b。如果这两个数是 p 和 q，那么 x² + bx + c = (x + p)(x + q)。
Difference of two squares: a² – b² = (a + b)(a – b)
平方差公式：a² – b² = (a + b)(a – b)
2. Linear Equations and Fractions | 线性方程与分式方程
A linear equation can be written as ax + b = c. To solve it, we isolate the variable by using inverse operations. Always do the same to both sides: if you subtract 3 from one side, subtract 3 from the other as well.
线性方程可写作 ax + b = c。求解时，我们通过逆运算移项求未知数。始终要保持等式两边平衡：如果一边减去 3，另一边也要减去 3。
When the equation contains fractions, multiply every term by the lowest common denominator (LCD) to clear the denominators. For example, to solve x/3 + 2 = x/2, multiply through by 6 to obtain 2x + 12 = 3x, giving x = 12.
当方程中含有分数时，将每一项乘以最小公分母 (LCD) 以去掉分母。例如，解 x/3 + 2 = x/2，两边乘以 6 得到 2x + 12 = 3x，解得 x = 12。
Always check your solution by substituting the value back into the original equation. This verifies that both sides produce the same number.
务必把求得的解代回原方程进行检验，以证实两边算出的数值相等。
General form: ax + b = c → x = (c – b) / a
一般形式：ax + b = c → x = (c – b) / a
3. Forming and Solving Inequalities | 列不等式与求解不等式
Inequalities use the symbols < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). They are solved similarly to equations, but with one crucial rule: if you multiply or divide by a negative number, you must reverse the inequality sign.
不等式使用符号 <（小于）、>（大于）、≤（小于或等于）和 ≥（大于或等于）。解法与方程类似，但有一条至关重要的规则：若乘以或除以一个负数，必须反转不等号方向。
To represent the solution on a number line, use an open circle for < or > and a closed circle for ≤ or ≥. The inequality –3 ≤ x < 5 means x is between –3 (inclusive) and 5 (exclusive).
在数轴上表示解集时，< 或 > 用空心圆，≤ 或 ≥ 用实心圆。不等式 –3 ≤ x < 5 表示 x 在 –3（含）到 5（不含）之间。
When solving compound inequalities such as 2 < 3x – 1 ≤ 8, deal with all three parts at the same time. Add 1 everywhere: 3 < 3x ≤ 9, then divide by 3: 1 < x ≤ 3.
求解复合不等式，如 2 < 3x – 1 ≤ 8，要同时对三个部分进行操作。先加 1：3 < 3x ≤ 9，再除以 3：1 < x ≤ 3。
4. Laws of Indices and Standard Form | 指数定律与标准形式
Indices (or powers) follow strict rules that simplify calculations. When multiplying powers with the same base, add the exponents: am × an = am+n. When dividing, subtract the exponents: am ÷ an = am–n.
指数（或幂）遵循严格的运算法则，可简化计算。同底数幂相乘时，指数相加：am × an = am+n。同底数幂相除时，指数相减：am ÷ an = am–n。
Any number raised to the power of zero equals 1 (a0 = 1, a ≠ 0). Negative exponents produce reciprocals: a–n = 1 / an. Fractional exponents indicate roots: a½ = √a and a1/3 = ³√a.
任何非零数的零次幂等于 1（a0 = 1，a ≠ 0）。负指数得到倒数：a–n = 1 / an。分数指数表示方根：a½ = √a，a1/3 = ³√a。
Standard form writes very large or very small numbers as A × 10n, where 1 ≤ A < 10 and n is an integer. For example, 0.00047 becomes 4.7 × 10–4, and 62 000 000 becomes 6.2 × 107.
标准形式将极大或极小的数写成 A × 10n，其中 1 ≤ A < 10 且 n 为整数。例如，0.00047 写作 4.7 × 10–4，62 000 000 写作 6.2 × 107。
(am)n = amn
幂的乘方：底数不变，指数相乘。
5. Sequences and the nth Term | 数列与第 n 项
An arithmetic sequence has a constant difference, d, between consecutive terms. The nth term is given by the formula: un = a + (n – 1)d, where a is the first term.
等差数列中相邻两项的差 d 为常数。第 n 项公式为：un = a + (n – 1)d，其中 a 为首项。
To find any term, substitute the position number n into the formula. For the sequence 4, 7, 10, 13, …, a = 4, d = 3, so the 20th term is u20 = 4 + 19 × 3 = 61.
要求某项，将位置编号 n 代入公式。数列 4, 7, 10, 13, … 中，a = 4, d = 3，因此第 20 项为 u20 = 4 + 19 × 3 = 61。
Some sequences are quadratic. The second difference is constant. For further study, you will learn to find an nth term of the form an² + bn + c, but at Year 8 you can often spot the pattern by building a term-to-term rule.
有些数列是二次数列，二阶差为常数。高阶学习中，你将找到形如 an² + bn + c 的第 n 项表达式；在八年级，通常可以通过递推规则发现规律。
Sum of the first n terms of an arithmetic series: Sn = n/2 (2a + (n – 1)d) or Sn = n/2 (a + l), where l is the last term.
等差数列前 n 项和：Sn = n/2 (2a + (n – 1)d) 或 Sn = n/2 (a + l)，其中 l 为末项。
6. Coordinate Geometry and Straight-Line Graphs | 坐标几何与直线图像
The gradient (slope) of a line passing through two points (x₁, y₁) and (x₂, y₂) is m = (y₂ – y₁) / (x₂ – x₁). A positive gradient means the line goes uphill; a negative gradient means it goes downhill.
通过两点 (x₁, y₁) 和 (x₂, y₂) 的直线的斜率（坡度）为 m = (y₂ – y₁) / (x₂ – x₁)。正斜率表示直线向上倾斜，负斜率表示向下倾斜。
The equation of a straight line can be written as y = mx + c, where m is the gradient and c is the y-intercept (the point where the line crosses the y-axis).
直线方程可写作 y = mx + c，其中 m 为斜率，c 为 y 轴截距（直线与 y 轴交点的纵坐标）。
Two lines are parallel if they have the same gradient. Two lines are perpendicular if the product of their gradients is –1 (m₁ × m₂ = –1).
若两条直线斜率相同，则它们平行。若两条直线斜率的乘积为 –1（m₁ × m₂ = –1），则它们互相垂直。
To find the midpoint of a segment joining (x₁, y₁) and (x₂, y₂), average the x‑coordinates and average the y‑coordinates: Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between the points is √[(x₂ – x₁)² + (y₂ – y₁)²].
求连接 (x₁, y₁) 与 (x₂, y₂) 的线段中点，将 x 坐标和 y 坐标分别取平均值：中点 = ((x₁ + x₂)/2, (y₁ + y₂)/2)。两点间距离为 √[(x₂ – x₁)² + (y₂ – y₁)²]。
7. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角学
Pythagoras’ theorem applies to right‑angled triangles. If c is the hypotenuse (longest side), then a² + b² = c². This allows you to find a missing side when the other two sides are known.
勾股定理适用于直角三角形。若 c 为斜边（最长边），则 a² + b² = c²。已知两边可求第三边。
The three trigonometric ratios in a right‑angled triangle are defined as: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Use SOH CAH TOA to remember them.
直角三角形中三个三角比的定义为：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。可用 SOH CAH TOA 帮助记忆。
When you know an angle and a side, you can use these ratios to find another side. When you know two sides, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to find the angle.
已知一角一边时，可用这些比值求另一边。已知两边时，使用反三角函数（sin⁻¹, cos⁻¹, tan⁻¹）求角度。
Exact trigonometric values to remember: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
需记忆的特殊角三角函数值：sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2；cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½；tan 30° = 1/√3, tan 45° = 1, tan 60° = √3。
8. Circle Theorems and Circular Measurements | 圆定理与圆的度量
The angle at the centre of a circle is twice the angle at the circumference when both subtend the same arc. This is one of several circle theorems you will use to solve angle problems within circles.
圆心角是同一弧所对圆周角的两倍。这是多个圆定理之一，用于解决圆内的角度问题。
Angles in the same segment are equal. The angle in a semicircle is always a right angle (90°). A tangent to a circle is perpendicular to the radius drawn to the point of contact.
同弧上的圆周角相等。半圆内的圆周角恒为直角（90°）。圆的切线与过切点的半径垂直。
For any circle, the circumference C is given by C = 2πr, and the area A is A = πr², where r is the radius. The arc length for a sector of angle θ° is (θ/360) × 2πr, and the sector area is (θ/360) × πr².
对任意圆，周长 C = 2πr，面积 A = πr²，其中 r 为半径。圆心角 θ° 对应的扇形弧长为 (θ/360) × 2πr，扇形面积为 (θ/360) × πr²。
Cyclic quadrilateral theorem: opposite angles sum to 180°.
圆内接四边形定理：对角之和为 180°。
9. Surface Area and Volume of 3D Solids | 三维立体的表面积和体积
The volume of a prism is found using V = base area × height. For a cylinder of radius r and height h, base area is πr², so V = πr²h. The curved surface area of a cylinder is 2πrh; total surface area includes the two circular ends: TSA = 2πrh + 2πr².
棱柱的体积公式为 V = 底面积 × 高。对于半径为 r、高为 h 的圆柱，底面积为 πr²，故 V = πr²h。圆柱的侧面积为 2πrh；总表面积包括两个底面：TSA = 2πrh + 2πr²。
For a cone with radius r, slant height l and vertical height h, volume V = ⅓πr²h and curved surface area = πrl. For a sphere of radius R, volume = ⁴⁄₃πR³ and surface area = 4πR².
对于半径为 r、斜高为 l、竖直高度为 h 的圆锥，体积 V = ⅓πr²h，侧面积 = πrl。半径为 R 的球体，体积 = ⁴⁄₃πR³，表面积 = 4πR²。
Composite shapes can be broken down into simpler solids. Calculate the volume or surface area of each part separately, then add or subtract as appropriate, being careful about overlapping faces.
组合体可分解为简单立体分别计算体积和表面积，再根据情况相加或相减，注意不要重复计算重叠的面。
Pyramid volume: V = ⅓ × base area × height.
棱锥体积：V = ⅓ × 底面积 × 高。
10. Probability and Statistical Diagrams | 概率与统计图表
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). For a fair trial, P(event) = number of favourable outcomes / total number of equally likely outcomes.
概率衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。在等可能结果试验中，P(事件) = 有利结果数 / 所有等可能结果总数。
The complement rule states that P(not A) = 1 – P(A). When events are mutually exclusive, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).
互补事件规则：P(非 A) = 1 – P(A)。两个互斥事件中，P(A 或 B) = P(A) + P(B)。两个独立事件中，P(A 且 B) = P(A) × P(B)。
Sample space diagrams and tree diagrams help list all possible outcomes. Frequency trees and two-way tables are used to organise data and calculate experimental probabilities. The mean of a data set is the sum of the values divided by the number of values; the median is the middle value when ordered; the mode is the most frequent.
样本空间图和树状图有助于列出所有可能结果。频率树和双向表用于整理数据和计算实验概率。一组数据的平均数等于总和除以数据个数；中位数是排序后处于中间的值；众数是出现最频繁的值。
Scatter graphs show the relationship between two variables. If points roughly follow a straight line, there is correlation. A line of best fit can be drawn to model the trend and make predictions.
散点图显示两个变量间的关系。若点大致沿直线分布，则存在相关性。可画一条最佳拟合线来模拟趋势并进行预测。
11. Quadratic Expressions and the Quadratic Formula | 二次表达式与求根公式
A quadratic expression has the general form ax² + bx + c where a ≠ 0. Expanding two linear brackets (x + p)(x + q) produces a quadratic: x² + (p + q)x + pq. Factorising reverses this process.
二次表达式的一般形式为 ax² + bx + c，其中 a ≠ 0。展开两个一次括号 (x + p)(x + q) 得到二次式 x² + (p + q)x + pq。因式分解则是这一过程的逆运算。
When the quadratic is set equal to zero, we have a quadratic equation: ax² + bx + c = 0. The solutions are called the roots. The quadratic formula gives these roots directly:
当二次式设为零时，我们得到一个二次方程：ax² + bx + c = 0。其解称为根。求根公式可直接给出这些解：
x = (–b ± √(b² – 4ac)) / (2a)
x = (–b ± √(b² – 4ac)) / (2a)
The expression under the square root, Δ = b² – 4ac, is called the discriminant. If Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one real root (a repeated root); if Δ < 0, there are no real roots.
根号下的部分 Δ = b² – 4ac 称为判别式。若 Δ > 0，方程有两个不等实根；若 Δ = 0，有一个实根（重根）；若 Δ < 0，无实根。
Completing the square rewrites a quadratic as a(x + p)² + q. This form makes it easy to find the turning point of its graph. The graph of y = ax² + bx + c is a parabola; if a > 0 it opens upwards, if a < 0 it opens downwards.
配方法将二次式重写为 a(x + p)² + q 的形式。这种形式便于求其图像顶点。y = ax² + bx + c 的图像是一条抛物线；若 a > 0，开口向上，若 a < 0，开口向下。
12. Vectors and Transformations | 向量与变换
A vector describes a translation: a movement in a given direction. In 2D, a vector is written as a column (x over y) or in component form xi + yj. Adding vectors: (a over b) + (c over d) = (a+c over b+d).
向量描述一种平移：在给定方向上的移动。在二维中，向量写作列向量 (x, y)T 或分量形式 xi + yj。向量加法：(a, b)T + (c, d)T = (a+c, b+d)T。
Multiplying a vector by a scalar stretches or shrinks it and may reverse its direction. k × (x over y) = (kx over ky). The magnitude (length) of a vector v = (x over y) is |v| = √(x² + y²).
向量乘以标量将伸长或缩短向量，并可能反转方向。k × (x, y)T = (kx, ky)T。向量 v = (x, y)T 的模（长度）为 |v| = √(x² + y²)。
In transformation geometry, translations are described by vectors. Reflections are defined by mirror lines; rotations require a centre, angle and direction; enlargements use a scale factor and centre of enlargement. These transformations can be combined.
在变换几何中，平移由向量描述；反射由对称轴定义；旋转需要指定中心、角度和方向；放大变换使用比例因子和放大中心。这些变换可以组合。
Invariant points are points that do not move under a transformation. You can be asked to find the image of a shape after a sequence of transformations. Always label the vertices clearly.
不变点是在变换下保持不动的点。考题可能要求你求出一个图形经过一系列变换后的像。一定要清晰地标记各顶点。
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