IGCSE Maths: Quadratic Functions Key Points | IGCSE 数学：二次函数 考点精讲

📚 IGCSE Maths: Quadratic Functions Key Points | IGCSE 数学：二次函数 考点精讲

Quadratic functions are fundamental in IGCSE Mathematics, appearing in various forms from solving equations to graphing parabolas. This article summarises the essential concepts, methods, and common pitfalls that every IGCSE student should master to excel in exams.

二次函数是 IGCSE 数学的核心内容之一，从解方程到绘制抛物线，形式多样。本文总结了每一位 IGCSE 学生必须熟练掌握的重要概念、解题方法和常见易错点，助你在考试中取得高分。

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1. What is a Quadratic Function? | 什么是二次函数？

A quadratic function is a polynomial function of degree 2, generally expressed as f(x) = ax² + bx + c, where a, b, c are constants and a ≠ 0. The graph of a quadratic function is a parabola that can open upwards (a > 0) or downwards (a < 0).

二次函数是最高次为 2 的多项式函数，一般表示为 f(x) = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。其图像是一条抛物线，开口可能向上（a > 0）或向下（a < 0）。

Plotting a quadratic function requires identifying its key features: vertex, axis of symmetry, intercepts, and the direction of opening. These features help sketch the graph without plotting many points.

绘制二次函数图像时，需确定其关键特征：顶点、对称轴、截距和开口方向。这些特征能让我们无需计算大量点即可画出草图。

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2. Standard Form and the Role of ‘a’ | 标准形式及 ‘a’ 的作用

In the standard form y = ax² + bx + c, the coefficient a determines the width and direction of the parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point. The larger |a|, the narrower the parabola; the smaller |a|, the wider it is.

在标准式 y = ax² + bx + c 中，系数 a 决定了抛物线的开口方向和宽窄。若 a > 0，抛物线开口向上，有最小值点；若 a < 0，开口向下，有最大值点。|a| 越大，抛物线越窄；|a| 越小，则越宽。

The coefficient c gives the y-intercept of the graph, as f(0) = c. The coefficient b affects the horizontal position of the vertex but its interpretation is less direct.

系数 c 给出图像在 y 轴上的截距，因为 f(0) = c。系数 b 影响顶点在水平方向的位置，但它的几何意义并不直接。

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3. Finding the Vertex and Axis of Symmetry | 求顶点与对称轴

The vertex of a parabola y = ax² + bx + c can be found using the formula x = -b/(2a) for the axis of symmetry. Then substitute this x-value into the function to get the y-coordinate of the vertex. The vertex form is y = a(x – h)² + k, where (h, k) is the vertex.

抛物线 y = ax² + bx + c 的顶点可利用对称轴的公式 x = -b/(2a) 求得。然后将此 x 值代入原函数，即可得到顶点的纵坐标。顶点式写作 y = a(x – h)² + k，其中 (h, k) 就是顶点坐标。

Completing the square converts the standard form to vertex form. For example, y = x² + 6x + 5 can be rewritten as y = (x + 3)² – 4, revealing the vertex (-3, -4).

通过配方法可将标准式转化为顶点式。例如，y = x² + 6x + 5 可改写为 y = (x + 3)² – 4，从而看出顶点为 (-3, -4)。

Vertex ( -b/(2a) , f( -b/(2a) ) )

顶点 ( -b/(2a) , f( -b/(2a) ) )

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4. The Discriminant (Δ) | 判别式（Δ）

The discriminant is given by Δ = b² – 4ac. It tells us the nature of the roots of the quadratic equation ax² + bx + c = 0 without solving it. If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated real root (the parabola touches the x-axis). If Δ < 0, there are no real roots (the parabola does not intersect the x-axis).

判别式定义为 Δ = b² – 4ac。它使我们不需求解方程就能判断二次方程 ax² + bx + c = 0 的根的性质。若 Δ > 0，方程有两个不相等的实根；若 Δ = 0，有一个重根（抛物线与 x 轴相切）；若 Δ < 0，没有实根（抛物线与 x 轴无交点）。

In IGCSE, the discriminant is often tested alongside sketching questions or to determine the number of intersection points between a line and a curve.

在 IGCSE 考试中，判别式常与草图题一起出现，或用于判断直线与曲线的交点个数。

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5. Solving Quadratic Equations by Factorising | 因式分解解二次方程

When a quadratic can be written as (x + p)(x + q) = 0, the solutions are x = -p or x = -q. To factorise x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c with a ≠ 1, methods like splitting the middle term or using the “ac” method are used.

当二次式能写成 (x + p)(x + q) = 0 时，解为 x = -p 或 x = -q。要对 x² + bx + c 进行因式分解，需找到两个数，其积为 c，其和为 b。对于 a ≠ 1 的 ax² + bx + c，可用十字相乘法或 “ac” 方法进行分解。

Always check by expanding the factors. Not all quadratics can be factorised with integers; in such cases, use the formula or completing the square.

分解后一定要展开检验。并非所有二次式都能用整数因式分解，这时需使用求根公式或配方法。

Example: Solve x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0 → x = 2 or x = 3.

示例：解 x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0 → x = 2 或 x = 3。

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6. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. For x² + bx, add and subtract (b/2)²: x² + bx = (x + b/2)² – (b/2)². This technique is essential for deriving the quadratic formula and for finding the minimum or maximum value.

配方法将二次式改写为

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