📚 Quadratic Functions: Key Points for IGCSE CIE Mathematics | IGCSE CIE 数学：二次函数 考点精讲
Quadratic functions form one of the most important building blocks in the IGCSE CIE Mathematics syllabus. Understanding their graphs, algebraic manipulation, and real-world applications is essential for success in both Paper 2 and Paper 4. This article systematically covers every core concept, from recognising the standard form to solving quadratic equations and interpreting discriminant values. Each section pairs English and Mandarin explanations to help bilingual learners solidify their knowledge and exam technique.
二次函数是 IGCSE CIE 数学课程中最重要的基础知识板块之一。掌握其图像特征、代数变换和实际应用对在 Paper 2 和 Paper 4 中取得高分至关重要。本文系统梳理了每一个核心概念，从识别标准式到求解二次方程，再到解读判别式的值。每节均以中英双语讲解，帮助双语学习者夯实知识并提升应试技巧。
1. What is a Quadratic Function? | 什么是二次函数？
s graph is always a curve called a parabola. The simplest quadratic function is f(x) = x².
二次函数是最高次项次数为 2 的多项式函数。在 IGCSE 中通常以 x 为变量，其图像总是一条称为抛物线的曲线。最简单的二次函数是 f(x) = x²。
nction reduces to a linear function, which is not quadratic. The domain of a quadratic function is all real numbers.
二次函数的一般形式为 f(x) = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。若 a = 0，函数退化为一次函数，不再属于二次函数。二次函数的定义域为全体实数。
2. Standard Form and Coefficients | 标准式与系数
s the standard form. Here a is the quadratic coefficient, b the linear coefficient, and c the constant term. The value of a determines the width and direction of the parabola, while c gives the y-intercept (0, c).
表达式 y = ax² + bx + c 被称为标准式。其中 a 是二次项系数，b 是一次项系数，c 是常数项。a 的值决定了抛物线的宽窄和开口方向，c 则给出了 y 轴截距 (0, c)。
When a > 0, the parabola opens upwards, forming a ∪ shape. When a < 0, it opens downwards, forming an ∩ shape. The magnitude of a affects the steepness: a larger |a| makes the parabola narrower, while a smaller |a| makes it wider.
当 a > 0 时，抛物线开口向上，呈 ∪ 形。当 a < 0 时，开口向下，呈 ∩ 形。|a| 的大小影响陡峭程度：|a| 越大，抛物线越窄；|a| 越小，抛物线越宽。
y = 2x² + 3x − 5 (a = 2, b = 3, c = −5)
3. Shape of a Parabola and Key Features | 抛物线的形状与关键特征
Every quadratic graph is symmetric. The line of symmetry is called the axis of symmetry, which is a vertical line passing through the vertex. The vertex is the maximum or minimum point of the parabola depending on the sign of a.
每一条二次函数图像都是对称的。对称线称为对称轴，是一条穿过顶点的竖直线。顶点是抛物线的最高点或最低点，取决于 a 的符号。
x² + bx + c = 0. IGCSE questions frequently ask you to identify these features from an equation or a given graph.
抛物线总有一个 y 轴截距，可能有 0 个、1 个或 2 个 x 轴截距。x 轴截距是方程 ax² + bx + c = 0 的实数解（根）。IGCSE 考题常要求你从方程或给定图像中识别这些特征。
To sketch a parabola, you must show the correct shape, label the vertex, axis of symmetry, and intercepts clearly. Use dashed lines for the axis of symmetry if needed.
要画抛物线草图，必须展示正确的形状，清楚标出顶点、对称轴和截距。必要时可用虚线表示对称轴。
4. Vertex Form and Completing the Square | 顶点式与配方法
tremely useful because it directly gives the turning point and the axis of symmetry x = h.
二次函数的顶点式为 y = a(x − h)² + k，其中 (h, k) 是顶点坐标。这种形式非常有用，因为它直接给出了转折点和对称轴 x = h。
add and subtract (b/2)² to create a perfect square trinomial. This skill is tested frequently in CIE IGCSE.
你可以通过配方法将标准式转化为顶点式。对于 y = x² + bx + c，通过加上并减去 (b/2)² 来构造完全平方三项式。这是 CIE IGCSE 中经常考查的技能。
Example: y = x² + 6x + 5 → y = (x + 3)² − 4, vertex (−3, −4)
5. Axis of Symmetry and Vertex from Standard Form | 由标准式求对称轴与顶点
as the same x-coordinate, and you can find the y-coordinate by substituting this x-value back into the original equation.
即使不配方，你也可以使用公式 x = −b/(2a) 求出对称轴。顶点的 x 坐标与此相同，再将此 x 值代回原方程即可求得 y 坐标。
vertex is (2, 2(2)² − 8(2) + 3) = (2, −5). This method is faster for many exam questions.
对于二次函数 y = 2x² − 8x + 3，a = 2，b = −8。对称轴为 x = −(−8)/(2×2) = 2。顶点为 (2, 2(2)² − 8(2) + 3) = (2, −5)。在许多考试题目中，这种方法更快捷。
Remember that the axis of symmetry always passes through the vertex. Once you know the vertex, you can confidently state whether it is a maximum (a < 0) or minimum (a > 0).
请记住对称轴总是穿过顶点。一旦知道了顶点，你就能自信地判断它是最大值点 (a < 0) 还是最小值点 (a > 0)。
6. Factorised Form and Roots | 因式分解形式与根
ts (roots) of the corresponding equation. The graph crosses the x-axis at x = p and x = q.
二次式有时可以写成因式分解形式：y = a(x − p)(x − q)。其中 p 和 q 是对应方程的 x 轴截距（根）。图像在 x = p 和 x = q 处与 x 轴相交。
dd to b. For example, x² + 5x + 6 = (x + 2)(x + 3). Always expand to check your factoring.
要对形如 x² + bx + c 的二次式进行因式分解，需寻找两个数，其积为 c，其和为 b。例如 x² + 5x + 6 = (x + 2)(x + 3)。务必展开括号检查分解是否正确。
If a ≠ 1, you may need to use the ac-method or factor by grouping. In IGCSE exams, factorising skills are essential for solving quadratic equations efficiently.
若 a ≠ 1，可能需要使用 ac 法或分组分解法。在 IGCSE 考试中，因式分解技能是高效求解二次方程的关键。
7. The Quadratic Formula and Solving Equations | 求根公式与解方程
c equations using the quadratic formula. For ax² + bx + c = 0, the solutions are given by:
当因式分解困难或无法进行时，你可以使用求根公式来解二次方程。对于 ax² + bx + c = 0，解由下式给出：
x = (−b ± √b² − 4ac) / (2a)
The symbol '±' s. Always write the formula accurately and substitute the values of a, b, and c carefully.
符号“±”表示通常有两个解：一个取加号，另一个取减号。务必准确书写公式，并细心代入 a、b、c 的值。
Remember to consider the order of operations. Calculate the discriminant b² − 4ac first, then find the square root, and finally divide by 2a. Leave answers in simplified surd form or to required decimal places as specified in the question.
注意运算顺序。先计算判别式 b² − 4ac，然后求平方根，最后除以 2a。答案可保留为简化根式形式，或按题目要求取指定小数位数。
8. The Discriminant and Nature of Roots | 判别式与根的性质
nt. It reveals the nature of the roots without solving the equation fully. IGCSE questions often ask you to use the discriminant to find unknown coefficients or to comment on the graph.
根号下的表达式 Δ = b² − 4ac 称为判别式。它无需完全解出方程即可揭示根的性质。IGCSE 题目常要求利用判别式求未知系数，或对图像作出评述。
s exactly one real root (a repeated root), and the graph touches the x-axis at the vertex. If Δ < 0, there are no real roots, and the graph does not intersect the x-axis.
若 Δ > 0，二次式有两个不同的实根，图像在两点与 x 轴相交。若 Δ = 0，求有一个实根（重根），图像在顶点处与 x 轴相切。若 Δ < 0，无实根，图像与 x 轴不相交。
| Discriminant (Δ) | Nature of roots | Graph behaviour |
| Δ > 0 | Two distinct real roots | Crosses x-axis twice |
| Δ = 0 | One repeated real root | Touches x-axis at vertex |
| Δ < 0 | No real roots | Does not intersect x-axis |
9. Graph Transformations of Quadratic Functions | 二次函数图像的变换
quare shifts the graph vertically. For example, y = x² + k moves the parabola up by k units if k > 0, down if k < 0.
理解函数方程的变化如何影响图像是一项关键技能。对于基本的抛物线 y = x²，在平方外进行加减会垂直平移图像。例如，y = x² + k 会将抛物线上移 k 个单位（k > 0），或下移 |k| 个单位（k < 0）。
x − h)² shifts the parabola to the right by h units. Note the minus sign: (x − 2)² moves the graph to the right, not left. This is a common misconception, so practise carefully.
改变平方项内的 x 会水平平移图像。y = (x − h)² 将抛物线右移 h 个单位。注意减号：(x − 2)² 使图像右移，而非左移。这是常见的误解，需仔细练习。
a(x − h)² + k, which represents a shift of h units horizontally, k units vertically, and a vertical stretch/compression with reflection if a is negative. Always describe transformations in the correct order: stretch first, then translations.
组合这些变换可得 y = a(x − h)² + k，表示水平移动 h 个单位，垂直移动 k 个单位，以及若 a 为负则进行垂直伸缩和翻转。务必按正确顺序描述变换：先伸缩，后平移。
10. Real-world Applications and Exam Tips | 实际应用与应试技巧
IGCSE frequently sets problems involving the maximum or minimum value of a quadratic model, such as the height of a projectile, area optimisation, or profit functions. Identify the quadratic and then find the vertex to solve these problems.
IGCSE 常会出现涉及二次模型最大值或最小值的问题，如抛射物的高度、面积优化或利润函数。识别出二次函数，然后求出顶点即可解决这类问题。
When reading word problems, define your variable, set up the quadratic equation, and clearly state what the solution represents. For example, the maximum height reached by a ball thrown upwards is given by the y-coordinate of the vertex of its height vs. time parabola.
在解答文字题时，定义变量、列出二次方程，并清楚说明解的含义。例如，上抛小球所能到达的最高高度，由其高度－时间抛物线的顶点的 y 坐标给出。
During the CIE exam, always show full algebraic steps even if you use a calculator. Marks are awarded for method, substitution, and correct simplification. Check your answers by substituting back or by examining if the graph behaviour matches the discriminant analysis.
在 CIE 考试中，即使使用计算器也要展示完整的代数步骤。方法、代入和正确化简均有分数。将答案代回原式，或检查图像行为是否与判别式分析相符，均可用于验证答案。
Finally, familiarise yourself with past paper questions on quadratic functions. Recognise patterns, such as questions that provide part of a graph and ask for the equation, or tasks that require you to prove a quadratic always positive using the discriminant.
最后，请熟悉有关二次函数的历年真题。识别出题模式，例如给出部分图像要求写出方程，或要求利用判别式证明某个二次式恒为正的题目。
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