📚 Mastering Quadratic Functions for IGCSE Maths | IGCSE数学二次函数完全攻略
Quadratic functions are one of the most frequently tested topics in the IGCSE Mathematics syllabus. Whether you are sitting for the Core or Extended paper, understanding the behaviour of parabolas, solving quadratic equations, and interpreting graphs will earn you significant marks across multiple sections. This article breaks down every essential concept into clear, exam-focused steps.
二次函数是IGCSE数学考纲中出现频率最高的考点之一。无论你参加Core还是Extended试卷，理解抛物线的性质、解二次方程以及解读图像，都会在多个题型中为你赢得可观的分数。这篇文章将每个核心概念拆解成清晰、紧扣考点的步骤，助你系统攻克这一专题。
1. What Is a Quadratic Function | 什么是二次函数
A quadratic function is any function that can be written in the form y = ax² + bx + c, where a, b and c are constants and a ≠ 0. The graph of a quadratic function is always a smooth, symmetric curve called a parabola. If a > 0, the parabola opens upwards and has a minimum point; if a < 0, it opens downwards and has a maximum point.
二次函数是任何可以写成 y = ax² + bx + c 形式的函数，其中 a、b、c 为常数，且 a ≠ 0。二次函数的图像总是一条平滑、对称的曲线，称为抛物线。当 a > 0 时，抛物线开口向上，存在最小值点；当 a < 0 时，抛物线开口向下，存在最大值点。
ower of x is 2. Meanwhile, y = 3x + 5 is not quadratic, since the highest power of x is 1.
例如，y = 2x² + 3x − 1 是二次函数，因为 x 的最高次数是 2。而 y = 3x + 5 不是二次函数，因为 x 的最高次数是 1。
General form: y = ax² + bx + c (a ≠ 0)
In IGCSE exams, you will often be asked to identify quadratic expressions, determine the values of a, b and c, or decide whether a given equation represents a quadratic function. Always check the coefficient of x² — if it is zero, the expression is not quadratic.
在IGCSE考试中，你常被要求识别二次表达式、确定 a、b、c 的值，或判断给定方程是否代表二次函数。务必检查 x² 的系数——如果它为0，则该表达式不是二次的。
2. Expanding and Factorising Quadratics | 展开与因式分解二次式
l for simplifying expressions before solving. In contrast, factorising is the reverse process — rewriting x² + 8x + 15 as (x + 3)(x + 5).
展开是去掉括号的过程：(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15。这项技能是在解方程前化简表达式的基础。相反，因式分解是逆过程——将 x² + 8x + 15 改写为 (x + 3)(x + 5)。
−4, since (−3) × (−4) = 12 and (−3) + (−4) = −7. Therefore, x² − 7x + 12 = (x − 3)(x − 4).
对于形如 x² + bx + c 的简单二次式，找两个数，它们相乘得 c，相加得 b。例如，分解 x² − 7x + 12：两个数是 −3 和 −4，因为 (−3) × (−4) = 12 且 (−3) + (−4) = −7。因此，x² − 7x + 12 = (x − 3)(x − 4)。
n as 2x² + 6x + x + 3, then factorise by grouping: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
当 x² 的系数不是1时，使用拆分中项法或箱型法。对于 2x² + 7x + 3，计算 a × c：2 × 3 = 6。找两个数相乘得6且相加得7：它们是6和1。将表达式改写为 2x² + 6x + x + 3，然后分组分解：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
3. Solving Quadratic Equations by Factorisation | 用因式分解法解二次方程
zero product property: if AB = 0, then A = 0 or B = 0. For example, solve x² − 5x + 6 = 0.
解二次方程就是找出使方程成立的 x 值。当二次方程等于零时，可以应用零积性质：若 AB = 0，则 A = 0 或 B = 0。例如，解 x² − 5x + 6 = 0。
r x = 3. Always check your solutions by substituting them back into the original equation.
因式分解：x² − 5x + 6 = (x − 2)(x − 3) = 0。然后令每个因子为零：x − 2 = 0 或 x − 3 = 0。因此，x = 2 或 x = 3。始终将解代回原方程验证。
ng x = 5 or x = −2. In an exam, write down the factorisation clearly to show your working.
如果方程不是标准形式，先重新整理。例如，x² = 3x + 10 变为 x² − 3x − 10 = 0，分解为 (x − 5)(x + 2) = 0，得 x = 5 或 x = −2。在考试中，清晰写下分解过程以展示你的解题步骤。
4. The Quadratic Formula | 二次求根公式
n such cases, use the quadratic formula. For ax² + bx + c = 0, the solutions are given by:
并非所有二次式都能轻松分解。在这种情况下，使用二次求根公式。对于 ax² + bx + c = 0，解为：
x = (−b ± √b² − 4ac) / 2a
into the formula: b² − 4ac = 9 + 16 = 25. Then x = (−3 ± 5) / 4, giving x = 0.5 or x = −2.
例如，解 2x² + 3x − 2 = 0，其中 a = 2，b = 3，c = −2。代入公式：b² − 4ac = 9 + 16 = 25。则 x = (−3 ± 5) / 4，得 x = 0.5 或 x = −2。
Be careful with signs — a common exam mistake is mishandling negative values of b or c. It is wise to write out the formula first, then substitute, and finally simplify step by step. The quadratic formula is always reliable, even when factorisation is not possible.
注意符号——常见的考试错误是处理 b 或 c 的负值时出错。建议先写出公式，再代入，最后逐步化简。即使无法因式分解，二次求根公式也始终可靠。
5. The Discriminant and the Nature of Roots | 判别式与根的性质
The expression b² − 4ac inside the quadratic formula is called the discriminant, often denoted as Δ. It determines the nature of the roots without actually solving the equation. This is a very common IGCSE question type.
二次公式中 b² − 4ac 这一表达式称为判别式，常用 Δ 表示。它无需实际解方程即可判断根的性质。这是IGCSE中非常常见的题型。
If b² − 4ac > 0: two distinct real roots. The parabola crosses the x-axis at two points.
若 b² − 4ac > 0：两个不同的实数根。抛物线在 x 轴上有两个交点。
exactly one repeated real root. The parabola touches the x-axis at one point (the vertex).
若 b² − 4ac = 0：一个重根。抛物线在 x 轴上只有一个切点（顶点处）。
If b² − 4ac < 0: no real roots. The parabola does not intersect the x-axis.
若 b² − 4ac < 0：没有实数根。抛物线不与 x 轴相交。
= 1 − 8 = −7, so there are no real roots. Understanding the discriminant helps you answer "show that" questions and multiple-choice questions quickly.
例如，对于 x² − 4x + 4 = 0，Δ = 16 − 16 = 0，因此方程有一个重根 x = 2。对于 x² + x + 2 = 0，Δ = 1 − 8 = −7，因此无实数根。理解判别式能帮你快速作答"证明"类问题和选择题。
6. The Graph of a Quadratic Function | 二次函数的图像
. Key features you must be able to identify include the vertex (turning point), the axis of symmetry, the y-intercept, and the x-intercepts (if any). In an exam, you may be asked to sketch the graph or read information from a given graph.
y = ax² + bx + c 的图像是一条抛物线。你必须能识别以下关键特征：顶点（转向点）、对称轴、y 截距和 x 截距（如果有）。在考试中，你可能会被要求画草图或从给定图像中读取信息。
y solving the equation. If the equation has no real roots, the parabola does not touch the x-axis at all.
令 x = 0 可得 y 截距，即 y = c。x 截距是方程 ax² + bx + c = 0 的实数根，通过解方程求得。如果方程无实数根，则抛物线完全不接触 x 轴。
-coordinate of the vertex. To find the y-coordinate of the vertex, substitute this value of x back into the original equation. The vertex represents the minimum (if a > 0) or maximum (if a < 0) value of the function.
对称轴是垂直线 x = −b/(2a)，它也给出了顶点的 x 坐标。要找到顶点的 y 坐标，将该 x 值代回原方程。顶点代表函数的最小值（a > 0 时）或最大值（a < 0 时）。
7. Completing the Square | 配方法
Completing the square is an essential skill for IGCSE. It rewrites a quadratic in the form a(x + h)² + k, which directly reveals the vertex and the line of symmetry. Start by factoring out the coefficient of x² from the first two terms, then add and subtract the square of half the coefficient of x inside the bracket.
配方法是IGCSE的核心技能。它将二次式改写为 a(x + h)² + k 的形式，直接显示顶点和对称轴。先从第一项和第二项提取 x² 的系数，然后在括号内加减 x 系数一半的平方。
x² + 6x + 7 = (x² + 6x + 9) − 9 + 7 = (x + 3)² − 2. This shows the vertex is at (−3, −2).
例如，对 x² + 6x + 7 配方。取6的一半得3，平方得9。则：x² + 6x + 7 = (x² + 6x + 9) − 9 + 7 = (x + 3)² − 2。这显示顶点在 (−3, −2)。
acket: 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 8 + 5 = 2(x + 2)² − 3. The vertex is at (−2, −3).
当 a ≠ 1 时，例如 2x² + 8x + 5，先提取2：2(x² + 4x) + 5。然后在括号内配方：2[(x + 2)² − 4] + 5 = 2(x + 2)² − 8 + 5 = 2(x + 2)² − 3。顶点在 (−2, −3)。
8. Solving Quadratic Equations by Completing the Square | 用配方法解二次方程
Completing the square can also be used to solve quadratic equations, especially when factorisation is difficult and you prefer an exact method over the formula. The key step is to isolate the squared term before taking square roots.
配方法也可以用于解二次方程，尤其是在因式分解困难而你希望用精确方法而非套公式时。关键步骤是先将平方项单独移到一边，再开平方根。
9 − 7 = 0, so (x + 3)² = 16. Taking square roots: x + 3 = ±4. Therefore, x = 1 or x = −7.
对于 x² + 6x − 7 = 0，配方：(x + 3)² − 9 − 7 = 0，即 (x + 3)² = 16。取平方根：x + 3 = ±4。因此，x = 1 或 x = −7。
both the positive and negative roots. In an exam, this is often where students lose a mark — the ± sign. Practise this method until you can complete the square quickly and accurately.
记住，对方程两边同时开平方根时，必须包含正负两个根。在考试中，这往往是学生丢分的地方——忘记 ± 号。反复练习这个方法，直到你能快速准确地完成配方。
9. Quadratic Graphs and Transformations | 二次函数图像的变换
orizontal and vertical shifts: h moves the graph left or right, and k moves it up or down.
在IGCSE Extended试卷中，你可能会被要求描述 y = x² 的图像如何变换为其他二次函数图像。形式 y = (x − h)² + k 显示了水平和平移：h 使图像左右移动，k 使图像上下移动。
y = (x − 3)²: the parabola shifts 3 units to the right.
y = (x − 3)²：抛物线向右平移3个单位。
y = x² + 2: the parabola shifts 2 units upward.
y = x² + 2：抛物线向上平移2个单位。
y = −x²: the parabola is reflected in the x-axis (opens downwards).
y = −x²：抛物线关于 x 轴翻转（开口向下）。
y = 3x²: the parabola becomes narrower (vertical stretch by factor 3).
y = 3x²：抛物线变得更窄（纵向拉伸3倍）。
Being able to describe these transformations with precise mathematical language is essential for gaining full marks on graph-related questions. Use terms like "translation", "reflection" and "stretch" with their exact values and directions.
能够用精确的数学语言描述这些变换，对于在图相关问题中获得满分至关重要。使用"平移""翻转""拉伸"等术语，并给出准确的值和方向。
10. Quadratic Inequalities | 二次不等式
and x = 3. Then sketch a quick parabola to determine which regions satisfy the inequality.
解二次不等式，如 x² − 5x + 6 < 0，是Extended级别的常见考点。首先解对应的二次方程以找到临界点。对于 x² − 5x + 6 = 0，根为 x = 2 和 x = 3。然后快速画出抛物线草图，确定哪些区间满足不等式。
rds. The graph is below the x-axis between the roots, so the solution is 2 < x < 3. For x² − 5x + 6 > 0, the solution would be x < 2 or x > 3.
由于 a = 1 > 0，抛物线开口向上。图像在两根之间位于 x 轴下方，因此解是 2 < x < 3。对于 x² − 5x + 6 > 0，解为 x < 2 或 x > 3。
question. Be careful with strict inequalities (<, >) versus non-strict inequalities (≤, ≥) — the latter include the roots themselves.
始终按题目要求使用集合符号或区间符号写出最终答案。注意严格不等式（<、>）与非严格不等式（≤、≥）的区别——后者包含根本身。
11. Applications: Word Problems and Modelling | 应用：文字题与建模
Quadratic functions often appear in real-world applications such as projectile motion, area optimisation and revenue problems. A classic example: the area of a rectangle is 50 m², and its length is 5 m longer than its width. Find the width.
二次函数常出现在实际应用中，如抛体运动、面积优化和收入问题。经典例题：一个矩形的面积为 50 m²，其长比宽长 5 m。求宽。
5) = 0. Hence, x = 5 (rejecting x = −10 since width cannot be negative). The width is 5 m.
设宽为 x，则长为 x + 5。面积方程为 x(x + 5) = 50，化简为 x² + 5x − 50 = 0。因式分解求解：(x + 10)(x − 5) = 0。因此，x = 5（舍去 x = −10，因为宽度不能为负数）。宽度为 5 m。
In word problems, always define your variable clearly, translate the conditions into a quadratic equation, solve it, and check whether your answers make sense in context. This final sanity check is critical: negative lengths, times or quantities should be rejected.
在文字题中，务必清晰定义变量，将条件转化为二次方程，求解，并检查答案在情境中是否合理。最后的合理性检查至关重要：负数长度、时间或数量应被舍去。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
e roots, incorrect sign handling when using the quadratic formula, and misidentifying the vertex when reading from a graph. Being aware of these pitfalls is the first step to avoiding them.
学生在二次函数题目中最常见的错误包括：开平方根时忘记 ± 号、使用二次公式时符号处理错误、以及从图像读取顶点时识别错误。意识到这些陷阱是避免它们的第一步。
Here are some practical exam tips to maximise your score: always rearrange the equation into standard form before solving; show all your working, as IGCSE marks are awarded for method as well as accuracy; sketch a quick graph to guide your reasoning; and check your answers by substitution.
以下是一些实用的考试技巧，助你最大化得分：解题前务必将方程重排为标准形式；展示所有步骤，因为IGCSE评分既看方法也看准确性；画一个快速草图来引导推理；并通过代入检查答案。
Finally, master all three solving methods — factorisation, the quadratic formula, and completing the square — because different questions reward different techniques. Factorisation is fastest when possible; the formula is a reliable fallback; and completing the square connects directly to graph properties. With consistent practice, quadratic functions become one of the easiest high-scoring topics on the paper.
最后，掌握所有三种解题方法——因式分解、二次求根公式和配方法——因为不同题目偏爱不同技巧。因式分解在可行时最快；公式是可靠的备用方案；配方法直接关联图像性质。坚持练习，二次函数将成为试卷中最容易拿高分的高频考点之一。
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