IGCSE AQA Mathematics: Mastering Quadratic Functions | IGCSE AQA 数学：二次函数 考点精讲

📚 IGCSE AQA Mathematics: Mastering Quadratic Functions | IGCSE AQA 数学：二次函数 考点精讲

Quadratic functions are a crucial topic in the IGCSE AQA Mathematics syllabus, appearing in both the Foundation and Higher tiers. This article covers all the essential concepts, techniques, and exam tips you need to master quadratics, from basic shapes to solving equations and applying them to real-world problems.

二次函数是 IGCSE AQA 数学大纲中的重要主题，出现在基础卷和高级卷中。本文涵盖了掌握二次函数所需的所有基本概念、解题技巧和应试贴士，从基本图像到解方程，再到解决实际问题。

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

A quadratic function is a polynomial function of degree two. Its general form is f(x) = 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.

二次函数是一个二次多项式函数。其一般形式为 f(x) = ax² + bx + c，其中 a、b、c 是常数，且 a ≠ 0。二次函数的图像是一条平滑的曲线，称为抛物线。

If a > 0, the parabola opens upwards (⋃ shape), and the function has a minimum value. If a < 0, it opens downwards (⋂ shape) and has a maximum value.

如果 a > 0，抛物线开口向上（⋃ 形），函数有一个最小值。如果 a < 0，开口向下（⋂ 形），函数有一个最大值。

The simplest quadratic is y = x², which passes through the origin (0,0) and is symmetric about the y-axis.

最简单的二次函数是 y = x²，它经过原点 (0,0)，且关于 y 轴对称。

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2. Standard Form and Graph Shape | 标准形式与图形形状

The standard form y = ax² + bx + c instantly tells you the y-intercept: when x = 0, y = c. So the curve crosses the y-axis at (0, c).

标准形式 y = ax² + bx + c 能立即告诉你 y 轴截距：当 x = 0 时，y = c。因此曲线与 y 轴交于点 (0, c)。

The coefficient a determines the width as well. Larger |a| makes the parabola narrower, while smaller |a| (0 < |a| < 1) makes it wider.

系数 a 还决定了抛物线的宽度。|a| 越大，抛物线越窄；|a| 越小（0 < |a| < 1），抛物线越宽。

Example: Compare y = 2x² (narrow) and y = ½x² (wide). Both have vertex at (0,0) and open upwards.

例如：比较 y = 2x²（窄）和 y = ½x²（宽）。两者的顶点都在 (0,0) 且开口向上。

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3. Finding the Vertex and Axis of Symmetry | 求顶点和对称轴

The vertex is the turning point of the parabola. For a quadratic in form y = a(x – h)² + k, the vertex is (h, k). The axis of symmetry is the vertical line x = h.

顶点是抛物线的转折点。对于形如 y = a(x – h)² + k 的二次函数，顶点为 (h, k)。对称轴是垂直线 x = h。

To find the vertex from y = ax² + bx + c, use the formula for the x-coordinate: x = -b / (2a). Then substitute this x into the function to find y.

要从 y = ax² + bx + c 求顶点，可使用 x 坐标公式：x = -b / (2a)。然后将此 x 代入函数求出 y。

For example, y = x² – 4x + 3: x = -(-4) / (2×1) = 2. Then y = (2)² – 4(2) + 3 = 4 – 8 + 3 = -1. Vertex is (2, -1). The axis of symmetry is x = 2.

例如，y = x² – 4x + 3：x = -(-4)/(2×1) = 2。然后 y = (2)² – 4(2) + 3 = 4 – 8 + 3 = -1。顶点为 (2, -1)，对称轴为 x = 2。

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4. Factoring Quadratics | 因式分解二次式

Factoring (or factorising) a quadratic expression ax² + bx + c is the reverse of expanding double brackets. The most common type is when a = 1, so you look for two numbers that multiply to c and add to b.

因式分解二次式 ax² + bx + c 是展开双括号的逆运算。最常见的类型是 a = 1，此时需要寻找两个数，乘积为 c，和为 b。

For x² + 5x + 6, the numbers 2 and 3 multiply to 6 and add to 5. So (x + 2)(x + 3).

对于 x² + 5x + 6，数字 2 和 3 乘积为 6，和为 5。因此分解为 (x + 2)(x + 3)。

When a > 1, use methods like splitting the middle term or trial and error. E.g., 2x² + 7x + 3: find two numbers that multiply to 2×3=6 and add to 7 (6 and 1). Split: 2x² + 6x + x + 3 = 2x(x+3)+1(x+3) = (2x+1)(x+3).

当 a > 1 时，可使用中间项拆分法或试错法。例如 2x² + 7x + 3：寻找乘积为 2×3=6、和为 7 的两个数（6 和 1）。拆分：2x² + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3)。

Always check by expanding. Remember the difference of two squares: a² – b² = (a – b)(a + b), such as x² – 9 = (x – 3)(x + 3).

务必通过展开来检查。记住平方差公式：a² – b² = (a – b)(a + b)，例如 x² – 9 = (x – 3)(x + 3)。

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5. Completing the Square | 配方法

Completing the square transforms y = ax² + bx + c into the vertex form y = a(x – h)² + k. This is useful for finding the vertex and solving equations.

配方法将 y = ax² + bx + c 转化为顶点式 y = a(x – h)² + k。这在求顶点和解方程时非常有用。

For a = 1: take x² + bx, add and subtract (b/2)². Example: x² + 6x + 5. Half of 6 is 3, square is 9. Write (x + 3)² – 9 + 5 = (x + 3)² – 4. Vertex (-3, -4).

当 a = 1 时：对于 x² + bx，加上并减去 (b/2)²。例如：x² + 6x + 5。6 的一半是 3，平方是 9。写作 (x + 3)² – 9 + 5 = (x + 3)² – 4。顶点为 (-3, -4)。

When a ≠ 1, factor out a from the x² and x terms first, then complete the square inside the bracket. For 2x² + 8x + 3 = 2[x² + 4x] + 3 = 2[(x+2)² – 4] + 3 = 2(x+2)² – 8 + 3 = 2

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