Quadratic Functions & Equations | 二次函数与方程

📚 Quadratic Functions & Equations | 二次函数与方程

Quadratic functions and equations form one of the most essential topics in the IGCSE Mathematics syllabus. From plotting parabolas to solving real-world optimisation problems, mastery of quadratics is not only exam-critical but also foundational for higher-level mathematics including calculus and physics.

二次函数与方程是 IGCSE 数学大纲中最核心的主题之一。从绘制抛物线到解决现实中的最优化问题，掌握二次函数不仅对考试至关重要，更是高阶数学（如微积分和物理）的基础。

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

A quadratic function is any function that can be written in the standard form:

f(x) = ax² + bx + c, where a ≠ 0

Here, a, b and c are real constants. The coefficient a determines the “width” and direction of the parabola; b affects the position of the axis of symmetry; c represents the y-intercept. A quadratic equation arises when we set f(x) = 0:

二次函数是指可以写成标准形式的函数：

f(x) = ax² + bx + c，其中 a ≠ 0

其中 a、b、c 为实数常数。系数 a 决定抛物线的“宽窄”和开口方向；b 影响对称轴的位置；c 为 y 轴截距。当令 f(x) = 0 时，就得到二次方程：

ax² + bx + c = 0

2. Graphs of Quadratic Functions | 二次函数图像

The graph of a quadratic function is a curve called a parabola. When a > 0, the parabola opens upward and has a minimum point (vertex). When a < 0, it opens downward and has a maximum point.

二次函数的图像是一条称为抛物线的曲线。当 a > 0 时，抛物线开口向上，存在最低点（顶点）；当 a < 0 时，开口向下，存在最高点。

Key features of a parabola include:

Vertex: the turning point of the curve, located at x = −b / (2a).
Axis of symmetry: the vertical line x = −b / (2a) passing through the vertex.
y-intercept: the point (0, c) where the curve crosses the y-axis.
x-intercepts (roots): the points where the curve crosses the x-axis, corresponding to the solutions of ax² + bx + c = 0.

顶点：曲线的转折点，位于 x = −b / (2a) 处。
对称轴：穿过顶点的竖直线 x = −b / (2a)。
y 轴截距：曲线与 y 轴的交点 (0, c)。
x 轴截距（根）：曲线与 x 轴的交点，对应方程 ax² + bx + c = 0 的解。

3. Solving by Factorisation | 因式分解法求解

Factorisation is often the fastest method when the quadratic has rational roots. The idea is to express ax² + bx + c as a product of two linear factors.

当二次方程有有理数根时，因式分解法通常是最快捷的方法。其核心思想是将 ax² + bx + c 写成两个一次因式的乘积。

Example: Solve x² − 5x + 6 = 0

示例：解方程 x² − 5x + 6 = 0

Step 1: Find two numbers that multiply to 6 and add to −5. These are −2 and −3.

步骤 1：寻找两个数，使它们的乘积为 6，和为 −5，这两个数是 −2 和 −3。

Step 2: Write the factorised form: (x − 2)(x − 3) = 0

步骤 2：写出因式分解形式：(x − 2)(x − 3) = 0

Step 3: Apply the zero product property: x = 2 or x = 3

步骤 3：应用零因子性质得：x = 2 或 x = 3

4. Completing the Square | 配方法

Completing the square is a powerful technique that transforms any quadratic into vertex form. This method is particularly useful for finding the maximum or minimum value of a function and for solving equations that do not factorise easily.

配方法是一种强大的技巧，它可以将任意二次函数转化为顶点式。这种方法在求函数最大值或最小值以及解不容易因式分解的方程时尤为有效。

The general procedure for x² + bx + c:

x² + bx + c 的一般操作步骤如下：

(x + b/2)² − (b/2)² + c

Example: Express f(x) = x² + 6x + 5 in the form (x + p)² + q.

示例：将 f(x) = x² + 6x + 5 表示为 (x + p)² + q 的形式。

f(x) = (x + 3)² − 9 + 5 = (x + 3)² − 4

Thus the vertex is at (−3, −4) and the minimum value of the function is −4.

因此顶点坐标为 (−3, −4)，函数的最小值为 −4。

5. The Quadratic Formula | 二次求根公式

The quadratic formula is a universal tool that solves any quadratic equation, regardless of whether the roots are rational, irrational or complex (in IGCSE, we focus on real roots).

二次求根公式是解决任何二次方程的万能工具，无论根是有理数、无理数还是复数（在 IGCSE 中我们主要关注实数根）。

x = (−b ± √(b² − 4ac)) / (2a)

This formula is derived by completing the square on the general form ax² + bx + c = 0. It is essential to memorise it and understand how to substitute values correctly.

该公式通过对一般形式 ax² + bx + c = 0 配方推导得出。务必熟记并正确代入系数进行计算。

Example: Solve 2x² − 4x − 3 = 0 using the quadratic formula.

示例：使用求根公式解方程 2x² − 4x − 3 = 0。

Here a = 2, b = −4, c = −3. Substituting:

其中 a = 2，b = −4，c = −3。代入得：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (2 ± √10) / 2

So the two solutions are x ≈ 2.58 and x ≈ −0.58.

因此两个解约为 x ≈ 2.58 和 x ≈ −0.58。

6. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant (denoted Δ). It tells us about the nature of the roots without actually solving the equation.

表达式 b² − 4ac 被称为判别式（记作 Δ）。它不需要实际解方程就能判断根的性质。

Δ = b² − 4ac	Nature of Roots \| 根的性质
Δ > 0	Two distinct real roots \| 两个不相等的实数根
Δ = 0	One repeated real root \| 两个相等实数根（一个重根）
Δ < 0	No real roots \| 没有实数根

For example, for the equation x² + 2x + 5 = 0, we have Δ = 4 − 20 = −16 < 0, so the graph does not intersect the x-axis at all.

例如，对于方程 x² + 2x + 5 = 0，Δ = 4 − 20 = −16 < 0，因此抛物线完全不与 x 轴相交。

7. Vertex Form and Transformations | 顶点式与图像变换

It is often useful to rewrite a quadratic function in vertex form: f(x) = a(x − h)² + k, where (h, k) is the vertex. This form directly reveals the vertex and makes graph transformations easier to apply.

将二次函数写成顶点式是非常有用的：f(x) = a(x − h)² + k，其中 (h, k) 为顶点坐标。这种形式直接给出了顶点，也使图像变换更加清晰。

Graph transformations:

f(x) + k: shift upward by k units | 图像向上平移 k 个单位
f(x + h): shift left by h units | 图像向左平移 h 个单位
−f(x): reflection across the x-axis | 图像关于 x 轴对称
af(x): vertical stretch/compression by factor a | 图像按比例 a 纵向伸缩

For example, the graph of y = 2(x − 3)² + 1 is obtained from y = x² by: shifting right 3 units, stretching vertically by factor 2, and moving up 1 unit.

例如，y = 2(x − 3)² + 1 的图像可由 y = x² 经过以下变换得到：向右平移 3 个单位、纵向拉伸 2 倍、向上平移 1 个单位。

8. Quadratic Inequalities | 二次不等式

Solving quadratic inequalities is a common IGCSE topic that builds on the graphical understanding of parabolas. The key is to find the roots first, then determine the sign of the quadratic expression in each interval.

解二次不等式是 IGCSE 常考内容，它建立在对抛物线图像的理解之上。关键是先求出根，再判断各区间内二次表达式的正负号。

Example: Solve x² − x − 6 < 0

示例：解不等式 x² − x − 6 < 0

First, factorise: (x − 3)(x + 2) < 0

首先因式分解：(x − 3)(x + 2) < 0

The roots are x = 3 and x = −2. Since the parabola opens upward, the expression is negative between the roots.

根为 x = 3 和 x = −2。由于抛物线开口向上，表达式在两根之间为负。

−2 < x < 3

For ≥ or ≤, include the endpoints: e.g., x² − x − 6 ≤ 0 gives −2 ≤ x ≤ 3.

对于 ≥ 或 ≤，需要包含端点：例如 x² − x − 6 ≤ 0 的解为 −2 ≤ x ≤ 3。

9. Applications in Real-World Problems | 实际应用问题

Quadratic functions are widely used to model real-world phenomena, such as projectile motion, area optimisation and economic profit functions.

二次函数被广泛应用于模拟现实世界中的各种现象，如抛体运动、面积优化和利润函数等。

Example 1: Projectile motion — A ball is thrown upward with height h(t) = −5t² + 20t + 2 metres. Find the maximum height.

示例 1：抛体运动 —— 小球上抛高度 h(t) = −5t² + 20t + 2（单位：米），求最大高度。

The maximum occurs at t = −b / (2a) = −20 / (2 × (−5)) = 2 seconds.

当 t = −b / (2a) = −20 / (2 × (−5)) = 2 秒时达到最大值。

h(2) = −5(4) + 40 + 2 = 22 metres

Example 2: Rectangular fencing — A farmer has 100 m of fencing to enclose a rectangular area. The maximum area is found by setting width x, length (50 − x), then maximising A = x(50 − x) = 50x − x². The vertex occurs at x = 25 m, giving a maximum area of 625 m².

示例 2：矩形围栏 —— 农夫用 100 m 的篱笆围出一块矩形区域。设宽为 x，长为 (50 − x)，面积 A = x(50 − x) = 50x − x² 在 x = 25 m 时取最大值，最大面积为 625 平方米。

10. Common Exam Tips and Pitfalls | 常见考点提示与易错点

Based on years of IGCSE exam analysis, the following points are frequently tested:

根据多年 IGCSE 考题分析，以下几点为高频考点：

Reading roots from the graph — always state the x-coordinates where the curve crosses the x-axis.
Checking the sign of a — it determines whether the vertex is a maximum or minimum.
Substitution errors — when using the quadratic formula, be careful with negative b values.
Not simplifying surds — leave answers as √ values when required, e.g., (2 ± √10)/2.
Interpreting word problems — convert the context into an equation before solving.

从图像中读出根 —— 注意标明曲线与 x 轴交点的横坐标。
检查 a 的符号 —— 它决定了顶点是最大值还是最小值。
代入错误 —— 使用求根公式时，注意负 b 值的正确处理。
未化简根式 —— 按要求保留根号形式，例如 (2 ± √10)/2。
文字题理解 —— 先将实际问题转化为方程再求解。

Mastering quadratic functions and equations is a journey: start with the graph, then conquer factorisation, completing the square and the quadratic formula, and finally apply these tools to inequalities and word problems. With consistent practice, this topic will become one of your strongest areas in the IGCSE Mathematics exam.

掌握二次函数与方程的过程是一场循序渐进的旅程：从图像入手，逐步攻克因式分解、配方法和求根公式，最终将这些工具灵活运用于不等式和实际应用题。通过持续练习，这个主题将成为你在 IGCSE 数学考试中最拿分的板块之一。

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Quadratic Functions & Equations | 二次函数与方程