Quadratic Functions | 二次函数

📚 Quadratic Functions | 二次函数

Quadratic functions lie at the heart of IB Mathematics, appearing in topics ranging from algebra and functions to calculus and modelling. Mastering quadratics means being able to recognise their forms, manipulate their equations, and interpret their graphs. This guide takes you through the key concepts, methods, and applications you need for success in both Analysis & Approaches and Applications & Interpretation.

二次函数是 IB 数学的核心内容，出现在从代数、函数到微积分和建模的多个主题中。掌握二次函数意味着能够识别其形式、操作其方程并解释其图像。本指南将带你梳理在分析与方法和应用与解释课程中取得好成绩所需的核心概念、方法和应用。

1. Definition and Standard Form | 定义与标准形式

A quadratic function is a polynomial function of degree 2. Its standard (or general) form is f(x) = ax² + bx + c, where a, b, c are real numbers and a ≠ 0. The coefficient a determines the direction and width of the parabola.

二次函数是一个二次多项式函数。其标准（或称一般）形式为 f(x) = ax² + bx + c，其中 a、b、c 是实数且 a ≠ 0。系数 a 决定了抛物线的开口方向和宽窄。

The term ax² is the quadratic term, bx is the linear term, and c is the constant term. If a = 0, the function degenerates to a linear function, which is why the condition a ≠ 0 is essential. For example, f(x) = 2x² − 4x + 1 is a quadratic with a = 2, b = −4, c = 1.

ax² 是二次项，bx 是一次项，c 是常数项。若 a = 0，函数就退化为一次函数，因此条件 a ≠ 0 必不可少。例如，f(x) = 2x² − 4x + 1 是一个二次函数，其中 a = 2，b = −4，c = 1。

2. The Parabola: Shape and Key Features | 抛物线：形状与关键特征

The graph of a quadratic function is a parabola. If a > 0, the parabola opens upwards (U-shaped), and the vertex is a minimum point. If a < 0, the parabola opens downwards (∩-shaped), and the vertex is a maximum point.

二次函数的图像是一条抛物线。若 a > 0，抛物线开口向上（呈 U 形），顶点为最小值点；若 a < 0，抛物线开口向下（呈 ∩ 形），顶点为最大值点。

The axis of symmetry is a vertical line passing through the vertex. For the standard form, its equation is x = −b/(2a). The y-intercept is easily found by setting x = 0, giving the point (0, c). The x-intercepts (roots or zeros) are the solutions of the equation ax² + bx + c = 0 and can be found by factorisation, completing the square, or using the quadratic formula.

对称轴是一条穿过顶点的垂直线。对于标准形式，其方程为 x = −b/(2a)。y 轴截距可通过令 x = 0 轻松求得，即点 (0, c)。x 轴截距（根或零点）是方程 ax² + bx + c = 0 的解，可通过因式分解、配方法或求根公式找到。

The vertex coordinates can be obtained by substituting x = −b/(2a) back into the function: the vertex is (−b/(2a), f(−b/(2a))).

顶点坐标可通过将 x = −b/(2a) 代回函数得到：顶点为 (−b/(2a), f(−b/(2a)))。

3. Vertex Form and Completing the Square | 顶点式与配方法

The vertex form of a quadratic function is f(x) = a(x − h)² + k, where (h, k) is the vertex and the axis of symmetry is x = h. This form reveals transformations and the extreme value directly.

二次函数的顶点式为 f(x) = a(x − h)² + k，其中 (h, k) 为顶点，对称轴为 x = h。该形式直接展现了变换和极值。

To convert standard form to vertex form we complete the square. Take f(x) = 2x² + 8x + 5. First, factor ‘2’ from the first two terms: 2(x² + 4x) + 5. Inside the bracket, add and subtract (4/2)² = 4 to create a perfect square: 2[(x + 2)² − 4] + 5. Distribute and simplify: 2(x + 2)² − 8 + 5 = 2(x + 2)² − 3. The vertex is (−2, −3).

要将标准形式转换为顶点式，我们需要配方法。以 f(x) = 2x² + 8x + 5 为例。首先从前两项中提出“2”：2(x² + 4x) + 5。在括号内加上并减去 (4/2)² = 4 以构成完全平方：2[(x + 2)² − 4] + 5。展开并化简：2(x + 2)² − 8 + 5 = 2(x + 2)² − 3。顶点为 (−2, −3)。

Completing the square is essential for deriving the quadratic formula and for finding maximum or minimum values in optimisation problems without calculus.

配方法对于推导求根公式以及在不使用微积分的情况下解决优化问题中的最值求解至关重要。

4. Factored Form and Zeros | 因式分解形式与零点

If a quadratic factorises over the reals, it can be written in factored form: f(x) = a(x − p)(x − q), where p and q are the roots (zeros) of the function. The x-intercepts are (p, 0) and (q, 0).

若二次函数可在实数范围内因式分解，便可写成因式分解形式：f(x) = a(x −

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