📚 Year 12 CAIE Mathematics: Formula & Theorem Quick Reference Guide | 公式定理速查手册
This quick reference handbook compiles all essential formulae and theorems covered in the Year 12 CAIE Mathematics (9709) syllabus, focusing on Pure Mathematics 1 along with a selection from Probability & Statistics 1. Keep this guide handy for revision and exam preparation.
本速查手册汇总了 Year 12 CAIE 数学 (9709) 大纲涵盖的所有核心公式和定理，重点包括纯数学 1 以及概率与统计 1 的部分内容。复习备考时请随时查阅本手册。
1. Algebraic Identities & Factor Theorem | 代数恒等式与因式定理
The square of a binomial: (a + b)² = a² + 2ab + b².
两项和的平方：(a + b)² = a² + 2ab + b²。
The square of a difference: (a – b)² = a² – 2ab + b².
两项差的平方：(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)。
Sum of cubes: a³ + b³ = (a + b)(a² – ab + b²).
立方和：a³ + b³ = (a + b)(a² – ab + b²)。
Difference of cubes: a³ – b³ = (a – b)(a² + ab + b²).
立方差：a³ – b³ = (a – b)(a² + ab + b²)。
Remainder Theorem: When a polynomial f(x) is divided by (x – a), the remainder is f(a).
余式定理：多项式 f(x) 除以 (x – a) 时，余数为 f(a)。
Factor Theorem: (x – a) is a factor of f(x) if and only if f(a) = 0.
因式定理：(x – a) 是 f(x) 的因式当且仅当 f(a) = 0。
2. Quadratic Equations & Functions | 二次方程与函数
General form: ax² + bx + c = 0, with a ≠ 0.
一般式：ax² + bx + c = 0，a ≠ 0。
The discriminant Δ = b² – s: Δ > 0 (two distinct real roots), Δ = 0 (one repeated real root), Δ < 0 (no real roots).
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），Δ < 0（无实根）。
Quadratic formula for the roots:
x = (–b ± √b² – 4ac) / (2a)
求根公式：x = (–b ± √b² – 4ac) / (2a)。
Sum of roots: α + β = –b/a; product of roots: αβ = c/a.
根的和：α + β = –b/a；根的积：αβ = c/a。
Completed square form: y = a(x – h)² + k, where the vertex is (h, k).
配方式：y = a(x – h)² + k，其中顶点坐标为 (h, k)。
The axis of symmetry is the line x = h.
对称轴为直线 x = h。
3. Inequalities & Modulus | 不等式与绝对值
Solving quadratic inequalities: sketch the graph of ax² + bx + c to identify intervals where the expression is > 0 or < 0.
解二次不等式：画出 ax² + bx + c 的图像，确定表达式大于 0 或小于 0 的区间。
For |ax + b| = c, solve ax + b = c or ax + b = –c.
解 |ax + b| = c：ax + b = c 或 ax + b = –c。
For |ax + b| < c, rewrite as –c < ax + b < c.
对于 |ax + b| < c，重写为 –c < ax + b < c。
For |ax + b| > c, solve ax + b < –c or ax + b > c.
对于 |ax + b| > c，解 ax + b < –c 或 ax + b > c。
Always check critical values when multiplying or dividing by a negative quantity in an inequality.
在不等式两端乘或除负数时务必检查临界值并反转不等号。
4. Graphs & Transformations | 函数图像与变换
Vertical translation: y = f(x) + a moves the graph up by a units.
垂直平移：y = f(x) + a 将图像向上平移 a 个单位。
Horizontal translation: y = f(x + a) moves the graph left by a units (a > 0).
水平平移：y = f(x + a) 将图像向左平移 a 个单位（a > 0）。
= a f(x) stretches the graph vertically by factor a (a > 1) or compresses it (0 < a < 1).
垂直伸缩：y = a f(x) 将图像沿垂直方向拉伸 a 倍 (a > 1) 或压缩 (0 < a < 1)。
f(ax) compresses the graph horizontally by factor 1/a (a > 1) or stretches it (0 < a < 1).
水平伸缩：y = f(ax) 将图像沿水平方向压缩至 1/a 倍 (a > 1) 或拉伸 (0 < a < 1)。
Reflection in the x‑axis: y = –f(x). Reflection in the y‑axis: y = f(–x).
沿 x 轴反射：y = –f(x)。沿 y 轴反射：y = f(–x)。
Know the basic shapes of y = xⁿ, y = √x, y = 1/x, y = |x| and their transformed versions.
熟记 y = xⁿ、y = √x、y = 1/x、y = |x| 的基本图像及其变换后的形状。
5. Coordinate Geometry | 坐标几何
Distance between two points (x₁, y₁) and (x₂, y₂): d = √[(x₂ – x₁)² + (y₂ – y₁)²].
两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。
Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
中点：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
Gradient (slope) of the line through the two points: m = (y₂ – y₁)/(x₂ – x₁).
过这两点的直线的斜率：m = (y₂ – y₁)/(x₂ – x₁)。
Equation of a straight line: y – y₁ = m(x – x₁), or y = mx + c.
直线方程：y – y₁ = m(x – x₁)，或 y = mx + c。
If two lines are perpendicular, the product of their gradients is –1: m₁ m₂ = –1.
若两直线垂直，其斜率之积为 –1：m₁ m₂ = –1。
Equation of a circle with centre (a, b) and radius r: (x – a)² + (y – b)² = r².
圆心为 (a, b)、半径为 r 的圆的方程：(x – a)² + (y – b)² = r²。
The general form x² + y² + 2gx + 2fy + c = 0 has centre (–g, –f) and radius √g² + f² – c.
一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (–g, –f)，半径为 √g² + f² – c。
6. Trigonometry | 三角学
Conversion between degrees and radians: π radians = 180°.
度与弧度换算：π 弧度 = 180°。
Fundamental Pythagorean identity: sin²θ + cos²θ = 1.
基本毕达哥拉斯恒等式：sin²θ + cos²θ = 1。
Definition of tangent: tan θ = sin θ / cos θ.
正切定义：tan θ = sin θ / cos θ。
Exact values for key angles:
| θ (radians) | 0 | π/6 | π/4 | π/3 | π/2 |
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
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