📚 Edexcel Maths Year 1 Pure: Complete AS-Level Revision Guide | 爱德思数学 Year 1 Pure 完整复习指南
This guide covers the core Year 1 Pure Mathematics content for the Edexcel AS specification. Each section explains a key topic, records the formulas you must memorise, and highlights the most common exam pitfalls.
本指南涵盖爱德思 AS 数学 Year 1 Pure 的核心内容。每节讲解一个关键主题，列出必须记忆的公式，并指出最常见的考试易错点。
1. Algebraic Expressions, Surds and Indices | 代数表达式、根式与指数
In Year 1 Pure, you must be confident with index laws: xm × xn = xm+n, (xm)n = xmn, x0 = 1 and x−n = 1 / xn. Fractional powers link to roots, so x1/2 = √x and xm/n = (x1/n)m.
在 Year 1 Pure 中，你必须熟练掌握指数律：xm × xn = xm+n，(xm)n = xmn，x0 = 1，以及 x−n = 1 / xn。分数指数与根式相关，因此 x1/2 = √x，xm/n = (x1/n)m。
When working with surds, simplify √a × √b = √(ab) and rationalise denominators such as 1/(√a + b) by multiplying the numerator and denominator by the conjugate √a − b. Always write surds in their simplest form unless the question says otherwise.
处理根式时，化简 √a × √b = √(ab)，并通过将分子和分母同乘共轭式 √a − b 来有理化如 1/(√a + b) 的分母。除非题目另有说明，否则始终把根式化为最简形式。
Expanding brackets and taking out common factors are tested in almost every paper. Always check whether a quadratic, cubic or rational expression factorises before applying a heavier method, and watch for the difference of two squares: a² − b² = (a + b)(a − b).
展开括号和提取公因式几乎在每份试卷中都会出现。在使用更复杂的方法之前，始终检查二次、三次或有理式是否可以因式分解，并留意平方差公式：a² − b² = (a + b)(a − b)。
2. Quadratics and the Discriminant | 二次函数与判别式
A quadratic can be written as y = ax² + bx + c. Completing the square gives y = a(x + b/(2a))² + (c − b²/(4a)), which reveals the vertex and is essential for solving hidden quadratics and proving minimising or maximising values.
二次函数可以写成 y = ax² + bx + c。配方法可将其化为 y = a(x + b/(2a))² + (c − b²/(4a))，由此可以直接看出顶点，并且在解隐藏二次方程以及证明最小值或最大值时非常关键。
For the equation ax² + bx + c = 0, the quadratic formula is:
对于方程 ax² + bx + c = 0，求根公式为：
x = (−b ± √(b² − 4ac)) / (2a)
The discriminant D = b² − 4ac decides the nature of the roots: D > 0 gives two distinct real roots, D = 0 gives one repeated real root, and D < 0 gives no real roots. You may also need to find a range of k such that kx² + 3x + k = 0 has real roots.
判别式 D = b² − 4ac 决定根的性质：D > 0 有两个不相等的实根，D = 0 有一个重根，D < 0 没有实根。你可能还需要求出使 kx² + 3x + k = 0 有实根的 k 的取值范围。
Hidden quadratics appear when a power is double another, such as x⁴ − 5x² + 4 = 0. Substitute u = x² to form u² − 5u + 4 = 0, solve for u, then reverse the substitution and reject impossible values.
当出现一个幂是另一个幂的两倍时，就会遇到隐藏二次方程，例如 x⁴ − 5x² + 4 = 0。令 u = x² 得到 u² − 5u + 4 = 0，解出 u 后再回代，并舍去不可能的值。
3. Equations and Inequalities | 方程与不等式
For simultaneous equations where one is linear and one is quadratic, use substitution. Rearrange the linear equation into y = mx + c or x = py + q, substitute into the quadratic, then solve the resulting quadratic. Always check that both solutions satisfy the original equations.
对于一个是线性、一个是二次的联立方程组，使用代入法。将线性方程改写为 y = mx + c 或 x = py + q，代入二次方程，然后解所得二次方程。始终检验两组解是否都满足原方程。
Quadratic inequalities should be rearranged so that one side is 0. Factorise, find the critical values, and use a sign diagram or sketch. If (x − a)(x − b) > 0, the solution is x < a or x > b; if (x − a)(x − b) < 0, the solution is a < x < b.
解二次不等式时，应先将一边化为 0。因式分解，求出临界值，并使用符号表或草图。若 (x − a)(x − b) > 0，解为 x < a 或 x > b；若 (x − a)(x − b) < 0，解为 a < x < b。
For rational inequalities, include critical values from both the numerator and denominator. Remember that values making the denominator zero must be excluded from the solution set, and use strict or weak inequality signs carefully.
对于分式不等式，要同时考虑分子和分母的临界值。记住使分母为零的值必须从解集中排除，并且要仔细区分严格不等式和带等号的不等式。
4. Graphs and Transformations | 函数图像与变换
You need to recognise the shapes of y = x³, y = x⁴, y = 1/x, y = 1/x² and their translations. Use roots, y-intercept and end behaviour to sketch graphs quickly, especially for cubic and quartic functions.
你需要识别 y = x³、y = x⁴、y = 1/x、y = 1/x² 及其平移后的图像。利用零点、y 轴截距和图像在无穷远处的趋势快速绘制函数草图，尤其是三次和四次函数。
Transformations of f(x) follow standard rules. The table below summarises the effects; note that horizontal shifts and stretches use the reciprocal or opposite sign to what appears inside the bracket.
f(x) 的变换遵循标准规则。下表总结了变换效果；注意括号内的水平平移和水平伸缩使用相反的数字或倒数。
| Transformation / 变换 | Equation / 方程 | Effect / 效果 |
|---|---|---|
| Vertical shift / 竖直平移 | y = f(x) + a | Moves graph up by a / 图像上移 a |
| Horizontal shift / 水平平移 | y = f(x + a) | Moves graph left by a / 图像左移 a |
| Vertical stretch / 竖直伸缩 | y = a f(x) | Stretch by scale factor a in y-direction / 在 y 方向拉伸 a 倍 |
| Horizontal stretch / 水平伸缩 | y = f(ax) | Stretch by scale factor 1/a in x-direction / 在 x 方向拉伸 1/a 倍 |
| Reflection / 对称 | y = −f(x) or y = f(−x) | Reflect in x-axis or y-axis / 关于 x 轴或 y 轴对称 |
When sketching transformed graphs, apply transformations step by step: inside the bracket first, then stretches and reflections, then translations. Clearly label any intercepts and asymptotes.
绘制变换后的图像时，要逐步进行变换：先处理括号内的变换，再处理伸缩和对称，最后进行平移。清楚地标出所有截距和渐近线。
5. Straight Line Graphs | 直线方程
The gradient of a line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). You can write the equation as y − y₁ = m(x − x₁) or y = mx + c, where c is the y-intercept.
经过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ − y₁)/(x₂ − x₁)。直线方程可以写成 y − y₁ = m(x − x₁) 或 y = mx + c，其中 c 是 y 轴截距。
Parallel lines have equal gradients, so m₁ = m₂. Perpendicular lines satisfy m₁ × m₂ = −1. This is often tested when finding a perpendicular bisector or a line normal to a curve.
平行直线的斜率相等，即 m₁ = m₂。垂直直线满足 m₁ × m₂ = −1。在求垂直平分线或曲线的法线时，经常考查这一关系。
The midpoint of (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between the two points is √((x₂ − x₁)² + (y₂ − y₁)²).
(x₁, y₁) 和 (x₂, y₂) 的中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。两点间距离为 √((x₂ − x₁)² + (y₂ − y₁)²)。
6. Circles | 圆
The standard equation of a circle with centre (a, b) and radius r is:
以 (a, b) 为圆心、r 为半径的圆的标准方程为：
(x − a)² + (y − b)² = r²
The general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c). If g² + f² − c is negative, the equation does not represent a real circle.
一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (−g, −f)，半径为 √(g² + f² − c)。如果 g² + f² − c 为负，则该方程不代表实圆。
A tangent to a circle is perpendicular to the radius at the point of contact. To find a tangent, first find the gradient of the radius from the centre to the point, then use the perpendicular gradient with y − y
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