📚 MA01 Pure Mathematics Unit 1 Key Topics Explained | MA01 纯数单元1 核心知识点精讲
This article revises the core concepts from the International AS Pure Mathematics Unit 1 (MA01) specification. Each section pairs concise English explanations with matched Chinese translations, helping you build a solid foundation in algebra, coordinate geometry, trigonometry, and calculus for the Oxford AQA 9660 examination.
本文系统梳理国际AS纯数单元1（MA01）的核心知识点。每个小节采用中英对照精讲模式，帮助你在代数、坐标几何、三角学与微积分模块打下扎实基础，从容应对牛津AQA 9660考试。
1. Quadratic Functions and the Discriminant | 二次函数与判别式
The general quadratic is f(x) = ax² + bx + c (a ≠ 0). Its discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots (only complex). This tool is essential for analysing intersection points and solving quadratic inequalities.
一般二次函数为 f(x)=ax²+bx+c (a≠0)。判别式 Δ=b²−4ac 决定根的性质：Δ>0 有两个不等实根，Δ=0 有一个重实根，Δ<0 无实根。这一工具常用于分析图像交点与解二次不等式。
- Completing the square: ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a)). It reveals the vertex and axis of symmetry.
- 配方法：ax²+bx+c = a(x + b/(2a))² + (c − b²/(4a))，可直接读出顶点坐标和对称轴。
2. Quadratic Inequalities and Region Testing | 二次不等式与区域检验
To solve ax² + bx + c > 0, first find the roots r₁, r₂ (real and distinct). Sketch the parabola: if a > 0, the curve is ∪‑shaped, so the inequality holds for x < r₁ or x > r₂. If a < 0, it holds between the roots. Use a sign table or test points to confirm regions.
解 ax²+bx+c>0 时，先求出两实根 r₁, r₂。若 a>0 抛物线开口向上，解集为 x
Always rearrange the inequality to have zero on one side before solving. Never multiply by a negative expression without reversing the inequality sign.
解题前务必将不等式一侧化为零。乘除负数表达式时必须反转不等号方向。
3. Polynomials and the Factor Theorem | 多项式与因子定理
For a polynomial p(x), if p(a) = 0 then (x − a) is a factor. This factor theorem helps factorise cubics and quartics. Combined with polynomial division (long division or synthetic division), we can fully factorise and find all roots.
对于多项式 p(x)，若 p(a)=0，则 (x−a) 是其一个因子。利用因子定理结合长除法或综合除法，可对三次、四次多项式完全分解并求出所有根。
Remainder theorem: When p(x) is divided by (x − a), the remainder is p(a). This avoids full division when only the remainder is needed.
余数定理：p(x) 除以 (x−a) 的余数等于 p(a)，只需求余数时无需完整除法。
4. Binomial Expansion for Rational n | 有理指数的二项展开
The binomial theorem for positive integer n gives (a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ. For |x| < 1 and rational n, (1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … . This infinite series expansion is valid only when |x| < 1.
正整数指数二项定理：(a+b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ。在 |x|<1 且 n 为有理数时，(1+x)ⁿ 可展开为无穷级数 1+nx+[n(n−1)/2!]x²+[n(n−1)(n−2)/3!]x³+…，需注意收敛域。
Approximations: Often we use only the first few terms to estimate values, e.g., √(1+x) ≈ 1 + ½ x − ⅛ x² for small x.
近似计算：常用前几项对根式估值，如当 x 很小时 √(1+x) ≈ 1+½ x−⅛ x²。
5. Coordinate Geometry: Straight Lines | 坐标几何：直线
The gradient between (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). Equations of a straight line: y − y₁ = m(x − x₁), y = mx + c, ax + by + c = 0. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
两点间斜率 m=(y₂−y₁)/(x₂−x₁)。直线方程有三种形式：点斜式 y−y₁=m(x−x₁)，斜截式 y=mx+c，一般式 ax+by+c=0。平行线斜率相等，垂直线斜率乘积为−1。
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Distance: √[(x₂−x₁)² + (y₂−y₁)²]. These are fundamental tools for geometric proofs.
中点坐标与距离公式分别是 ((x₁+x₂)/2, (y₁+y₂)/2) 和 √[(x₂−x₁)²+(y₂−y₁)²]，是几何证明的基础工具。
6. Coordinate Geometry: Circles | 坐标几何：圆
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². The expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c). Completing the square retrieves centre and radius from the general form.
圆心 (a,b)、半径 r 的圆方程为 (x−a)²+(y−b)²=r²。一般式 x²+y²+2gx+2fy+c=0 对应的圆心为 (−g,−f)，半径为 √(g²+f²−c)。通过配方法可从一般式求出圆心与半径。
The discriminant method applied to a line and a circle gives the number of intersection points (2, 1 for tangent, 0). A tangent is perpendicular to the radius at the point of contact.
用判别式处理直线与圆的交点：Δ>0 两个交点，Δ=0 相切，Δ<0 无交点。切线垂直于过切点的半径。
7. Trigonometry: Ratios and Identities | 三角学：比值与恒等式
Define sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Key identities: tan θ ≡ sin θ / cos θ, sin²θ + cos²θ ≡ 1. These help simplify expressions and solve equations in a given interval.
定义 sinθ=对边/斜边，cosθ=邻边/斜边，tanθ=对边/邻边。核心恒等式：tanθ ≡ sinθ/cosθ；sin²θ+cos²θ ≡ 1。利用它们可化简三角式并在给定区间内解方程。
Exact values for 30°, 45°, 60° and related angles must be memorised, often expressed as surds.
必须熟记 30°、45°、60°及其相关角的精确三角值，通常以根式表示。
8. Radian Measure and Sector Geometry | 弧度制与扇形几何
Radians are defined so that π rad = 180°. Arc length s = rθ, sector area A = ½ r²θ, where θ is in radians. This simplifies calculus derivatives of trig functions and provides a natural measure for circular motion.
弧度制定义 π rad = 180°。弧长公式 s=rθ，扇形面积公式 A=½ r²θ，其中 θ 必须以弧度为单位。这简化了三角函数的微积分运算，也更适合描述圆周运动。
Converting degrees to radians: multiply by π/180. Always check the mode on your calculator.
角度转弧度乘以 π/180，解题时务必注意计算器模式设置。
9. Differentiation: First Principles and Rules | 微分：第一原理与求导法则
The derivative f'(x) = lim(h→0) [f(x+h)−f(x)]/h. For xⁿ, d/dx (xⁿ) = n xⁿ⁻¹ (n real). Constant multiple, sum, and difference rules apply. Tangents and normals: gradient of tangent is f'(x₁); normal gradient is −1/f'(x₁) if f'(x₁) ≠ 0.
导数定义为 f'(x)=lim(h→0)[f(x+h)−f(x)]/h。幂函数的导数公式 d/dx (xⁿ)=n xⁿ⁻¹ (n 为实数)。可推广至常数倍、和差法则。切线的斜率即 f'(x₁)，法线斜率为 −1/f'(x₁)（若 f'(x₁)≠0）。
Increasing/decreasing functions: f'(x) > 0 ⇒ increasing; f'(x) < 0 ⇒ decreasing. Stationary points occur where f'(x) = 0; use second derivative f''(x) to classify maxima (f''(x) < 0) and minima (f''(x) > 0).
函数增减性：f'(x)>0 递增，f'(x)<0 递减。驻点满足 f'(x)=0，用二阶导数 f''(x) 判别极大值 (f''(x)<0) 与极小值 (f''(x)>0)。
10. Integration: Reverse Differentiation and Area | 积分：反微分与面积
Integration is the reverse process: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1). Definite integrals calculate the signed area between a curve and the x‑axis. Area = ∫ₐᵇ f(x) dx. For areas below the axis, take absolute value or split the interval.
积分是微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1)+C (n ≠ −1)。定积分计算曲线与 x 轴之间的带符号面积：面积=∫ₐᵇ f(x) dx。若曲线在 x 轴下方，应对各区段取绝对值或分段处理。
The fundamental theorem connects differentiation and integration: d/dx [∫ₐˣ f(t) dt] = f(x). Indefinite integrals must include the constant of integration C.
微积分基本定理联系微分与积分：d/dx[∫ₐˣ f(t) dt]=f(x)。不定积分必须添加积分常数 C。
11. Functions, Graphs and Transformations | 函数、图像与变换
A function maps each input x to exactly one output f(x). Domain and range must be clearly identified. Composite functions fg(x) = f(g(x)) exist when the range of g lies within the domain of f. Inverse functions f⁻¹(x) reverse the mapping; graphs of f and f⁻¹ are reflections in y = x.
函数将每个输入 x 映射到唯一输出 f(x)，需明确定义域与值域。复合函数 fg(x)=f(g(x)) 存在的前提是 g 的值域包含于 f 的定义域。反函数 f⁻¹(x) 逆转映射关系，其图像与 f(x) 关于直线 y=x 对称。
Transformations: y = f(x) + a (vertical shift), y = f(x + a) (horizontal shift), y = a f(x) (vertical stretch), y = f(ax) (horizontal stretch), y = −f(x) (reflection in x‑axis), y = f(−x) (reflection in y‑axis). Combine in the correct order (stretches, reflections, then translations).
图像变换：y=f(x)+a 上下平移，y=f(x+a) 左右平移，y=a f(x) 纵向伸缩，y=f(ax) 横向伸缩，y=−f(x) 关于 x 轴对称，y=f(−x) 关于 y 轴对称。复合变换顺序通常为先伸缩反射、再平移。
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