Year 9 CAIE Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 9 CAIE 进阶数学：公式定理速查手册

📚 Year 9 CAIE Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 9 CAIE 进阶数学：公式定理速查手册

This revision handbook provides a concise collection of the essential formulas and theorems required for Year 9 CAIE Further Mathematics. It is designed as a quick reference to support your daily practice, homework, and exam preparation. Each section presents key results with clear notation, enabling you to locate the rule you need without flipping through lengthy textbooks.

本速查手册汇集了 Year 9 CAIE 进阶数学所必需的核心公式与定理。它旨在成为一份快捷参考，辅助日常练习、作业与考试复习。每个版块以清晰的符号罗列重要结论，让你不必翻阅厚重的课本就能迅速找到所需的法则。

1. Laws of Indices and Basic Algebra | 指数定律与代数基础

For any non‑zero real numbers a, b and rational exponents m, n the following index laws hold. They are fundamental for simplifying expressions and solving equations.

对于任意非零实数 a, b 以及有理指数 m, n，以下指数定律成立。它们是化简表达式与解方程的基础。

Product of powers: aᵐ × aⁿ = aᵐ⁺ⁿ

同底数幂的乘法：aᵐ × aⁿ = aᵐ⁺ⁿ

Quotient of powers: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

同底数幂的除法：aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Power of a power: (aᵐ)ⁿ = aᵐⁿ

幂的乘方：(aᵐ)ⁿ = aᵐⁿ

Power of a product: (ab)ⁿ = aⁿbⁿ

积的乘方：(ab)ⁿ = aⁿbⁿ

Zero exponent: a⁰ = 1 (a ≠ 0)

零指数：a⁰ = 1 (a ≠ 0)

Negative exponent: a⁻ⁿ = 1/aⁿ

负指数：a⁻ⁿ = 1/aⁿ

Key expansion identities:

重要展开恒等式：

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²

Difference of two squares: a² − b² = (a + b)(a − b)

平方差公式：a² − b² = (a + b)(a − b)

---

2. Quadratic Equations and Factorising | 二次方程与因式分解

lutions can be found by factorising, completing the square or using the quadratic formula.

二次方程的一般形式为 ax² + bx + c = 0，其中 a ≠ 0。其解可通过因式分解、配方法或求根公式求得。

Quadratic formula:

x = (−b ± √b² − 4ac) / 2a

根的判别式 (discriminant): Δ = b² − 4ac

Δ = b² − 4ac

Discriminant determines the nature of roots:
Δ > 0 → two distinct real roots
Δ = 0 → one repeated real root
Δ < 0 → no real roots

判别式决定根的情况：
Δ > 0 → 两个不等实根
Δ = 0 → 一个重根
Δ < 0 → 无实根

Factorising quadratics: x² + bx + c = (x + p)(x + q) where p + q = b and pq = c.

因式分解二次三项式：x² + bx + c = (x + p)(x + q)，其中 p + q = b 且 pq = c。

Sum and product of roots: if α and β are roots, then α + β = −b/a, αβ = c/a.

根与系数的关系：若 α 与 β 为根，则 α + β = −b/a，αβ = c/a。

---

3. Inequalities and Interval Notation | 不等式与区间表示

Inequalities express the relative size of two expressions. Solutions to linear and quadratic inequalities are often represented on a number line or using interval notation.

不等式表示两个表达式之间的大小关系。一次及二次不等式的解常借助数轴或用区间符号表示。

Basic rules: Adding or subtracting the same number from both sides preserves the inequality. Multiplying or dividing by a positive number preserves direction; multiplying or dividing by a negative number reverses the direction.

基本法则：两边同加或同减一个数，不等号方向不变；两边同乘或同除正数，方向不变；同乘或同除负数，不等号反向。

Interval notation:
(a, b) = {x | a < x < b}
[a, b] = {x | a ≤ x ≤ b}
(a, ∞) = {x | x > a}

区间表示法：
(a, b) 表示 a < x < b
[a, b] 表示 a ≤ x ≤ b
(a, ∞) 表示 x > a

Solving quadratic inequalities: factorise, find critical values, then use a sign diagram.

解二次不等式：先因式分解，求出临界值，再利用符号表确定解集。

---

4. Sequences and Series | 数列与级数

Arithmetic and geometric sequences are foundational patterns in further mathematics. Being able to find the nth term and the sum of the first n terms is essential.

等差数列与等比数列是进阶数学中的基础模式。求出第 n 项和前 n 项和的公式必须熟练掌握。

Arithmetic sequence: first term a, common difference d.

等差数列：首项 a，公差 d。

nth term: uₙ = a + (n − 1)d

第 n 项：uₙ = a + (n − 1)d

Sum of first n terms: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.

前 n 项和：Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)，其中 l 为末项。

Geometric sequence: first term a, common ratio r (r ≠ 0).

等比数列：首项 a，公比 r (r ≠ 0)。

nth term: uₙ = arⁿ⁻¹

第 n 项：uₙ = arⁿ⁻¹

Sum of first n terms (r ≠ 1): Sₙ = a(1 − rⁿ) / (1 − r)

前 n 项和（r ≠ 1）：Sₙ = a(1 − rⁿ) / (1 − r)

Sum to infinity (|r| < 1): S∞ = a / (1 − r)

无穷项和（当 |r| < 1）：S∞ = a / (1 − r)

---

5. Functions and Their Graphs | 函数及其图像

A function f maps each input x from the domain to exactly one output f(x) in the range. Notation and transformation skills are tested regularly.

函数 f 将定义域中的每一个输入 x 唯一地对应到值域中的一个输出 f(x)。函数的表示法及图像变换是经常考查的内容。

Domain and range: domain is the set of all possible x‑values; range is the set of all possible f(x)‑values.

定义域与值域：定义域是所有可能输入 x 的集合；值域是所有可能输出 f(x) 的集合。

Composite function: (f ∘ g)(x) = f(g(x)). Apply g first, then f.

复合函数：(f ∘ g)(x) = f(g(x))，先作用 g，再作用 f。

function f⁻¹(x) satisfies f(f⁻¹(x)) = x. Graphically it is a reflection in the line y = x.

反函数 f⁻¹(x) 满足 f(f⁻¹(x)) = x。其图像关于直线 y = x 对称。

Graph transformations (c > 0):
y = f(x) + c → translation up by c
y = f(x + c) → translation left by c
y = −f(x) → reflection in the x‑axis
y = af(x) → vertical stretch by factor a

图像变换（c > 0）：
y = f(x) + c → 向上平移 c 个单位
y = f(x + c) → 向左平移 c 个单位
y = −f(x) → 关于 x 轴反射
y = af(x) → 垂直方向伸缩 a 倍

---

广告 / Sponsored

6. Trigonometry | 三角学

Trigonometry connects angles and side lengths in right‑angled triangles and, via the sine and cosine rules, in any triangle. Special angles and fundamental identities are vital.

三角学将直角三角形中的角度与边长联系起来，通过正弦定理和余弦定理可推广至任意三角形。特殊角及其基本恒等式至关重要。

Right‑angled triangle definitions:
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent

直角三角形中的定义：
sin θ = 对边 / 斜边
cos θ = 邻边 / 斜边
tan θ = 对边 / 邻边

Exact values for special angles:

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

精确值速查表：

Pythagorean identity: sin²θ + cos²θ = 1

毕达哥拉斯恒等式：sin²θ + cos²θ = 1

Sine rule: a / sin A = b / sin B = c / sin C

正弦定理：a / sin A = b / sin B = c / sin C

Cosine rule: a² = b² + c² − 2bc cos A

余弦定理：a² = b² + c² − 2bc cos A

Area of triangle: Area = ½ ab sin C

三角形面积公式：面积 = ½ ab sin C

---

7. Coordinate Geometry | 坐标几何

Coordinate geometry provides algebraic tools to describe points, lines and shapes on the Cartesian plane. Key formulas allow us to compute distances, midpoints and gradients with ease.

坐标几何用代数工具描述笛卡尔平面上的点、线及图形。关键公式帮助我们轻松计算距离、中点与斜率。

Distance between two points (x₁, y₁) and (x₂, y₂):
d = √[(x₂ − x₁)² + (y₂ − y₁)²]

两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Midpoint: M = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

中点坐标：M = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

Gradient (slope) of a line: m = (y₂ − y₁) / (x₂ − x₁)

直线斜率：m = (y₂ − y₁) / (x₂ − x₁)

Equation of a straight line:
Slope‑intercept form: y = mx + c
Point‑slope form: y − y₁ = m(x − x₁)

直线方程：
斜截式：y = mx + c
点斜式：y − y₁ = m(x − x₁)

Parallel lines: m₁ = m₂. Perpendicular lines: m₁ × m₂ = −1.

两直线平行：m₁ = m₂；两直线垂直：m₁ × m₂ = −1。

---

8. Vectors | 向量

Vectors represent quantities that have both magnitude and direction. They are written in column form or using unit vectors i, j. Arithmetic operations follow specific rules.

向量表示既有大小又有方向的量。通常用列向量或单位向量 i, j 表示。向量的加、减及数乘遵循特定的法则。

Notation: vector a = (x, y) or a = xi + yj.

表示法：向量 a = (x, y) 或 a = xi + yj。

Addition: a + b = (x₁ + x₂, y₁ + y₂)

加法：a + b = (x₁ + x₂, y₁ + y₂)

Scalar multiplication: k a = (kx, ky)

数乘：k a = (kx, ky)

Magnitude of a vector: |a| = √x² + y²

向量的模：|a| = √x² + y²

Position vector: the vector from the origin to a point P(x, y) is OP = (x, y).

位置向量：从原点指向点 P(x, y) 的向量为 OP = (x, y)。

Displacement vector AB = b − a, where a and b are position vectors of A and B.

位移向量 AB = b − a，其中 a, b 分别为 A, B 的位置向量。

---

9. Statistics and Probability | 统计与概率

Descriptive statistics summarise a data set using measures of central tendency and spread. Probability rules quantify the chance of events and underpin further topics like tree diagrams.

描述性统计通过集中趋势和离散程度的度量概括数据集。概率法则量化事件发生的可能性，并支撑树形图等后续内容。

Mean = (sum of all data values) ÷ (number of values).
Median = middle value when data is ordered.
Mode = most frequent value.
Range = maximum − minimum.

平均数 = 所有数据之和 ÷ 数据个数
中位数 = 排序后中间的值
众数 = 出现频率最高的值
全距 = 最大值 − 最小值

of the lower half, Q₃ is the median of the upper half. Interquartile range IQR = Q₃ − Q₁.

四分位数：下四分位数 Q₁ 及上四分位数 Q₃ 分别为下半部分和上半部分的中位数；四分位距 IQR = Q₃ − Q₁。

robability of an event A: P(A) = number of favourable outcomes / total number of outcomes.

事件 A 的概率：P(A) = 有利结果数 / 可能结果总数。

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
For mutually exclusive events P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
若 A 与 B 互斥，则 P(A ∩ B) = 0，此时 P(A ∪ B) = P(A) + P(B)。

Multiplication rule for independent events: P(A ∩ B) = P(A) × P(B).

独立事件的乘法法则：P(A ∩ B) = P(A) × P(B)。

Tree diagrams multiply probabilities along branches and sum outcomes for combined events.

树形图：沿分支相乘概率，将所需联合事件的结果相加。

---

Published by TutorHao | Further Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）