Pre-Calculus: Formula Compendium | 预备微积分：公式大全

📚 Pre-Calculus: Formula Compendium | 预备微积分：公式大全

Welcome to your ultimate Pre-Calculus formula reference. This compendium brings together essential algebraic manipulations, function properties, trigonometric identities, conic section equations, sequences, vectors, and limit rules — all the building blocks you need before stepping into calculus. Use it for quick revision or as a dependable cheat sheet during practice.

欢迎来到预备微积分公式大全。本合集汇集了代数运算、函数性质、三角恒等式、圆锥曲线方程、数列、向量与极限法则等重要公式，是你踏入微积分之前的必备基石。可供快速复习或练习时参照。

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1. Algebraic Foundations | 代数基础

Exponent rules: aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ (a ≠ 0). For any non‑zero a: a⁰ = 1, a⁻ⁿ = 1 / aⁿ. Rational exponents: a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ.

指数运算法则：aᵐ · aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，(ab)ⁿ = aⁿbⁿ，aᵐ / aⁿ = aᵐ⁻ⁿ (a ≠ 0)。对任何非零 a：a⁰ = 1，a⁻ⁿ = 1 / aⁿ。有理指数：a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ。

Radical properties: √(ab) = √a · √b, √(a / b) = √a / √b (a, b ≥ 0). The expression √(a²) = |a|.

根式性质：√(ab) = √a · √b， √(a / b) = √a / √b (a, b ≥ 0)。表达式 √(a²) = |a|。

Logarithm laws (a > 0, a ≠ 1, M, N > 0): logₐ(MN) = logₐM + logₐN; logₐ(M / N) = logₐM − logₐN; logₐ(Mᵖ) = p logₐM. Change of base: logₐb = (log꜀b) / (log꜀a). Natural log: ln e = 1, ln 1 = 0.

对数法则 (a > 0, a ≠ 1, M, N > 0)：logₐ(MN) = logₐM + logₐN；logₐ(M / N) = logₐM − logₐN；logₐ(Mᵖ) = p logₐM。换底公式：logₐb = (log꜀b) / (log꜀a)。自然对数：ln e = 1， ln 1 = 0。

Key factorisations: a² − b² = (a − b)(a + b); a² ± 2ab + b² = (a ± b)²; a³ ± b³ = (a ± b)(a² ∓ ab + b²).

重要因式分解：a² − b² = (a − b)(a + b)；a² ± 2ab + b² = (a ± b)²；a³ ± b³ = (a ± b)(a² ∓ ab + b²)。

Quadratic Formula: x = [−b ± √(b² − 4ac)] / (2a)

二次公式：x = [−b ± √(b² − 4ac)] / (2a)

Discriminant Δ = b² − 4ac: Δ > 0 → two distinct real roots; Δ = 0 → one real double root; Δ < 0 → two non‑real complex roots.

判别式 Δ = b² − 4ac：Δ > 0 → 两个不同实根；Δ = 0 → 一个实重根；Δ < 0 → 两个非实复根。

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2. Functions and Graphs | 函数与图像

A function f is even if f(−x) = f(x) (symmetric about the y‑axis). It is odd if f(−x) = −f(x) (symmetric about the origin).

函数 f 为偶函数若 f(−x) = f(x)（关于 y 轴对称）。为奇函数若 f(−x) = −f(x)（关于原点对称）。

Vertical shift: y = f(x) + k (up) / y = f(x) − k (down). Horizontal shift: y = f(x − h) shifts right by h. Vertical stretch/compression: y = a·f(x) (|a| > 1 stretch; 0 < |a| < 1 compression). Reflection: y = −f(x) reflects across x‑axis; y = f(−x) reflects across y‑axis.

垂直平移：y = f(x) + k（上移） / y = f(x) − k（下移）。水平平移：y = f(x − h) 向右移 h。垂直伸缩：y = a·f(x)（|a| > 1 拉伸；0 < |a| < 1 压缩）。反射：y = −f(x) 关于 x 轴反射；y = f(−x) 关于 y 轴反射。

Inverse function: f⁻¹(f(x)) = x and f(f⁻¹(y)) = y. Graphically, y = f⁻¹(x) is the reflection of y = f(x) in the line y = x.

反函数：f⁻¹(f(x)) = x 且 f(f⁻¹(y)) = y。图像上，y = f⁻¹(x) 是 y = f(x) 关于直线 y = x 的反射。

Composition: (f ∘ g)(x) = f(g(x)). Domain restriction: g(x) must be in the domain of f.

复合函数：(f ∘ g)(x) = f(g(x))。定义域限制：g(x) 必须在 f 的定义域内。

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3. Polynomial and Rational Functions | 多项式与有理函数

Factor Theorem: (x − c) is a factor of P(x) iff P(c) = 0. Remainder Theorem: when P(x) is divided by (x − c), the remainder is P(c).

因式定理：(x − c) 是 P(x) 的因式当且仅当 P(c) = 0。余式定理：P(x) 除以 (x − c) 的余式为 P(c)。

Rational Root Theorem: for P(x) = aₙxⁿ + … + a₀ with integer coefficients, any rational root p/q has p dividing a₀ and q dividing aₙ.

有理根定理：对于整系数多项式 P(x) = aₙxⁿ + … + a₀，任何有理根 p/q 满足 p 整除 a₀ 且 q 整除 aₙ。

Vertex form of a quadratic: y = a(x − h)² + k, vertex (h, k). Standard form: y = ax² + bx + c has vertex (−b/(2a), f(−b/(2a))).

二次函数顶点式：y = a(x − h)² + k，顶点 (h, k)。一般式：y = ax² + bx + c 的顶点为 (−b/(2a), f(−b/(2a)))。

Rational function asymptotes: For R(x) = P(x)/Q(x), vertical asymptotes at zeros of Q(x) (provided no common factor with P). Horizontal: if deg(P) < deg(Q) → y = 0; if deg(P) = deg(Q) → y = leading coefficient ratio; if deg(P) = deg(Q) + 1 → oblique asymptote (by polynomial division).

有理函数渐近线：对于 R(x) = P(x)/Q(x)，垂直渐近线出现在 Q(x) 的零点（且与 P 无公因式）。水平渐近线：若 deg(P) < deg(Q) → y = 0；若 deg(P) = deg(Q) → y = 首项系数之比；若 deg(P) = deg(Q) + 1 → 斜渐近线（由多项式除法求得）。

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4. Exponential and Logarithmic Functions | 指数与对数函数

Exponential model: y = a·bˣ (b > 0, b ≠ 1). Natural exponential base e ≈ 2.71828. Continuous growth/decay: A(t) = A₀e^(kt), where k > 0 for growth, k < 0 for decay.

指数模型：y = a·bˣ (b > 0, b ≠ 1)。自然指数底 e ≈ 2.71828。连续增长/衰减：A(t) = A₀e^(kt)，其中 k > 0 为增长，k < 0 为衰减。

Logarithmic form: logₐy = x ⇔ aˣ = y. Properties: logₐ(aˣ) = x, a^(logₐx) = x. Common log: log₁₀x; natural log: ln x.

对数形式：logₐy = x ⇔ aˣ = y。性质：logₐ(aˣ) = x，a^(logₐx) = x。常用对数：log₁₀x；自然对数：ln x。

Half‑life formula (decay): t₁/₂ = (ln 2) / |k|. Doubling time (growth): T₂ = (ln 2) / k.

半衰期公式（衰减）：t₁/₂ = (ln 2) / |k|。倍增时间（增长）：T₂ = (ln 2) / k。

pH = −log[H⁺]    Richter scale: R = log(I / I₀)

pH = −log[H⁺]    里氏震级：R = log(I / I₀)

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5. Trigonometric Functions | 三角函数

Radian measure: 180° = π rad. Conversion: rad = deg × (π / 180). Arc length: s = rθ. Sector area: A = ½ r²θ.

弧度制：180° = π rad。转换：弧度 = 度数 × (π / 180)。弧长：s = rθ。扇形面积：A = ½ r²θ。

Unit‑circle definitions: sin θ = y / r, cos θ = x / r, tan θ = y / x (x ≠ 0). Reciprocal functions: csc θ = 1 / sin θ, sec θ = 1 / cos θ, cot θ = 1 / tan θ.

单位圆定义：sin θ = y / r，cos θ = x / r，tan θ = y / x (x ≠ 0)。倒数函数：csc θ = 1 / sin θ，sec θ = 1 / cos θ，cot θ = 1 / tan θ。

Key right‑triangle ratios: sin = opp/hyp, cos = adj/hyp, tan = opp/adj. Special angle values (θ in radians): sin(π/6)=½, cos(π/3)=½, tan(π/4)=1.

直角三角形关键比值：sin = 对边/斜边，cos = 邻边/斜边，tan = 对边/邻边。特殊角值（弧度）：sin(π/6)=½

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