📚 Year 9 Cambridge Additional Mathematics: Formula & Theorems Quick Reference | 进阶数学公式定理速查手册
This concise handbook brings together the essential formulas, identities and theorems you will encounter in the Year 9 Cambridge Additional Mathematics course. Keep it handy for revision, homework and quick fact-checking.
本手册汇集了你在九年级剑桥附加数学课程中将会用到的所有核心公式、恒等式与定理，方便复习、写作业和快速查阅。
1. Algebra Essentials | 代数基础
Laws of indices form the backbone of algebraic manipulation.
指数定律是代数运算的根基。
Product rule: multiply powers with the same base by adding the exponents.
乘法法则：同底数幂相乘，指数相加。
am × an = am+n
Quotient rule: divide powers by subtracting the exponents.
除法法则：同底数幂相除，指数相减。
am ÷ an = am−n
Power of a power: multiply the exponents.
幂的乘方：指数相乘。
(am)n = amn
Zero exponent: any non‑zero base raised to zero equals 1.
零指数：任何非零数的零次幂等于 1。
a0 = 1 (a ≠ 0)
Negative exponent: reciprocal of the positive power.
负指数：等于正指数幂的倒数。
a−n = 1 / an
Expanding brackets often uses the distributive law and special products.
展开括号常应用分配律和特殊乘积。
(a + b)2 = a2 + 2ab + b2
(a − b)2 = a2 − 2ab + b2
Factorising reverses expansion. The difference of two squares is a key pattern.
因式分解是展开的逆过程。平方差是一个关键模型。
a2 − b2 = (a + b)(a − b)
2. Quadratic Equations & Theorems | 二次方程与定理
A quadratic equation has the form ax2 + bx + c = 0, with a ≠ 0.
二次方程的形式为 ax2 + bx + c = 0，其中 a ≠ 0。
The quadratic formula gives the solutions directly.
求根公式可直接给出方程的解。
x = [ −b ± √b2 − 4ac ] / 2a
The discriminant Δ = b2 − 4ac determines the nature of the roots.
判别式 Δ = b2 − 4ac 决定了根的性质。
Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.
若 Δ > 0，有两个不等实根；Δ = 0，一个重根；Δ < 0，无实根。
Sum and product of roots (Vieta's formulas) link the coefficients.
韦达定理将根的和与积同系数联系起来。
Sum of roots= −b/a, Product of roots= c/a
Completing the square transforms the quadratic and reveals the vertex.
配方法可转换二次式并揭示顶点。
ax2 + bx + c = a(x + b/2a)2 + (c − b2/4a)
3. Functions | 函数
Function notation: f(x) denotes the output of a function f for an input x.
函数记号：f(x) 表示函数 f 对于输入 x 的输出。
The domain is the set of all allowed inputs; the range is the set of all possible outputs.
定义域是所有允许输入的集合；值域是所有可能输出的集合。
A composite function applies one function after another.
复合函数是将一个函数接在另一个函数之后运用。
(f ∘ g)(x) = f(g(x))
An inverse function reverses the effect of the original function. It exists only if the function is one‑to‑one.
反函数逆转原函数的效果。只有单射函数才存在反函数。
If f(a) = b, then f−1(b) = a. f−1(f(x)) = x.
To find an inverse, write y = f(x), swap x and y, then solve for y.
求反函数时，设 y = f(x)，互换 x 与 y，然后解出 y。
4. Coordinate Geometry | 坐标几何
The distance between two points (x₁, y₁) and (x₂, y₂) comes from Pythagoras.
两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离由勾股定理得出。
d = √[(x₂ − x₁)2 + (y₂ − y₁)2]
The midpoint has coordinates that are the averages of the endpoints.
中点坐标是端点坐标的平均值。
M = ((x₁+x₂)/2 , (y₁+y₂)/2)
Gradient (slope) measures steepness.
斜率表示倾斜程度。
m = (y₂ − y₁) / (x₂ − x₁)
Equation of a straight line can be written in several forms.
直线方程有多种形式。
y = mx + c (slope‑intercept form)
y − y₁ = m(x − x₁) (point‑slope form)
Parallel lines have equal gradients; perpendicular lines have gradients whose product is −1.
平行线斜率相等；垂直线斜率之积为 −1。
m₁ = m₂ (parallel); m₁ × m₂ = −1 (perpendicular)
5. Sequences and Series | 数列与级数
An arithmetic progression (AP) has a common difference d.
等差数列有公差 d。
nth term: un = a + (n − 1)d
Sum of first n terms: Sn = n/2 [2a + (n−1)d] = n/2 (a + l)
A geometric progression (GP) has a common ratio r.
等比数列有公比 r。
nth term: un = a rn−1
Sum of first n terms: Sn = a(1 − rn)/(1 − r) (r ≠ 1)
For an infinite GP with |r| < 1, the sum converges.
当 |r| < 1 时，无穷等比级数收敛。
S∞ = a / (1 − r)
6. Trigonometry | 三角学
Basic trigonometric ratios for a right‑angled triangle:
直角三角形的基本三角比：
opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
Key identities bind the functions together.
关键恒等式将三角函数联系起来。
sin2θ + cos2θ = 1
tan θ = sin θ / cos θ
The sine rule is used for non‑right‑angled triangles.
正弦定理用于非直角三角形。
a / sin A = b / sin B = c / sin C
The cosine rule links sides and an included angle.
余弦定理将边与夹角关联起来。
a2 = b2 + c2 − 2bc cos A
Area of a triangle can be found using two sides and the included angle.
三角形面积可用两边及其夹角求出。
Area = ½ ab sin C
7. Exponentials and Logarithms | 指数与对数
Exponential functions have the form f(x) = ax (a > 0). The natural exponential is ex where e ≈ 2.718.
指数函数形如 f(x) = ax (a > 0)。自然指数为 ex，e ≈ 2.718。
Logarithms are the inverses of exponentials.
对数是指数函数的逆运算。
If ax = b, then logab = x. logaax = x.
Logarithm laws simplify complicated expressions.
对数运算法则可简化复杂表达式。
loga(mn) = logam + logan
loga(m/n) = logam − logan
loga(mk) = k logam
Change of base formula is useful for calculations.
换底公式在计算中很有用。
logab = logcb / logca
8. Differentiation Rules | 微分法则
The derivative f'(x) or dy/dx gives the gradient of a curve.
导数 f'(x) 或 dy/dx 表示曲线的斜率。
Basic derivatives to memorise:
需要牢记的基本导数：
d/dx (xn) = n xn−1
d/dx (ex) = ex； d/dx (ln x) = 1/x
d/dx (sin x) = cos x； d/dx (cos x) = −sin x
The chain rule handles composite functions.
链式法则用于复合函数。
dy/dx = dy/du × du/dx (or f'(g(x))·g'(x))
The product rule: derivative of u·v.
乘法法则：u·v 的导数。
d/dx (uv) = u'v + uv'
The quotient rule: derivative of u/v.
除法法则：u/v 的导数。
d/dx (u/v) = (u'v − uv') / v2
Stationary points occur where f'(x) = 0. Use second derivative to classify.
驻点满足 f'(x) = 0。用二阶导数判别极值。
If f''(x) > 0: minimum; f''(x) < 0: maximum.
9. Integration Fundamentals | 积分基础
Integration reverses differentiation and finds the area under a curve.
积分是微分的逆运算，并用于求曲线下面积。
∫ xn dx = xn+1/(n+1) + C (n ≠ −1)
∫ ex dx = ex + C ； ∫ 1/x dx = ln |x| + C
∫ sin x dx = −cos x + C ； ∫ cos x dx = sin x + C
The definite integral between limits a and b yields the net area.
定积分从 a 到 b 给出净面积。
∫ab f(x) dx = F(b) − F(a), where F'(x) = f(x).
Remember to add the constant of integration C for indefinite integrals.
不定积分务必要加上积分常数 C。
10. Vectors | 向量
A vector has magnitude and direction; it can be written as a column or using i, j notation.
向量有大小和方向；可用列向量或 i, j 表示法写出。
a = (x, y) = x i +
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