📚 GCSE AQA Maths: Quadratic Functions | 二次函数考点精讲
Quadratic functions are a cornerstone of GCSE Mathematics, appearing in both Foundation and Higher tier papers. They encompass everything from basic curve sketching to solving complex equations, and even real-life projectile problems. This guide systematically covers all the key concepts you need to master for AQA exams, including factorising, completing the square, the quadratic formula, discriminant, turning points, and inequalities.
二次函数是 GCSE 数学的基石，出现在基础卷与高阶卷中。它涵盖了从简单的曲线作图到解复杂方程，乃至现实中的抛射问题。本文系统梳理了 AQA 考试所需掌握的全部核心概念，包括因式分解、配方法、二次公式、判别式、顶点以及不等式。
1. What is a Quadratic Function? | 什么是二次函数？
= ax² + bx + c, where a, b and c are constants, and a ≠ 0 (otherwise it would be linear).
二次函数是 x 的最高次幂为 2 的多项式函数。一般形式为 f(x) = ax² + bx + c，其中 a、b、c 为常数，且 a ≠ 0（否则就成了线性函数）。
Quadratic functions produce a characteristic U-shaped or n-shaped curve called a parabola. When a > 0, the parabola opens upwards (smile shape); when a < 0, it opens downwards (frown shape).
二次函数产生特有的 U 形或倒 U 形曲线，称为抛物线。当 a > 0 时，抛物线开口向上（笑脸形状）；当 a < 0 时，开口向下（哭脸形状）。
2. Shape and Key Features of a Parabola | 抛物线的形状与关键特征
, the line of symmetry (a vertical line through the turning point), and the turning point (maximum or minimum point).
抛物线有几个关键特征：y 轴截距（x = 0 时的点）、根或 x 轴截距（f(x) = 0 的地方）、对称轴（经过顶点的垂直线）以及顶点（最高点或最低点）。
adratic given by y = ax² + bx + c is x = −b/(2a). This is a key formula you must remember.
对于二次函数 y = ax² + bx + c，其对称轴为 x = −b/(2a)。这是一个必须熟记的公式。
The turning point lies on the line of symmetry. Its coordinates can be found by substituting the x-coordinate into the function or by completing the square (see later).
顶点位于对称轴上。其坐标可以通过将 x 坐标代入函数求得，或者通过配方法（后文介绍）获得。
3. Plotting Quadratics | 绘制二次函数图像
To plot a quadratic graph, you need to create a table of values, usually for x from -3 to 3 (or a range that includes the turning point). Substitute each x into the function to calculate corresponding y values.
要画二次函数图像，需要建立数值表格，x 的取值范围通常从 -3 到 3（或者包含顶点的范围）。将每个 x 值代入函数计算对应的 y 值。
Plot the (x, y) points on a grid and connect them with a smooth, curved line – not straight line segments. The curve should be symmetrical.
在坐标网格中描出 (x, y) 点，并用平滑曲线连接——不能用折线段。曲线应对称。
Always label the curve, if asked, and clearly mark any intercepts and the turning point. AQA often awards marks for a smooth freehand curve passing through all plotted points.
如果题目要求，一定要标注曲线，并清楚标出所有截距和顶点。AQA 考试通常对穿过所有描点的手绘平滑曲线给分。
4. Solving by Factorising | 因式分解法
wo numbers that multiply to the constant term (6) and add to the coefficient of x (5). Those numbers are 2 and 3.
要解如 x² + 5x + 6 = 0 的二次方程，先用因式分解法，找两个数，乘积为常数项(6)，和为 x 的系数(5)。这两个数是 2 和 3。
)(x + 3) = 0. Set each bracket equal to zero: x + 2 = 0 or x + 3 = 0, giving solutions x = –2 and x = –3.
然后将二次式写成 (x + 2)(x + 3) = 0。让每个括号等于零：x + 2 = 0 或 x + 3 = 0，得出解 x = –2 和 x = –3。
If the coefficient of x² is not 1, for example 2x² + 3x – advanced method (ac method or grouping). Find two numbers that multiply to (a × c) = (2 × –2) = –4 and add to 3: numbers are 4 and –1. Rewrite: 2x² + 4x – x – 2 = 0, then factor by grouping: 2x(x + 2) – 1(x + 2) = (2x – 1)(x + 2) = 0, solutions x = 1/2, x = –2.
如果 x² 的系数不是 1，例如 2x² + 3x – 2 = 0，则需要更复杂的方法（ac 法或分组法）。找两个数乘积为 (a×c) = –4，和为 3：这两个数是 4 和 –1。重写为 2x² + 4x – x – 2 = 0，然后分组分解：2x(x + 2) – 1(x + 2) = (2x – 1)(x + 2) = 0，解得 x = 1/2, x = –2。
Published by TutorHao | GCSE 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