📚 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）：
- am × an = am + n | 同底数幂相乘，指数相加
- am ÷ an = am – n | 同底数幂相除，指数相减
- (am)n = am n | 幂的乘方，指数相乘
- (a b)n = an bn | 积的乘方，各因子分别乘方
- a0 = 1, a–n = 1 / an | 零指数和负指数
- am/n = n√(am) = (n√a)m | 分数指数与根式的互化
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 – 4ac determines the nature of the roots: Δ > 0 → two distinct real 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。
Quadratic 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。
Inverse 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：
- y = f(x) + k → vertical translation by k | 垂直平移 k 个单位
- y = f(x + h) → horizontal translation by –h | 水平平移 –h 个单位（左加右减）
- y = a f(x) → vertical stretch by factor a | 垂直方向伸缩 a 倍
- y = f(bx) → horizontal stretch by factor 1/b | 水平方向伸缩 1/b 倍
- y = –f(x) → reflection in x‑axis | 关于 x 轴对称
- y = f(–x) → reflection in y‑axis | 关于 y 轴对称
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。
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：
- sin(A ± B) = sin A cos B ± cos A sin B | sin(A ± B)
- cos(A ± B) = cos A cos B ∓ sin A sin B | cos(A ± B)
- tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B) | tan(A ± B)
- sin 2θ = 2 sin θ cos θ | 正弦倍角
- cos 2θ = cos² θ – sin² θ = 2 cos² θ – 1 = 1 – 2 sin² θ | 余弦倍角
- tan 2θ = 2 tan θ / (1 – tan² θ) | 正切倍角
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）：
- logₐ (MN) = logₐ M + logₐ N | 积的对数等于对数的和
- logₐ (M/N) = logₐ M – logₐ N | 商的对数等于对数的差
- logₐ (Mk) = k logₐ M | 幂的对数等于指数与底数对数的积
- logₐ a = 1, logₐ 1 = 0 | 底数的对数为 1，1 的对数为 0
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：
- d/dx (ex) = ex
- d/dx (ln x) = 1/x
- d/dx (sin x) = cos x
- d/dx (cos x) = –sin x
- d/dx (tan x) = sec² x
Rules of differentiation:
导数运算法则：
- Constant multiple: d/dx [k f(x)] = k f'(x) | 常数倍法则
- Sum/Difference: d/dx [f(x) ± g(x)] = f'(x) ± g'(x) | 加减法则
- Product rule: d/dx [f(x) g(x)] = f'(x) g(x) + f(x) g'(x) | 乘法法则
- Quotient rule: d/dx [f(x)/g(x)] = [f'(x)g(x) – f(x)g'(x)] / [g(x)]² | 除法法则
- Chain rule: d/dx f(g(x)) = f'(g(x)) ·
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