📚 AS Mathematics: High-Frequency Exam Topics Summary | AS 数学：高频考点总结
Getting ready for your AS Mathematics exam? Whether you are taking Pure Mathematics 1 alone or alongside Statistics or Mechanics, certain topics consistently appear in exams. This summary revisits the high-frequency concepts, formulas, and typical question types to help you focus your revision.
准备AS数学考试？不管你是考纯数1还是搭配统计或力学，某些主题总是高频出现。本总结回顾这些常考概念、公式和典型题型，助你集中复习。
1. Quadratic Equations and Inequalities | 二次方程与不等式
are solved by factorising, completing the square, or using the quadratic formula x = [−b ± √b² − 4ac] / (2a). Always check whether the question requires exact form or decimal answers.
形如ax² + bx + c = 0的二次方程可通过因式分解、配方法或求根公式 x = [−b ± √b² − 4ac] / (2a) 求解。务必看清题目要求精确值还是小数答案。
root; if Δ < 0, no real roots. You may be asked to find the range of k for which a quadratic has real roots.
判别式Δ = b² − 4ac给出根的信息：Δ > 0时有两个不等实根；Δ = 0时有一个重根；Δ < 0时无实根。常考求k的范围使二次方程有实根。
To solve a quadratic inequality, e.g. x² − 5x + 6 > 0, first find the roots and then sketch the parabola to identify the intervals where the inequality holds. Use critical values and test intervals.
解二次不等式，如x² − 5x + 6 > 0，先求根，然后画抛物线草图来确定满足不等式的区间。使用关键值并检查区间。
2. Functions and Graph Transformations | 函数与图像变换
ues y. Watch out for restrictions like square roots (argument ≥ 0) and denominators (≠ 0).
函数f(x)的定义域是所有可能输入x的集合，值域是所有可能输出y的集合。注意限制条件，如平方根内≥0，分母≠0。
Composite functions such as fg(x) mean applying g then f. For inverse function f⁻¹(x), swap x and y, then solve for y. Only one-to-one functions have inverses over their whole domain.
复合函数fg(x)意为先g后f。求反函数f⁻¹(x)时，交换x和y再解出y。只有一一映射函数在整个定义域上才有反函数。
= f(ax) stretches horizontally by factor 1/a. Reflections include y = −f(x) and y = f(−x).
记住常见变换：y = f(x) + a 上下平移a；y = f(x + a) 左右平移−a；y = a f(x) 垂直拉伸a倍；y = f(ax) 水平拉伸1/a倍。反射变换有y = −f(x)和y = f(−x)。
3. Coordinate Geometry of Straight Lines | 直线坐标几何
c (slope-intercept), y − y₁ = m(x − x₁) (point-slope), and ax + by + c = 0 (general form).
两点间梯度m = (y₂ − y₁)/(x₂ − x₁)。直线方程形式：y = mx + c（斜截式），y − y₁ = m(x − x₁)（点斜式），以及ax + by + c = 0（一般式）。
ar lines satisfy m₁ × m₂ = −1. Always convert to the form y = mx + c to compare gradients.
平行线梯度相等(m₁ = m₂)。垂直线满足m₁ × m₂ = −1。总是化为y = mx + c形式来比较梯度。
√[(x₂ − x₁)² + (y₂ − y₁)²]. These are often used in circle geometry and coordinate proofs.
中点为((x₁ + x₂)/2, (y₁ + y₂)/2)。两点间距离为√[(x₂ − x₁)² + (y₂ − y₁)²]。这些常用于圆的几何和坐标证明。
4. Circles | 圆的方程
² = r². Expanding gives x² + y² + 2gx + 2fy + c = 0, where centre is (−g, −f) and radius = √g² + f² − c.
圆心(a,b)、半径r的圆的方程为(x − a)² + (y − b)² = r²。展开得x² + y² + 2gx + 2fy + c = 0，圆心为(−g, −f)，半径= √g² + f² − c。
t. To find intersection of a line and circle, substitute the line equation into the circle's equation and solve; discriminant determines tangency (Δ = 0).
圆的切线只接触圆于一点。切线在切点处垂直于半径。求直线与圆的交点，将直线方程代入圆方程求解；判别式Δ=0时相切。
5. Sequences: Arithmetic and Geometric | 数列：等差与等比
(a + l) where l is the last term. Recognise word problems involving arithmetic sequences.
等差数列首项为a，公差为d。第n项: uₙ = a + (n − 1)d。前n项和: Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，l为末项。识别涉及等差数列的应用题。
um of first n terms: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. Sum to infinity (convergent GP when |r| < 1): S∞ = a/(1 − r). Common exam questions ask for the least n for Sₙ to exceed a value.
等比数列首项a，公比r。第n项: uₙ = a rⁿ⁻¹。前n项和: Sₙ = a(1 − rⁿ)/(1 − r) (r≠1)。无穷和（|r|<1时收敛）: S∞ = a/(1 − r)。常见考题要求求最小n使得Sₙ超过某值。
6. Trigonometry | 三角函数
sin² θ + cos² θ = 1, tan θ = sin θ / cos θ. Be comfortable with both degrees and radians.
sin θ = 对边/斜边, cos θ = 邻边/斜边, tan θ = 对边/邻边。关键恒等式：sin² θ + cos² θ = 1, tan θ = sin θ / cos θ。熟悉度和弧度制。
r graph to find all solutions. Remember to adjust the interval if the argument is 2x or (x + 30°).
解方程如sin x = 0.5，在0°≤x≤360°或0≤x≤2π内。用CAST图或图像求所有解。若参数为2x或(x+30°)，记得调整区间。
y = sin x, y = cos x, y = tan x. Amplitude and period changes: y = a sin(bx) has amplitude |a| and period 2π/b. Phase shifts and vertical shifts may be tested.
掌握y = sin x, y = cos x, y = tan x的图像。振幅和周期变化：y = a sin(bx)振幅为|a|，周期为2π/b。可考相移和垂直平移。
7. Differentiation Basics | 微分基础
erentiate term by term. Constants differentiate to zero. The derivative of a constant times a function is constant times the derivative.
若y = xⁿ，dy/dx = n xⁿ⁻¹。对求和逐项求导。常数
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