📚 Edexcel Year 1 Pure Maths: Quadratic Functions and Equations | 爱德思 Year 1 纯数学：二次函数与方程
Quadratic functions appear throughout Edexcel Year 1 Pure Mathematics. They are the foundation for curve sketching, optimisation and later topics such as functions and transformations. This revision guide covers recognising quadratics, factorising, completing the square, the quadratic formula, the discriminant, graph sketching, inequalities, modelling and simultaneous equations.
二次函数贯穿爱德思 Year 1 纯数学的始终。它们是图像绘制、最优化以及后续函数与变换等内容的基础。本复习指南涵盖二次函数的识别、因式分解、配方法、求根公式、判别式、图像绘制、不等式、建模以及二次联立方程。
1. Recognising Quadratic Functions | 认识二次函数
re real constants and a ≠ 0. The term ax² is called the quadratic term, bx the linear term and c the constant term.
二次函数的一般形式为 f(x) = ax² + bx + c，其中 a、b、c 是实常数且 a ≠ 0。ax² 称为二次项，bx 称为一次项，c 称为常数项。
0, the expression becomes bx + c, which is a linear function. Year 1 exam questions often ask you to identify the values of a, b and c before applying a method.
条件 a ≠ 0 至关重要：如果 a = 0，表达式就变成 bx + c，这是一次函数。Year 1 考试题常要求你先识别 a、b、c 的值，再应用相应方法。
f(x) = ax² + bx + c, a ≠ 0
2. Expanding and Factorising Quadratics | 二次式的展开与因式分解
When expanding a product of two linear brackets (x + p)(x + q), the result is x² + (p + q)x + pq. The middle coefficient is the sum of p and q, and the constant term is their product.
展开两个一次括号的乘积 (x + p)(x + q) 时，结果是 x² + (p + q)x + pq。中间项的系数是 p 与 q 的和，常数项是 p 与 q 的积。
Factorising reverses this process. To factorise x² + bx + c, look for two numbers that multiply to c and add to b. For example, x² − 5x + 6 factorises to (x − 2)(x − 3) because −2 and −3 multiply to 6 and add to −5.
因式分解是上述过程的逆运算。要对 x² + bx + c 进行因式分解，需要找到两个数，它们的积等于 c，和等于 b。例如，x² − 5x + 6 可分解为 (x − 2)(x − 3)，因为 −2 与 −3 的积为 6，和为 −5。
es: x² − a² = (x − a)(x + a). This is particularly useful when the linear term is missing.
特殊乘积包括平方差公式：x² − a² = (x − a)(x + a)。当一次项缺失时，这一公式特别有用。
3. Completing the Square | 配方法
Completing the square rewrites x² + bx + c in the form (x + p)² + q. To do this, take half of the coefficient of x, square it, and adjust the constant.
配方法将 x² + bx + c 改写为 (x + p)² + q 的形式。具体做法是取 x 项系数的一半、平方，然后调整常数项。
square form is (x + b/2)² + c − b²/4. The vertex of the parabola is therefore at x = −b/2.
对于一般表达式 x² + bx + c，配方后的形式为 (x + b/2)² + c − b²/4。因此抛物线的顶点位于 x = −b/2。
Example: x² + 6x + 5 = (x + 3)² − 9
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