📚 Year 9 WJEC Further Maths: Formula & Theorem Quick Reference | Year 9 WJEC 进阶数学：公式定理速查手册
This quick reference handbook brings together all the essential formulas and theorems needed for Year 9 WJEC Further Mathematics. It is designed to help you revise efficiently and build confidence before your assessments. Keep this guide handy whenever you are practising problems or doing last-minute revision.
这本速查手册汇集了 Year 9 WJEC 进阶数学所需的所有关键公式和定理，旨在帮助你高效复习并在评估前建立信心。无论是在练习题目还是最后冲刺复习时，都可以随时查阅本指南。
1. Algebraic Manipulation | 代数运算
ctorising is the reverse process, pulling out common factors or recognising patterns such as the difference of two squares.
进阶数学的基石从这里开始。去括号利用分配律：a(b + c) = ab + ac。因式分解是相反的过程，提取公因式或识别平方差等模式。
a² - b² = (a + b)(a - b)
o factorise into two binomials. The identity (x + p)(x + q) = x² + (p+q)x + pq is crucial.
对于二次表达式，我们经常需要将其因式分解为两个二项式。恒等式 (x + p)(x + q) = x² + (p+q)x + pq 至关重要。
s. Also, be careful with the laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ.
在化简代数分式时，记住先因式分解再约去公因式。同时，注意指数运算律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。
2. Linear Equations and Inequalities | 一次方程与不等式
c, isolate x by performing inverse operations on both sides. Always maintain balance: whatever you do to one side, do to the other.
解形如 ax + b = c 的方程时，通过对两边进行逆运算来分离 x。始终保持平衡：对一边做什么，对另一边也要做同样的计算。
For inequalities, the rules are identical except that multiplying or dividing by a negative number reverses the inequality sign. Representing solutions on a number line is a common requirement.
对于不等式，规则相同，但乘以或除以一个负数时会反转不等号的方向。在数轴上表示解是常见的要求。
Simultaneous linear equations can be tackled by elimination or substitution. The aim is to reduce two equations in two unknowns to one equation in one unknown.
联立一次方程可以通过消元法或代入法来求解。目标是将两个未知数的两个方程转化为一个未知数的一个方程。
ax + by = c and dx + ey = f
3. Quadratic Equations and Functions | 二次方程与函数
be solved by factorising, completing the square, or using the quadratic formula. The formula always works, even when factorising is difficult.
二次方程 ax² + bx + c = 0 可以通过因式分解、配方法或使用求根公式来求解。这个公式总是有效，即使因式分解很困难时也能用。
x = (-b ± √b² - 4ac) / (2a)
al roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are no real roots.
判别式 Δ = b² - 4ac 告诉我们根的性质。如果 Δ > 0，有两个不同的实根；如果 Δ = 0，有一个重根；如果 Δ < 0，没有实根。
vertex form, y = a(x - h)² + k, reveals the vertex (h, k) and the axis of symmetry x = h.
y = ax² + bx + c 的图像是一条抛物线。其顶点式 y = a(x - h)² + k 揭示了顶点 (h, k) 和对称轴 x = h。
4. Sequences and Series | 数列与级数
1)d, where a is the first term. The sum of the first n terms is Sₙ = n/2 [2a + (n - 1)d].
等差数列有一个公差 d。第 n 项由 uₙ = a + (n - 1)d 给出，其中 a 是首项。前 n 项和为 Sₙ = n/2 [2a + (n - 1)d]。
term is uₙ = arⁿ⁻¹. The sum of the first n terms, when r ≠ 1, is Sₙ = a(1 - rⁿ) / (1 - r).
等比数列有一个公比 r。其第 n 项为 uₙ = arⁿ⁻¹。当 r ≠ 1 时，前 n 项和为 Sₙ = a(1 - rⁿ) / (1 - r)。
Recognising patterns like triangular numbers (n(n+1)/2) or square numbers (n²) is also part of the syllabus, helping with problem solving and building generalised rules from term-to-term rules.
识别三角数 (n(n+1)/2) 或平方数 (n²) 等数列模式也是课程的一部分，有助于解决问题，并能够从递推规则推导出通项公式。
5. Properties of Shapes | 图形性质
Angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. In parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior angles sum to 180°.
直线上的角之和为 180°，一点周围的角之和为 360°，对顶角相等。在平行线中，内错角相等，同位角相等，同旁内角之和为 180°。
Angle sum of a triangle is 180°, and of a quadrilateral is 360°. The exterior angle of a triangle equals the sum of the two opposite interior angles, which is also a key theorem.
三角形的内角和为 180°，四边形的内角和为 360°。三角形的外角等于与它不相邻的两个内角之和，这也是一个关键定理。
| Polygon | Sum of interior angles | Each interior (regular) |
| n-sided | (n - 2) × 180° | (n - 2) × 180° / n |
Knowing the properties of special quadrilaterals (parallelogram, rhombus, trapezium, kite) is vital: side lengths, angle bisectors, diagonal properties and symmetries must be understood.
了解特殊四边形（平行四边形、菱形、梯形、风筝形）的性质至关重要：需要理解边长、角平分线、对角线性质以及对称性。
6. Pythagoras' Theorem and Trigonometry | 勾股定理与三角学
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This is the foundation for distance calculations and coordinate geometry.
在直角三角形中，斜边的平方等于另外两条直角边的平方和。这是距离计算和坐标几何的基础。
c² = a² + b²
otenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Remember SOH CAH TOA.
三个基本三角比将角度和边长联系起来：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。记住 SOH CAH TOA。
or 0°, 30°, 45°, 60° and 90° should be memorised. For example, sin 30° = 1/2, cos 45° = 1/√2, tan 60° = √3. These are frequently used in non-calculator papers.
必须记住 0°、30°、45°、60° 和 90° 的精确三角值。例如，sin 30° = 1/2，cos 45° = 1/√2，tan 60° = √3。这些在非计算器试卷中经常使用。
7. Perimeter, Area and Volume | 周长、面积与体积
Common 2D area formulas: rectangle (bh), triangle (½bh), parallelogram (bh), trapezium (½(a+b)h), circle (πr²). Circumference of a circle is 2πr or πd.
常见二维面积公式：矩形 (bh)，三角形 (½bh)，平行四边形 (bh)，梯形 (½(a+b)h)，圆 (πr²)。圆的周长为 2πr 或 πd。
is a prism with circular base, so volume = πr²h. Surface area of a cylinder = 2πrh + 2πr².
对于三维图形，棱柱的体积 = 横截面积 × 长度。圆柱是以圆为底的棱柱，因此体积 = πr²h。圆柱的表面积 = 2πrh + 2πr²。
3 πr²h. Sphere volume = 4/3 πr³ and surface area = 4πr². These should be used confidently.
棱锥的体积 = 1/3 × 底面积 × 高，圆锥的体积 = 1/3 πr²h。球体体积 = 4/3 πr³，表面积 = 4πr²。需要熟练运用这些公式。
roportional to the central angle: arc length = (θ/360) × 2πr, sector area = (θ/360) × πr².
复合图形需要分解成更简单的组成部分。弧长和扇形面积与圆心角成比例：弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。
8. Statistics and Data Handling | 统计与数据处理
Measures of central tendency include mean, median, and mode. The mean of a frequency table is Σfx / Σf. The median is the middle value when data is ordered; for grouped data, linear interpolation is used.
集中趋势的度量包括平均数、中位数和众数。频数表的平均数为 Σfx / Σf。中位数是数据排序后的中间值；对于分组数据，使用线性插值法。
clude the interquartile range (IQR = Q₃ - Q₁), which describes the spread of the middle 50% of data.
极差给出了简单的离散度量：最大值 - 最小值。更高级的度量包括四分位距 (IQR = Q₃ - Q₁)，它描述了中间 50% 数据的分布范围。
Box plots visually represent the five-number summary: minimum, Q₁, median, Q₃, maximum. They are useful for comparing distributions and identifying skewness.
箱线图直观地表示了五数汇总：最小值、Q₁、中位数、Q₃、最大值。它们有助于比较分布和识别偏态。
Scatter graphs show relationships between two variables. Correlation can be positive, negative or none. A line of best fit can be drawn to make predictions, though caution is needed for extrapolation.
散点图显示两个变量之间的关系。相关性可以是正相关、负相关或无相关。可以画出最佳拟合线来进行预测，但在外推时需要谨慎。
9. Probability | 概率
ally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes.
概率的度量范围从 0（不可能）到 1（必然）。对于等可能结果，P(事件) = 有利结果的数量 / 结果总数。
P(A or B) = P(A) + P(B) - P(A and B). If mutually exclusive, then P(A or B) = P(A) + P(B).
一个实验中所有互斥结果的概率之和为 1。对于两个事件 A 和 B，P(A 或 B) = P(A) + P(B) - P(A 和 B)。如果互斥，则 P(A 或 B) = P(A) + P(B)。
Tree diagrams are essential for compound events. Multiply along branches for 'and', add probabilities of relevant outcomes for 'or'. Conditional probability updates the likelihood based on prior outcomes.
树状图对于复合事件至关重要。沿分支相乘表示“且”的关系，将相关结果的概率相加表示“或”的关系。条件概率根据已有结果更新可能性。
P(A|B) = P(A and B) / P(B)
10. Ratio, Proportion and Rates | 比、比例与率
Ratios compare quantities and can be simplified just like fractions. Divide all parts by their greatest common factor. A ratio in the form 1:n is often useful for scaling.
比用来比较数量，可以像分数一样化简。将所有部分除以它们的最大公约数。1:n 形式的比对于缩放比例经常很有用。
Direct proportion means y ∝ x, so y = kx for a constant k. Inverse proportion means y ∝ ct proportion are straight lines through the origin; inverse proportion gives a hyperbola.
正比例意味着 y ∝ x，因此对于常数 k 有 y = kx。反比例意味着 y ∝ 1/x，因此 y = k/x。正比例的图像是过原点的直线；反比例给出双曲线。
are key compound measures. Ensure units are consistent before substituting into formulas.
速度 = 距离 / 时间，密度 = 质量 / 体积，压强 = 力 / 面积，这些都是关键的复合度量。在代入公式之前，要确保单位一致。
00). Repeated percentage change uses a multiplier raised to a power, forming geometric connections.
百分比增减：新值 = 原值 × (1 ± 百分比/100)。重复的百分比变化使用乘数的幂次方，这构成了等比数列的联系。
11. Coordinate Geometry | 坐标几何
A horizontal line has gradient 0, a vertical line has undefined gradient. Parallel lines have equal gradients; perpendicular lines have gradients that multiply to -1.
连接 (x₁, y₁) 和 (x₂, y₂) 的直线的斜率为 m = (y₂ - y₁) / (x₂ - x₁)。水平线的斜率为 0，垂直线的斜率不存在。平行线斜率相等；垂直线斜率的乘积为 -1。
e point-gradient form y - y₁ = m(x - x₁) is handy when a point and the gradient are known.
直线的方程可以表示为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。当已知一点和斜率时，点斜式 y - y₁ = m(x - x₁) 很有用。
stance between two points is √((x₂-x₁)² + (y₂-y₁)²), which comes directly from Pythagoras.
两点的中点为 ((x₁+x₂)/2, (y₁+y₂)/2)。两点间的距离为 √((x₂-x₁)² + (y₂-y₁)²)，这直接来自勾股定理。
12. Transformations | 变换
There are four fundamental geometric transformations: translation, reflection, rotation, and enlargement. A translation moves a shape by a vector (x, y). Reflection flips it over a mirror line.
有四种基本的几何变换：平移、反射、旋转和位似。平移通过向量 (x, y) 移动形状。反射将其翻折过一条镜面线。
Rotation requires a centre, an angle, and a direction. Tracing paper is helpful in exams. Enlargement changes size by a scale factor, and if the scale factor is negative, the image appears inverted.
旋转需要一个中心、一个角度和一个方向。在考试中描图纸很有帮助。位似通过一个比例因子改变大小，如果比例因子为负，图形会倒立出现。
A shape and its transformation image are congruent for translation, reflection and rotation (they keep the same size and shape). Enlargement produces similar figures, where angles stay the same but lengths multiply by the scale factor.
在平移、反射和旋转后，形状与其变换图形全等（它们保持相同的大小和形状）。位似产生相似图形，角度不变但边长乘以比例因子。
The area scale factor of an enlargement is the square of the linear scale factor, and the volume scale factor is its cube. These principles link geometry with ratio and proportion topics.
位似的面积比例因子是线性比例因子的平方，体积比例因子是其立方。这些原理将几何与比和比例等主题联系起来。
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