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IB CCEA Mathematics: Formula Summary Handbook | IB CCEA 数学：公式汇总手册

2026-07-14 22:06 · 本文部分内容由AI辅助创作

📚 IB CCEA Mathematics: Formula Summary Handbook | IB CCEA 数学：公式汇总手册

This handbook brings together the essential formulas needed for IB and CCEA Mathematics courses. Whether you are revising for the IB Analysis & Approaches, Applications & Interpretation, or CCEA GCE Mathematics, these summaries will serve as a quick reference covering algebra, functions, trigonometry, calculus, vectors, and statistics. Keep this guide close during revision to memorise core relationships and improve problem‑solving speed.

本手册汇集了 IB 和 CCEA 数学课程所需的核心公式。无论你正在准备 IB 分析与方法、应用与解释，还是 CCEA 普通教育数学考试，这份速查表都能帮你快速回顾代数、函数、三角学、微积分、向量和统计等关键内容。复习时随身携带，强化记忆，提高解题效率。

1. Algebra and Identities | 代数与恒等式

Expanding two binomials: (a + b)(c + d) = ac + ad + bc + bd.

两项式展开：(a + b)(c + d) = ac + ad + bc + bd。

(a ± b)² = a² ± 2ab + b²

完全平方公式：两数和（差）的平方等于首平方、尾平方，首尾两倍放中央。

a² – b² = (a – b)(a + b)

平方差公式：两数平方差等于两数和与差的乘积。

Laws of indices for a > 0 and rational exponents:

指数定律（底数 a > 0，指数为有理数）：

The binomial expansion for positive integer n: (1 + x)n = 1 + n x + [n(n–1)/2!] x² + … + xn.

二项式展开（n 为正整数）：(1 + x)n = 1 + n x + [n(n–1)/2!] x² + … + xn。一般形式使用组合数。

2. Equations and Inequalities | 方程与不等式

For ax² + bx + c = 0, the solutions are given by the quadratic formula:

对于 ax² + bx + c = 0，求根公式为：

x = ( –b ± √b² – 4ac ) / (2a)

The discriminant Δ = b² – roots; Δ = 0 → one repeated real root; Δ < 0 → no real roots (complex conjugate pair).

判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 ⇔ 两个不等实根；Δ = 0 ⇔ 一个重根（两个相等实根）；Δ < 0 ⇔ 无实根（一对共轭复根）。

Sum and product of roots: if α and β are roots of ax² + bx + c = 0, then α + β = –b/a and αβ = c/a.

根与系数的关系：若 α、β 为 ax² + bx + c = 0 的根，则 α + β = –b/a，αβ = c/a。

Solving linear inequalities follows the same steps as equations, but reversing the sign when multiplying or dividing by a negative number.

解一元一次不等式时，步骤与方程相同，但乘或除以负数时需反转不等号方向。

c inequalities can be solved by sketching the graph or using a sign table. For example, x² – 5x + 6 < 0 → 2 < x < 3.

二次不等式可借助图像或符号表求解。如 x² – 5x + 6 < 0 的解为 2 < x < 3。

3. Coordinate Geometry | 坐标几何

Distance between two points (x₁, y₁) and (x₂, y₂):

两点间距离公式：

d = √[(x₂ – x₁)² + (y₂ – y₁)²]

Midpoint of a segment:

线段中点坐标：

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Gradient (slope) of a line through the two points:

过两点的直线斜率：

m = (y₂ – y₁) / (x₂ – x₁)

Equation of a straight line: y – y₁ = m(x – x₁) [point‑gradient form], or y = mx + c [slope‑intercept form]. General form: ax + by + d = 0.

直线方程的几种形式：点斜式 y – y₁ = m(x – x₁)；斜截式 y = mx + c；一般式 ax + by + d = 0。

Two lines with gradients m₁, m₂ are parallel if m₁ = m₂, and perpendicular if m₁ m₂ = –1.

两直线平行 ⇔ 斜率 m₁ = m₂；垂直 ⇔ m₁ × m₂ = –1。

Equation of a circle with centre (a, b) and radius r:

圆心在 (a, b)、半径为 r 的圆的标准方程：

(x – a)² + (y – b)² = r²

The expanded form x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (–g, –f) and radius √g² + f² – c, provided g² + f² > c.

一般方程 x² + y² + 2gx + 2fy + c = 0 表示圆，圆心 (–g, –f)，半径 √g² + f² – c，要求 g² + f² > c。

4. Functions and Transformations | 函数与变换

A function f maps each input from its domain to exactly one output in its range. The vertical line test can identify a function from a graph.

函数 f 将定义域中的每一个输入映射到值域中的唯一输出。可用竖直线检验判断图像是否表示函数。

Composite function: (f ◦ g)(x) = f(g(x)). The inner function g is applied first.

复合函数：(f ◦ g)(x) = f(g(x))，先作用内层函数 g。

function f⁻¹ reverses the effect of f, such that f(f⁻¹(x)) = x. It exists only if f is one‑to‑one. Graph of f⁻¹ is the reflection of y = f(x) in the line y = x.

反函数 f⁻¹ 满足 f(f⁻¹(x)) = x，只在一一对应的函数中存在。反函数图像与 y = f(x) 关于直线 y = x 对称。

Transformations of y = f(x):

函数 y = f(x) 的图像变换：

Modulus function: |x| = x if x ≥ 0, and –x if x < 0. The graph of y = |f(x)| reflects the negative parts of f(x) in the x‑axis.

绝对值函数：|x| = x (x≥0), 或 –x (x<0)。y = |f(x)| 的图像是将 f(x) 负值部分沿 x 轴翻折而成。

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5. Trigonometry | 三角学

Radian measure: π radians = 180°. To convert degrees to radians multiply by π/180.

弧度制：π 弧度 = 180°。角度转弧度乘 π/180。

Basic definitions on the unit circle: cos θ = x‑coordinate, sin θ = y‑coordinate, tan θ = sin θ / cos θ.

单位圆定义：cos θ 为 x 坐标，sin θ 为 y 坐标，tan θ = sin θ / cos θ。

Fundamental identities:

基本三角恒等式：

sin² θ + cos² θ = 1

1 + tan² θ = sec² θ

1 + cot² θ = csc² θ

Graphs of sin, cos and tan are periodic: sin and cos have period 2π, amplitude 1; tan has period π with asymptotes.

正弦、余弦和正切函数图像具有周期性：sin 和 cos 周期 2π，振幅 1；tan 周期 π，且存在竖直渐近线。

Sine rule (for any triangle): a / sin A = b / sin B = c / sin C.

正弦定理（适用于任意三角形）：a / sin A = b / sin B = c / sin C。

Cosine rule: a² = b² + c² – 2bc cos A; also cos A = (b² + c² – a²) / (2bc).

余弦定理：a² = b² + c² – 2bc cos A；变形可得 cos A = (b² + c² – a²) / (2bc)。

Area of a triangle: ½ ab sin C, or ½ bc sin A, or ½ ac sin B.

三角形面积公式：½ ab sin C、½ bc sin A 或 ½ ac sin B。

Compound and double angle formulae:

和差角公式与倍角公式：

6. Exponentials and Logarithms | 指数与对数

The exponential function f(x) = ax (a > 0, a ≠ 1) has domain ℝ and range (0, ∞). The natural exponential function is y = ex, where e ≈ 2.71828.

指数函数 f(x) = ax（a > 0, a ≠ 1）定义域为 ℝ，值域为 (0, ∞)。自然指数函数为 y = ex，其中 e ≈ 2.71828。

Logarithm definition: if y = ax then x = logₐ y. The natural logarithm is loge x = ln x.

对数定义：若 y = ax，则 x = logₐ y。自然对数 ln x = loge x。

Laws of logarithms (for any base >0, ≠1):

对数运算律（适用于任何大于 0 且不等于 1 的底数）：

Change of base: logₐ b = logc b / logc a. Commonly, logₐ b = ln b / ln a.

换底公式：logₐ b = logc b / logc a。常用自然对数换底：logₐ b = ln b / ln a。

The equation ax = ex ln a allows any exponential to be rewritten in base e, useful in calculus.

利用恒等式 ax = ex ln a 可将任意指数函数化为以 e 为底，这在微积分中非常方便。

Exponential growth and decay model: N(t) = N₀ ek t, where k > 0 gives growth and k < 0 gives decay.

指数增长与衰减模型：N(t) = N₀ ek t，k > 0 表示增长，k < 0 表示衰减。

7. Sequences and Series | 数列与级数

Arithmetic sequence: the difference d between consecutive terms is constant. n‑th term: uₙ = a + (n – 1)d, where a = u₁.

等差数列：相邻两项的差 d 为常数。第 n 项（通项公式）uₙ = a + (n – 1)d，其中 a 为首项。

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)，l 为末项。

Geometric sequence: the ratio r between consecutive terms is constant. n‑th term: uₙ = a rn–1.

等比数列：相邻两项的比 r 为常数。第 n 项 uₙ = a rn–1。

Sum of the first n terms of a geometric series (r ≠ 1): Sₙ = a (1 – rn) / (1 – r).

等比数列前 n 项和（r ≠ 1）：Sₙ = a (1 – rn) / (1 – r)。

Sum to infinity exists when |r| < 1: S∞ = a / (1 – r).

当 |r| < 1 时，无穷等比数列的和收敛：S∞ = a / (1 – r)。

Sigma notation: Σk=1n uk. Important sums: Σk=1n k = n(n+1)/2, Σk=1n k² = n(n+1)(2n+1)/6.

求和符号（西格玛）常用于表示级数。常用求和公式：Σ k = n(n+1)/2，Σ k² = n(n+1)(2n+1)/6。

8. Calculus | 微积分

Derivative of a power function: d/dx (xn) = n xn–1, for any real n.

幂函数求导：d/dx (xn) = n xn–1，n 为任意实数。

Basic derivatives:

基本导数公式：

Rules of differentiation:

导数运算法则：