📚 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。
rse 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。
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
emainder 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。
xample, (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 ₐ(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
Published by TutorHao | Pre-U Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图