📚 AS Mathematics: Algebra and Functions Key Points Revision | AS 数学：代数和函数 考点精讲
Algebra and functions form the core of AS Mathematics, demanding fluency in manipulation, equation solving, and graphical interpretation. This article walks you through every critical concept, common errors, and smart revision techniques, ensuring you build a strong foundation for exams.
代数与函数是 AS 数学的核心模块，需要熟练掌握代数运算、方程求解以及图像分析。本文将带你逐一攻克关键概念、常见易错点和高效的复习技巧，为考试打下坚实基础。
1. Simplifying Algebraic Expressions | 代数表达式的化简
Simplifying expressions correctly saves time and prevents sign errors. Always combine like terms, apply index laws, and expand brackets step by step.
正确化简表达式能节省时间并避免符号错误。务必逐步合并同类项、运用指数法则并展开括号。
Index laws must be second nature: am × an = am+n, (am)n = amn, am ÷ an = am−n (a ≠ 0), and a0 = 1 (a ≠ 0).
指数法则必须烂熟于心：am × an = am+n，(am)n = amn，am ÷ an = am−n (a ≠ 0)，以及 a0 = 1 (a ≠ 0)。
When expanding two brackets, apply (a + b)(c + d) = ac + ad + bc + bd; watch carefully for negative coefficients such as (x − 3)(2x + 1).
展开两个括号时，使用 (a + b)(c + d) = ac + ad + bc + bd；对于负系数如 (x − 3)(2x + 1) 要格外仔细。
2. Solving Quadratic Equations | 解二次方程
Three main methods exist: factorising, completing the square, and the quadratic formula. Choice depends on the equation structure.
主要有三种方法：因式分解法、配方法和公式法。方法的选择取决于方程的结构。
Factorising: rewrite ax2 + bx + c = 0 as (px + q)(rx + s) = 0, then set each bracket to zero.
因式分解：将 ax2 + bx + c = 0 改写为 (px + q)(rx + s) = 0，然后令每个括号等于零。
The quadratic formula works for any quadratic and is especially useful when coefficients are not integers:
公式法适用于任何二次方程，当系数不是整数时尤其有效：
x = (–b ± √(b2 – 4ac)) ÷ 2a
The discriminant Δ = b2 – 4ac tells you the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
判别式 Δ = b2 – 4ac 揭示了根的性质：Δ > 0 得两个不等实根，Δ = 0 得一个重根，Δ < 0 没有实根。
3. Quadratic Functions and Their Graphs | 二次函数及其图像
The graph of y = ax2 + bx + c is a parabola. The sign of a determines whether it opens upwards (a > 0, ∪-shaped) or downwards (a < 0, ∩-shaped).
y = ax2 + bx + c 的图像是一条抛物线。a 的符号决定开口方向：a > 0 时开口向上（∪ 形），a < 0 时开口向下（∩ 形）。
Complete the square to find the vertex form: y = a(x – h)2 + k, where (h, k) is the vertex. The axis of symmetry is the vertical line x = h.
通过配方得到顶点式：y = a(x – h)2 + k，其中 (h, k) 为顶点坐标。对称轴是垂直线 x = h。
The y-intercept occurs when x = 0, giving (0, c). The x-intercepts (roots) are found by solving ax2 + bx + c = 0.
y 轴截距在 x = 0 处，为 (0, c)。x 轴截距（根）通过求解 ax2 + bx + c = 0 得到。
4. Inequalities | 不等式
Linear inequalities are solved like equations, but multiplying or dividing by a negative number flips the inequality sign. Always show the solution on a number line or in set notation.
线性不等式的解法与方程类似，但乘以或除以负数时需翻转不等号。结果应标明在数轴上或用集合符号表示。
For quadratic inequalities, first rearrange to have 0 on one side, find the critical values (roots), then test intervals or use a sign chart. Remember that for (x – p)(x – q) > 0 the solution is outside the interval between p and q when a > 0.
对于二次不等式，先移项使一边为 0，求出临界值（根），然后检验区间或使用符号表。记住若 a > 0，则 (x – p)(x – q) > 0 的解在 p 和 q 区间之外。
5. Polynomials and the Factor Theorem | 多项式与因式定理
The factor theorem states: (x – p) is a factor of a polynomial f(x) if and only if f(p) = 0. The remainder theorem says that when f(x) is divided by (x – p), the remainder is f(p).
因式定理指出：(x – p) 是多项式 f(x) 的因式当且仅当 f(p) = 0。余数定理表明，f(x) 除以 (x – p) 的余数为 f(p)。
To factorise a cubic or higher-degree polynomial, use the factor theorem to find one linear factor, then perform polynomial division or compare coefficients to break it into a linear factor and a quadratic factor.
分解三次或更高次多项式时，先用因式定理找到一个线性因式，然后通过多项式除法或比较系数，将其分解为一个线性因式和一个二次因式。
6. Functions: Domain, Range, and Mappings | 函数：定义域、值域与映射
A function is a rule that assigns each input exactly one output. The domain is the set of all possible input values, and the range is the set of all possible output values.
函数是将每个输入唯一对应到一个输出的规则。定义域是所有可能输入值的集合，值域是所有可能输出值的集合。
Restrictions on domain often arise from denominators (cannot be zero) or square roots (radicand must be non-negative). Range can be found by considering the graph or the function’s behaviour.
定义域的限制常来自分母（不能为零）或平方根（被开方数须非负）。值域可通过图像或函数性质来确定。
Functions can be one-to-one (each y comes from exactly one x) or many-to-one (several x can give the same y). For a function to have an inverse, it must be one-to-one on its domain.
函数可以是一对一的（每个 y 只对应一个 x）或多对一的（多个 x 可给出相同 y）。函数要有反函数，其变换必须是定义域上的一对一映射。
7. Composite and Inverse Functions | 复合函数与反函数
A composite function fg(x) means applying g first, then f: fg(x) = f(g(x)). The order matters; fg(x) is generally not the same as gf(x).
复合函数 fg(x) 是先应用 g，再应用 f：fg(x) = f(g(x))。顺序至关重要；fg(x) 通常不等于 gf(x)。
To find the inverse function f−1(x), write y = f(x), swap x and y, then solve for y. The domain of f−1 is the range of f, and vice versa.
求反函数 f−1(x) 时，先写出 y = f(x)，交换 x 和 y，然后解出 y。f−1 的定义域是 f 的值域，反之亦然。
Graphically, the inverse is a reflection of the original function in the line y = x.
从图像上看，反函数是原函数关于直线 y = x 的镜像。
8. Exponential and Logarithmic Functions | 指数函数与对数函数
Exponential functions have the form y = ax (with a > 0, a ≠ 1). They all pass through (0,1) and have a horizontal asymptote y = 0. The natural exponential function y = ex is particularly important.
指数函数形如 y = ax（a > 0, a ≠ 1）。它们都经过 (0,1) 且有一条水平渐近线 y = 0。自然指数函数 y = ex 尤为重要。
Logarithms are the inverses of exponentials: if y = ax then x = loga y. The graph of y = loga x has a vertical asymptote at x = 0 and passes through (1,0).
对数是指数函数的反函数：若 y = ax，则 x = loga y。y = loga x 的图像有垂直渐近线 x = 0 且经过 (1,0)。
Key log laws: loga(xy) = loga x + loga y, loga(x/y) = loga x – loga y, loga(xn) = n loga x. Change of base: loga b = logc b ÷ logc a.
核心对数法则：loga(xy) = loga x + loga y，loga(x/y) = loga x – loga y，loga(xn) = n loga x。换底公式：loga b = logc b ÷ logc a。
9. Transformations of Functions | 函数图像的变换
Transformations allow you to sketch new graphs from basic ones. The main types are translations, stretches, and reflections.
通过图像变换，可以由基本函数快速绘制新图像。主要类型包括平移、伸缩和反射。
Translations: y = f(x) + a shifts the graph vertically by a (up if a > 0). y = f(x + a) shifts horizontally by –a (left if a > 0).
平移：y = f(x) + a 使图像垂直移动 a（a > 0 向上）。y = f(x + a) 使图像水平移动 –a（a > 0 向左）。
Stretches: y = k f(x) stretches vertically by factor k. y = f(kx) stretches horizontally by factor 1/k.
伸缩：y = k f(x) 沿纵向拉伸为原来的 k 倍。y = f(kx) 沿横向拉伸为原来的 1/k 倍。
Reflections: y = –f(x) reflects in the x-axis; y = f(–x) reflects in the y-axis.
反射：y = –f(x) 关于 x 轴对称；y = f(–x) 关于 y 轴对称。
10. Algebraic Fractions | 代数分式
Algebraic fractions are simplified by factorising both numerator and denominator and cancelling common factors. Always state restrictions where the denominator equals zero.
化简代数分式需将分子分母同时因式分解，并约去公因式。始终要注明使分母为零的限制条件。
To add or subtract algebraic fractions, find a common denominator, rewrite each fraction, then combine numerators. Simplify the result by factorising if possible.
加减代数分式时，先找到公分母，改写每个分数，然后合并分子。尽可能对结果进行因式分解并化简。
Multiplication: multiply numerators together and denominators together, then cancel common factors before expanding. Division: invert the second fraction and multiply.
乘法：分子相乘、分母相乘，展开前先约分。除法：将第二个分数倒置后相乘。
Always check for hidden factors like (x – 1) and (1 – x), where (1 – x) = –(x – 1).
注意隐藏因式，例如 (1 – x) = –(x – 1)，可以提取负号进行约分。
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