📚 Mastering Quadratic Functions for Edexcel IGCSE Maths | 掌握二次函数：爱德思 IGCSE 数学考点精讲
Quadratic functions form one of the most heavily tested topics in the Edexcel IGCSE Mathematics syllabus. From factorising and solving to sketching graphs and interpreting roots, a solid understanding of quadratics is essential for tackling both Paper 1 and Paper 2 with confidence. This article breaks down everything you need to know, with worked examples and exam-style strategies.
二次函数是爱德思 IGCSE 数学考纲中考查频率最高的专题之一。从因式分解、求解方程到绘制图像、解释根的意义，扎实掌握二次函数是顺利完成 Paper 1 和 Paper 2 的关键。本文将系统梳理所有必考点，配合典型例题和应试技巧，帮助你高效备考。
1. What Is a Quadratic Function? | 什么是二次函数？
A quadratic function is any expression of the form ax² + bx + c, where a, b and c are constants and a ≠ 0. The graph of a quadratic function is a smooth curve called a parabola. If a > 0, the parabola opens upwards, like a smile; if a < 0, it opens downwards, like a frown.
二次函数是形如 ax² + bx + c 的表达式，其中 a、b、c 为常数，且 a ≠ 0。二次函数的图像是一条平滑曲线，称为抛物线。若 a > 0，抛物线开口向上，形如微笑；若 a < 0，抛物线开口向下，形如皱眉。
- Standard form: ax² + bx + c | 标准形式：ax² + bx + c
- Vertex form: a(x – h)² + k, where (h, k) is the turning point | 顶点式：a(x – h)² + k，其中 (h, k) 为顶点坐标
- Factorised form: a(x – p)(x – q), where p and q are the roots | 因式分解形式：a(x – p)(x – q)，其中 p 和 q 为根
2. Expanding and Simplifying | 展开与化简
Expanding means removing brackets by multiplying terms. For two binomials, use the FOIL method: First, Outer, Inner, Last. For example, (x + 3)(x – 5) = x² – 5x + 3x – 15 = x² – 2x – 15.
展开是指通过逐项相乘去掉括号。对于两个二项式，使用 FOIL 方法：First（首项）、Outer（外项）、Inner（内项）、Last（末项）。例如，(x + 3)(x – 5) = x² – 5x + 3x – 15 = x² – 2x – 15。
(ax + b)(cx + d) = acx² + (ad + bc)x + bd
When expanding three brackets, first expand two, then multiply the result by the third. A common mistake is forgetting the middle term — always check that the coefficient of x is the sum of the outer and inner products.
展开三个括号时，先展开其中两个，再将结果与第三个相乘。常见错误是漏掉中间项——务必检查 x 的系数等于外项和内项乘积之和。
3. Factorising Quadratics | 因式分解二次式
Factorising is the reverse of expanding. To factorise x² + bx + c, look for two numbers that multiply to c and add to b. For example, x² + 7x + 12 = (x + 3)(x + 4), because 3 × 4 = 12 and 3 + 4 = 7.
因式分解是展开的逆运算。对于 x² + bx + c，寻找两个数，它们相乘等于 c，相加等于 b。例如，x² + 7x + 12 = (x + 3)(x + 4)，因为 3 × 4 = 12 且 3 + 4 = 7。
When the coefficient of x² is not 1, use the method of factoring by grouping: multiply a and c, find two numbers that multiply to ac and add to b, then split the middle term and factor pair-wise.
当 x² 的系数不为 1 时，使用分组分解法：将 a 和 c 相乘，找到两个数相乘等于 ac 且相加等于 b，然后拆分中间项并分组提取公因式。
Example: 2x² + 7x + 3 = 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)
Always check your factorisation by expanding the product. Factoring is a critical skill because it underpins solving quadratic equations and simplifying rational expressions.
始终通过重新展开来检验因式分解是否正确。因式分解是关键技能，它是解二次方程和化简有理式的基础。
4. Difference of Two Squares | 平方差公式
A special case of factorising is the difference of two squares: a² – b² = (a + b)(a – b). This pattern is common in IGCSE exams, so learn to spot it quickly. For example, x² – 25 = (x + 5)(x – 5), and 4x² – 9y² = (2x + 3y)(2x – 3y).
因式分解的一个特例是平方差公式：a² – b² = (a + b)(a – b)。这一模式在 IGCSE 考试中非常常见，务必学会快速识别。例如，x² – 25 = (x + 5)(x – 5)，4x² – 9y² = (2x + 3y)(2x – 3y)。
- Recognise perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 | 识别完全平方数：1、4、9、16、25、36、49、64、81、100
- Recognise squared variables: x², y², 4x² | 识别平方变量：x²、y²、4x²
- Note that a² + b² (sum of squares) cannot be factored over real numbers | 注意 a² + b²（平方和）在实数范围内无法分解
5. Completing the Square | 配方法
Completing the square rewrites x² + bx + c in the form (x + p)² + q. This is essential for finding the vertex of a parabola and solving equations that do not factor nicely.
配方法将 x² + bx + c 改写为 (x + p)² + q 的形式。这是求抛物线顶点和求解无法简便因式分解的方程的重要工具。
(x + p)² = x² + 2px + p²
To complete the square for x² + bx: take half of b, square it, then add and subtract that value. For example, x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4.
对 x² + bx 配方：取 b 的一半，平方后加回再减去这个值。例如，x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4。
For ax² + bx + c, factor out a before completing the square. The final form a(x – h)² + k directly reveals the turning point (h, k) and the line of symmetry x = h.
对于 ax² + bx + c，先提取 a，再进行配方。最终形式 a(x – h)² + k 直接给出顶点 (h, k) 和对称轴 x = h。
6. Solving Quadratic Equations | 解二次方程
A quadratic equation is typically written as ax² + bx + c = 0. There are three main methods: factorising, using the quadratic formula, and completing the square. Factorising is fastest when it works; the formula always works.
二次方程通常写作 ax² + bx + c = 0。有三种主要解法：因式分解法、求根公式法和配方法。因式分解在可行时最快；求根公式始终有效。
x = (−b ± √(b² − 4ac)) / 2a
The quadratic formula is derived from completing the square and is used when factorising is difficult or impossible. Ensure your equation is in standard form before identifying a, b and c.
求根公式由配方法推导而来，适用于因式分解困难或不可行的情况。使用前务必先将方程化为标准形式，再确定 a、b、c。
Example: Solve 2x² – 4x – 6 = 0. Dividing by 2 gives x² – 2x – 3 = 0, which factors to (x – 3)(x + 1) = 0, so x = 3 or x = -1.
例：解 2x² – 4x – 6 = 0。先除以 2，得 x² – 2x – 3 = 0，因式分解为 (x – 3)(x + 1) = 0，所以 x = 3 或 x = -1。
7. The Discriminant | 判别式
The discriminant is the part of the quadratic formula under the square root: b² – 4ac. It tells us how many real roots a quadratic equation has, without solving it fully.
判别式是求根公式中根号内的部分：b² – 4ac。它无需完全求解即可告诉我们二次方程有多少个实数根。
| Discriminant 判别式 | Nature of roots 根的性质 | Graph meaning 图像含义 |
| b² – 4ac > 0 | Two distinct real roots | 两个不等实数根 | Crosses x-axis twice | 与 x 轴有两个交点 |
| b² – 4ac = 0 | One repeated root | 一个重根 | Touches x-axis at vertex | 顶点与 x 轴相切 |
| b² – 4ac < 0 | No real roots | 无实数根 | Does not meet x-axis | 不与 x 轴相交 |
Exams often ask you to find the range of k for which an equation ax² + bx + c = 0 has real roots. Set the discriminant to be ≥ 0 and solve the resulting inequality.
考试常要求你求出使方程 ax² + bx + c = 0 有实数根时 k 的取值范围。令判别式 ≥ 0 并求解所得不等式即可。
8. Sketching Quadratic Graphs | 绘制二次函数图像
To sketch a quadratic graph accurately, you must identify: the direction of opening, the y-intercept, the roots (x-intercepts), the vertex, and the axis of symmetry.
要准确绘制二次函数图像，你必须确定：开口方向、y 截距、根（x 截距）、顶点和对称轴。
- Opening direction: sign of a | 开口方向：由 a 的正负决定
- y-intercept: (0, c) | y 截距：(0, c)
- Roots: solve ax² + bx + c = 0 | 根：解 ax² + bx + c = 0
- Axis of symmetry: x = −b/(2a) | 对称轴：x = −b/(2a)
- Vertex: x-coordinate is −b/(2a), substitute to find y | 顶点：横坐标为 −b/(2a)，代入求纵坐标
Label all intercepts and the turning point. Make sure the curve is smooth, not made of straight lines. A good sketch does not need to be to scale, but the key features must be correctly positioned.
标出所有截距和顶点。确保曲线平滑，不能用直线拼接。好的草图不需要严格按比例绘制，但关键特征必须位置正确。
9. Quadratic Inequalities | 二次不等式
Solving quadratic inequalities requires finding the roots first, then testing intervals. For example, to solve x² – x – 6 > 0, factor to get (x – 3)(x + 2) > 0. The critical points are x = 3 and x = -2.
解二次不等式需要先求出根，再对区间进行检验。例如，解 x² – x – 6 > 0，因式分解得 (x – 3)(x + 2) > 0。临界点为 x = 3 和 x = -2。
For a > 0, the graph is a smile. The expression is positive outside the roots and negative between them. Thus the solution is x < -2 or x > 3.
当 a > 0 时，图像开口向上。表达式在两根之外为正、两根之间为负。因此解集为 x < -2 或 x > 3。
If the inequality sign is ≤ or ≥, include the endpoints. A common mistake is writing the answer as -2 < x < 3 instead of a union of two intervals — draw the graph to avoid confusion.
如果不等号包含等号（≤ 或 ≥），则需包含端点。常见错误是把答案写成 -2 < x < 3 而不是两个区间的并集——画图可以避免混淆。
10. Solving Simultaneous Equations with Quadratics | 联立方程中的二次方程
Edexcel IGCSE frequently tests solving a linear and a quadratic equation simultaneously. The standard method is substitution: rearrange the linear equation to make one variable the subject, then substitute into the quadratic equation.
爱德思 IGCSE 经常考查联立求解一个线性方程和一个二次方程。标准方法是代入法：将线性方程变形为用一个变量表示另一个变量，然后代入二次方程。
y = mx + c 代入 ax² + bx + c = 0 → 得到关于 x 的二次方程
Example: Solve y = 2x + 1 and y = x² – 2x – 3. Substituting: 2x + 1 = x² – 2x – 3, so x² – 4x – 4 = 0. Solve using the quadratic formula to get two x-values, then substitute back to find the corresponding y-values.
例：解方程组 y = 2x + 1 和 y = x² – 2x – 3。代入得：2x + 1 = x² – 2x – 3，即 x² – 4x – 4 = 0。用求根公式解出两个 x 值，再代回求出对应的 y 值。
The solutions represent the intersection points of the line and the parabola. If the discriminant is negative, the line does not intersect the curve at all.
解对应直线与抛物线的交点。若判别式为负，则直线与曲线没有交点。
11. Word Problems with Quadratics | 二次函数应用题
Exam problems often frame quadratics in real-world contexts such as projectile motion, area optimisation, and number puzzles. The key is to translate the problem into an equation carefully.
考试常将二次函数置于现实情境中，如抛体运动、面积优化和数字谜题。关键在于将题目准确转化为方程。
For area problems, draw a diagram and label unknown sides with variables. For number problems, define one number as x and express the other number in terms of x. For projectile motion, height h = -gt²/2 + ut + h₀.
面积问题中，画图并用变量标出未知边。数字问题中，设一个数为 x，用 x 表示另一个数。抛体运动中，高度 h = -gt²/2 + ut + h₀。
Always check that your solutions make sense in the context. A negative length or time indicates an extraneous root that should be discarded. Write your final answer with suitable units.
务必检查解在情境中是否合理。负数长度或时间通常意味着增根，应舍去。最终答案要带上合适的单位。
12. Exam Tips and Common Traps | 考试技巧与常见陷阱
Here are tried-and-tested strategies to maximise your marks when tackling quadratic questions in the IGCSE exam.
以下是在 IGCSE 考试中处理二次函数问题时经过验证的提分策略。
- Always write the equation in standard form ax² + bx + c = 0 before solving | 解题前务必先将方程化为标准形式 ax² + bx + c = 0
- If factorising is not obvious within 10 seconds, use the quadratic formula | 如果 10 秒内看不出因式分解，直接使用求根公式
- Show your working clearly — Edexcel awards method marks, not just final answers | 清晰展示解题步骤——爱德思给方法分，不只是答案分
- When sketching a graph, label every intercept with its coordinates | 画图时，标出每个截距的坐标
- Check solutions by substituting back into the original equation | 将解代回原方程进行验证
- Beware of minus signs when substituting into b² – 4ac | 将负数代入 b² – 4ac 时小心符号
Quadratics reward practice. Work through past paper questions until the methods become instinctive, and you will convert this topic from a challenge into a confident scoring area.
二次函数题重在练习。反复刷历年真题，直到解题方法成为本能，你就能将这一专题从挑战转变为稳稳的得分项。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导