Pre-U WJEC Mathematics: Core Knowledge Points Review | Pre-U WJEC 数学：核心知识点梳理

📚 Pre-U WJEC Mathematics: Core Knowledge Points Review | Pre-U WJEC 数学：核心知识点梳理

This article provides a comprehensive review of the essential core topics in Pre-U WJEC Mathematics, covering pure mathematics, statistics, and mechanics. Mastering these concepts is crucial for success in the examination.

本文全面梳理了 Pre-U WJEC 数学的核心知识点，涵盖纯数学、统计和力学。掌握这些概念对考试成功至关重要。

1. Algebra and Functions | 代数与函数

Functions map an input to exactly one output. The domain is the set of all permissible inputs, and the range is the set of all possible outputs.

函数将输入映射到唯一的输出。定义域是所有允许的输入的集合，值域是所有可能输出的集合。

The composition fg(x) means apply g first, then f: f(g(x)). The inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and can be found by swapping x and y and solving for y.

复合函数 fg(x) 表示先应用 g，再应用 f：f(g(x))。反函数 f⁻¹(x) 满足 f(f⁻¹(x)) = x，可以通过交换 x 和 y 并解出 y 来求得。

For a quadratic f(x) = ax² + bx + c, the discriminant Δ = b² − 4ac determines the nature of the roots: two distinct real roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.

对于二次函数 f(x) = ax² + bx + c，判别式 Δ = b² − 4ac 决定了根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。

f(x) = a(x − h)² + k → vertex (h, k)

Graph transformations: f(x − a) shifts the graph a units to the right; f(x) + a shifts it a units upward. f(ax) stretches horizontally by factor 1/a when 0 < a < 1.

图像变换：f(x − a) 将图像向右平移 a 个单位；f(x) + a 向上平移 a 个单位。f(ax) 当 0 < a < 1 时水平拉伸为原来的 1/a。

2. Polynomials and Rational Functions | 多项式与有理函数

The Remainder Theorem: when polynomial f(x) is divided by (x − a), the remainder is f(a). The Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.

余数定理：多项式 f(x) 除以 (x − a) 时，余数为 f(a)。因式定理：(x − a) 是 f(x) 的因式当且仅当 f(a) = 0。

Polynomial long division is used to divide a polynomial by another, obtaining a quotient and a remainder. Synthetic division provides a quicker method for linear divisors.

多项式长除法用于将多项式除以另一个多项式，得到商和余数。综合除法为一次因式提供了一种更快捷的方法。

Rational expressions can be split into partial fractions. For distinct linear factors, write, for example, (3x + 5)/((x + 1)(x − 2)) = A/(x + 1) + B/(x − 2) and solve for constants A and B.

有理表达式可以拆分为部分分式。对于不同的线性因式，例如 (3x + 5)/((x + 1)(x − 2)) 可写为 A/(x + 1) + B/(x − 2) 并解出常数 A 和 B。

∫ A/(x + a) dx = A ln|x + a| + C

3. Exponentials and Logarithms | 指数与对数

Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.

指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，a⁰ = 1 (a ≠ 0)，a⁻ⁿ = 1/aⁿ。

Logarithms are inverses of exponentials: if aˣ = y then x = logₐ y. Key rules: logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x − logₐ y, logₐ(xⁿ) = n logₐ x. Change of base: logₐ b = log_c b / log_c a.

对数是指数的逆运算：若 aˣ = y，则 x = logₐ y。关键法则：logₐ(xy) = logₐ x + logₐ y，logₐ(x/y) = logₐ x − logₐ y，logₐ(xⁿ) = n logₐ x。换底公式：logₐ b = log_c b / log_c a。

The natural exponential function eˣ and natural log ln x are inverses. To solve e²ˣ = 5, take ln both sides: 2x = ln 5 → x = ½ ln 5.

自然指数函数 eˣ 和自然对数 ln x 互逆。要解 e²ˣ = 5，两边取 ln：2x = ln 5 → x = ½ ln 5。

d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C

4. Trigonometry | 三角学

Angles are measured in radians

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