📚 Year 10 SQA Advanced Mathematics: Core Topics Summary | Year 10 SQA 进阶数学：核心知识点梳理
Mastering the core topics in Year 10 SQA Advanced Mathematics is essential for building a strong foundation for Higher and beyond. This article systematically reviews the essential concepts across algebra, functions, trigonometry, calculus, vectors, matrices, sequences, probability, and more. Each section presents key definitions, formulas, and solving strategies to help you consolidate your understanding.
掌握 Year 10 SQA 进阶数学的核心主题，是为高级课程及更远学习打下坚实基础的关键。本文系统梳理了代数、函数、三角学、微积分、向量、矩阵、数列、概率等重点领域的基本概念、公式与解题策略，帮助你巩固理解。
1. Algebraic Manipulation and Quadratic Equations | 代数运算与二次方程
Algebraic fluency involves expanding brackets, factorising expressions, and simplifying rational expressions. For quadratics, the three main solution methods are factorising, completing the square, and using the quadratic formula.
代数运算的流畅性包括展开括号、因式分解和化简有理表达式。对于二次方程，三种主要解法为因式分解、配方法和使用求根公式。
The quadratic formula for ax² + bx + c = 0 is:
二次方程 ax² + bx + c = 0 的求根公式为：
x = [-b ± √b² - 4ac] / (2a)
ac determines the nature of roots: positive Δ gives two distinct real roots, zero Δ gives one repeated real root, and negative Δ gives two complex roots.
判别式 Δ = b² - 4ac 决定根的性质：Δ 为正有两个不等实根，Δ 为零有一个重根，Δ 为负有两个共轭复根。
Completing the square rewrites ax² + bx + c as a(x + b/(2a))² + (c - b²/(4a)). This is useful for finding vertices of parabolas and solving equations.
配方法将 ax² + bx + c 写成 a(x + b/(2a))² + (c - b²/(4a))，这有助于求抛物线的顶点以及解方程。
2. Functions and Graphs | 函数与图像
A function maps each input (x) to exactly one output f(x). The domain is the set of permissible inputs, and the range is the set of possible outputs. Composite functions such as f(g(x)) apply one function to the output of another.
函数将每个输入 x 唯一映射到一个输出 f(x)。定义域是允许输入的集合，值域是可能输出的集合。复合函数如 f(g(x)) 是将一个函数作用于另一函数的输出。
Inverse functions 'undo' ne y = x. The horizontal line test determines whether the inverse will also be a function.
反函数“撤销”原函数的作用：若 f(a) = b，则 f⁻¹(b) = a。从图像上看，反函数是原函数关于直线 y = x 的对称。水平线检验可判断反函数是否仍是函数。
zontally by -h, y = kf(x) stretches vertically by factor k, and y = f(kx) compresses horizontally by factor 1/k.
图像变换包括平移、伸缩和反射。对于 y = f(x)，y = f(x) + k 沿纵向平移 k，y = f(x + h) 沿横向平移 -h，y = kf(x) 纵向拉伸 k 倍，y = f(kx) 横向压缩为 1/k 倍。
3. Polynomials and Factor/Remainder Theorems | 多项式与因式/剩余定理
Theorem states that R = P(a). If the remainder is zero, then (x - a) is a factor of P(x).
多项式 P(x) 除以 (x - a) 可得商式 Q(x) 和余数 R：P(x) = (x - a)Q(x) + R。剩余定理指出 R = P(a)。若余数为零，则 (x - a) 是 P(x) 的一个因式。
The Factor Theorem is often used with integer root testing to factorise cubic or higher-degree polynomials. Synthetic division provides a quick way to perform the division and factorise.
因式定理常与整数根检验结合使用，用于分解三次或更高次多项式。综合除法可以快速进行除法并完成因式分解。
iding yields x² + x - 2 = (x - 1)(x + 2), giving the full factorised form (x - 1)²(x + 2).
例如，对于 P(x) = x³ - 3x + 2，检验 a = 1 得 P(1) = 0，故 (x - 1) 为因式。除法得 x² + x - 2 = (x - 1)(x + 2)，最终完全分解为 (x - 1)²(x + 2)。
4. Coordinate Geometry: Lines and Circles | 坐标几何：直线与圆
]. The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). The gradient m of a line through these points is (y₂ - y₁)/(x₂ - x₁).
点 (x₁, y₁) 和 (x₂, y₂) 之间的距离为 √[(x₂ - x₁)² + (y₂ - y₁)²]，中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。直线斜率为 m = (y₂ - y₁)/(x₂ - x₁)。
-intercept. Parallel lines have equal gradients; perpendicular lines satisfy m₁ · m₂ = -1.
直线方程：y - y₁ = m(x - x₁)，或 y = mx + c，其中 c 为 y 截距。平行线斜率相等；垂直线满足 m₁ · m₂ = -1。
². The intersection of a line and a circle can be found by substituting the line equation into the circle and solving the resulting quadratic.
以 (h, k) 为圆心、r 为半径的圆的方程为 (x - h)² + (y - k)² = r²。直线与圆的交点可通过将直线方程代入圆的方程，并解所得二次方程来求得。
5. Trigonometry: Radians, Identities, and Equations | 三角学：弧度、恒等式与方程
area = ½r²θ, where θ is in radians. These formulas are essential in calculus and geometry.
角度可用弧度计量：180° = π 弧度，1 弧度 ≈ 57.3°。弧长 = rθ，扇形面积 = ½r²θ，其中 θ 单位为弧度。这些公式在微积分和几何中至关重要。
uble-angle formulas: sin2θ = 2 sinθ cosθ, cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ.
重要的三角恒等式包括 sin²θ + cos²θ = 1，tanθ = sinθ / cosθ，以及倍角公式：sin2θ = 2 sinθ cosθ，cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ。
Solving trigonometric equations often requires using identities to rewrite in terms of a single trig function, adjusting for the given domain, and considering all quadrants using CAST or similar diagrams.
解三角方程常需利用恒等式化为单一三角函数形式，并依据给定区域，利用 CAST 图等考虑所有象限来求通解。
6. Exponentials and Logarithms | 指数与对数
Exponential functions are of the form aˣ, with base a > 0. Key index laws: aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a⁰ = 1.
指数函数形式为 aˣ，底数 a > 0。基本指数律：aᵐ · aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，a⁰ = 1。
The logarithm logₐb is the exponent to which a must be raised to obtain b: aˣ = b ⇔ x = logₐb. Common bases are 10 and e (natural log, ln).
对数 logₐb 表示为使 a 的幂等于 b 所需的指数：aˣ = b ⇔ x = logₐb。常用底数为 10 和 e（自然对数 ln）。
y, logₐ(x/y) = logₐx - logₐy, logₐ(xⁿ) = n logₐx. Change of base: logₐb = (logₓb)/(logₓa).
对数法则：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx - logₐy，logₐ(xⁿ) = n logₐx。换底公式：logₐb = (logₓb)/(logₓa)。
Equations like aˣ re solved by taking logs: x = logₐb. Exponential growth/decay models use the form y = A eᵏˣ.
解形如 aˣ = b 的方程可两边取对数：x = logₐb。指数增长与衰减模型常用 y = A eᵏˣ 形式。
7. Differentiation: Rules and Applications | 微分法：法则与应用
The derivative f' radient of the tangent to the curve y = f(x). For a power function y = xⁿ, dy/dx = n xⁿ⁻¹.
导数 f'(x) = dy/dx 表示瞬时变化率或曲线 y = f(x) 的切线斜率。对于幂函数 y = xⁿ，dy/dx = n xⁿ⁻¹。
Rules: Constant multiple: d/dx [k f(x)] = k f'(x). Sum/difference: d/dx [f(x) ± g(x)] = f'(x) ± g'(x). Product: d/dx [u v] = u' v + u v'. Quotient: d/dx [u/v] = (u' v - u v') / v². Chain rule: d/dx [f(g(x))] = f'(g(x)) · g'(x).
法则：常数倍：d/dx [k f(x)] = k f'(x)；和差法：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)；积的法则：d/dx [u v] = u' v + u v'；商的法则：d/dx [u/v] = (u' v - u v') / v²；链式法则：d/dx [f(g(x))] = f'(g(x)) · g'(x)。
Derivatives help find equations of tangents and normals: tangent gradient = f'(a), normal gradient = -1/f'(a). Stationary points occur where f'(x) = 0; determine maxima/minima using the second derivative or sign change.
导数可用于求切线和法线方程：切线斜率 = f'(a)，法线斜率 = -1/f'(a)。驻点出现在 f'(x) = 0 处，可通过二阶导数或符号变化判断极大值或极小值。
8. Integration: Anti-differentiation and Areas | 积分法：反导数与面积
Integration is the reverse of differentiation: if F') = f(x), then ∫ f(x) dx = F(x) + C. The basic rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1.
积分是微分的逆运算：若 F'(x) = f(x)，则 ∫ f(x) dx = F(x) + C。基本法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ -1。
the axis are positive, below are negative. The total bounded area requires splitting the interval at points where the curve crosses the axis.
定积分 ∫ₐᵇ f(x) dx 给出曲线与 x 轴之间从 x = a 到 x = b 的净面积。轴上方面积为正，下方面积为负。计算所围总面积需在曲线穿越 x 轴的点处分割区间。
curve) dx. This extends to solving practical problems in kinematics, such as finding displacement from velocity.
积分还可求两曲线间的面积：∫ (上曲线 - 下曲线) dx。这可以拓展到运动学中的实际应用，如由速度求位移。
9. Vectors: Magnitude, Direction, and Dot Product | 向量：模、方向与点积
ctor has both magnitude and direction. In 2D, a vector v = (x, y) = xi + yj. Its magnitude |v| = √x² + y². In 3D, v = (x, y, z) with |v| = √x² + y² + z².
向量既有大小又有方向。二维中，向量 v = (x, y) = xi + yj，其模 |v| = √x² + y²。三维中，v = (x, y, z)，模为 |v| = √x² + y² + z²。
The dot (scalar) product is defined as a · b = |a||b| e between vectors and to test perpendicularity: a · b = 0 means vectors are perpendicular.
点积（标量积）定义为 a · b = |a||b| cos θ，分量形式为 a · b = x₁x₂ +
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