Mastering Quadratic Functions | 精通二次函数

📚 Mastering Quadratic Functions | 精通二次函数

Quadratic functions are one of the most fundamental topics in IGCSE Mathematics. Understanding their graphs, roots, and transformations is essential for success in both Paper 1 and Paper 2. This guide breaks down everything you need to know, from the standard form to problem-solving strategies.

二次函数是 IGCSE 数学中最基础且最重要的主题之一。理解其图像、根与变换，是 Paper 1 和 Paper 2 取得高分的关键。本指南将带你全面掌握从标准式到解题策略的所有核心知识。

1. The Standard Form | 标准形式

A quadratic function is any function that can be written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola — a symmetrical U-shaped curve.

二次函数是指可以写成 f(x) = ax² + bx + c 形式的函数，其中 a、b、c 为常数，且 a ≠ 0。二次函数的图像是抛物线——一条对称的 U 形曲线。

The sign of the coefficient a determines the direction of the parabola:

系数 a 的正负决定了抛物线的开口方向：

a > 0: The parabola opens upwards (minimum point). | 开口向上（有最小值）。
a < 0: The parabola opens downwards (maximum point). | 开口向下（有最大值）。

f(x) = ax² + bx + c, a ≠ 0

For example, f(x) = 2x² − 3x + 5 has a = 2, so it opens upwards. In contrast, g(x) = −x² + 4x − 1 has a = −1, so it opens downwards.

例如，f(x) = 2x² − 3x + 5 中 a = 2，因此开口向上；而 g(x) = −x² + 4x − 1 中 a = −1，因此开口向下。

2. Solving Quadratic Equations | 解二次方程

A quadratic equation is formed when we set the function equal to zero: ax² + bx + c = 0. There are three main methods for solving such equations at IGCSE level.

将二次函数设为零即可得到二次方程：ax² + bx + c = 0。在 IGCSE 阶段，主要有三种解法。

Method 1: Factorisation | 方法一：因式分解法

If the quadratic expression can be factorised, we use the zero product property. For example, to solve x² − 5x + 6 = 0, we factorise as (x − 2)(x − 3) = 0. Therefore, x = 2 or x = 3.

如果二次表达式可以因式分解，我们利用零乘积性质求解。例如，解 x² − 5x + 6 = 0 时，分解为 (x − 2)(x − 3) = 0，因此 x = 2 或 x = 3。

Method 2: Quadratic Formula | 方法二：求根公式法

When factorisation is not possible, we use the quadratic formula:

当无法因式分解时，我们使用求根公式：

x = (−b ± √(b² − 4ac)) ÷ (2a)

For example, solve 2x² + 3x − 2 = 0. Here, a = 2, b = 3, c = −2. Substituting into the formula gives x = (−3 ± √(9 + 16)) ÷ 4 = (−3 ± 5) ÷ 4, so x = 0.5 or x = −2.

例如，解 2x² + 3x − 2 = 0。此处 a = 2，b = 3，c = −2。代入公式得 x = (−3 ± √(9 + 16)) ÷ 4 = (−3 ± 5) ÷ 4，因此 x = 0.5 或 x = −2。

Method 3: Completing the Square | 方法三：配方法

This method rewrites the equation as a perfect square plus a constant. For x² + 6x + 2 = 0, we write (x + 3)² − 9 + 2 = 0, giving (x + 3)² = 7, so x = −3 ± √7.

配方法将方程改写为完全平方加常数项的形式。对于 x² + 6x + 2 = 0，我们写成 (x + 3)² − 9 + 2 = 0，即 (x + 3)² = 7，因此 x = −3 ± √7。

3. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant, denoted by Δ. It tells us the nature of the roots without solving the equation fully.

表达式 b² − 4ac 称为判别式，记为 Δ。它可以在不完全解方程的情况下判断根的性质。

Δ > 0: Two distinct real roots. | 两个不相等的实数根。
Δ = 0: One repeated real root. | 两个相等的实数根（重根）。
Δ < 0: No real roots (two complex roots). | 无实数根（两个复数根）。

For example, the equation x² + 4x + 5 = 0 has Δ = 16 − 20 = −4 < 0, so it has no real roots. On the other hand, x² − 2x + 1 = 0 has Δ = 4 − 4 = 0, so it has exactly one distinct real root, x = 1.

例如，方程 x² + 4x + 5 = 0 的判别式 Δ = 16 − 20 = −4 < 0，因此无实数根。而 x² − 2x + 1 = 0 的判别式 Δ = 4 − 4 = 0，因此只有一个实数根 x = 1。

4. The Vertex and Axis of Symmetry | 顶点与对称轴

Every parabola has a vertical line of symmetry that passes through its vertex. The x-coordinate of the vertex is given by:

每条抛物线都有一条通过顶点的垂直对称轴。顶点的 x 坐标由下式给出：

x = −b ÷ (2a)

To find the y-coordinate of the vertex, substitute this x-value back into the original equation. For example, for f(x) = x² − 4x + 3, the vertex x-coordinate is x = 4 ÷ 2 = 2, and the y-coordinate is f(2) = 4 − 8 + 3 = −1. Hence, the vertex is (2, −1).

将顶点 x 坐标代回原方程即可求得 y 坐标。例如，对于 f(x) = x² − 4x + 3，顶点 x 坐标为 x = 4 ÷ 2 = 2，y 坐标为 f(2) = 4 − 8 + 3 = −1，因此顶点为 (2, −1)。

If a > 0, this point is the minimum; if a < 0, it is the maximum. The axis of symmetry is the vertical line x = −b ÷ (2a).

若 a > 0，该点为最小值点；若 a < 0，该点为最大值点。对称轴是垂直线 x = −b ÷ (2a)。

5. Roots and the Graph | 根与图像的关系

The roots of a quadratic equation correspond to the x-intercepts of its graph — points where the parabola crosses the x-axis.

二次方程的根对应其图像与 x 轴的交点（即 x 截距）。

There are three possible scenarios for the graph of y = ax² + bx + c:

y = ax² + bx + c 的图像有三种可能情形：

The parabola crosses the x-axis at two distinct points (Δ > 0). | 抛物线与 x 轴相交于两个不同的点（Δ > 0）。
The parabola touches the x-axis at exactly one point (Δ = 0). | 抛物线与 x 轴相切于一个点（Δ = 0）。
The parabola does not touch the x-axis at all (Δ < 0). | 抛物线完全不接触 x 轴（Δ < 0）。

For instance, the graph of y = x² − 2x − 3 crosses the x-axis at x = −1 and x = 3, which we can find either by factorising (x − 3)(x + 1) = 0 or by observing the graph.

例如，y = x² − 2x − 3 的图像在 x = −1 和 x = 3 处穿过 x 轴，这可以通过因式分解 (x − 3)(x + 1) = 0 求出，也可以直接从图像观察得到。

6. Completed Square Form | 顶点式

Every quadratic function can be rewritten in the completed square form:

每个二次函数都可以改写为顶点式：

f(x) = a(x − h)² + k

In this form, the vertex is (h, k), and the axis of symmetry is x = h. This form is extremely useful because the vertex and the minimum/maximum value are immediately visible.

在这种形式中，顶点为 (h, k)，对称轴为 x = h。这种形式非常有用，因为顶点和最小值/最大值一目了然。

For example, f(x) = 2(x − 3)² + 5 has vertex (3, 5) and since a = 2 > 0, the minimum value of the function is 5.

例如，f(x) = 2(x − 3)² + 5 的顶点为 (3, 5)，因为 a = 2 > 0，所以函数的最小值为 5。

To convert from standard form to completed square form, we complete the square. For f(x) = x² + 8x + 15, we write (x + 4)² − 16 + 15 = (x + 4)² − 1, so the vertex is (−4, −1).

要将标准式转换为顶点式，需要使用配方法。例如 f(x) = x² + 8x + 15，可写成 (x + 4)² − 16 + 15 = (x + 4)² − 1，因此顶点为 (−4, −1)。

7. Transformations of Graphs | 图像的变换

Understanding how a quadratic graph transforms under changes to its equation is a common IGCSE examination topic. The completed square form f(x) = a(x − h)² + k reveals all transformations:

理解二次函数图像在方程变化下如何变换是 IGCSE 考试的常见考点。顶点式 f(x) = a(x − h)² + k 揭示了所有变换：

Translating vertically: Changing k shifts the graph up (k > 0) or down (k < 0). | 垂直平移：改变 k 会使图像向上（k > 0）或向下（k < 0）移动。
Translating horizontally: Changing h shifts the graph right (h > 0) or left (h < 0). | 水平平移：改变 h 会使图像向右（h > 0）或向左（h < 0）移动。
Reflecting: Changing the sign of a reflects the graph across the x-axis. | 反射：改变 a 的符号会使图像关于 x 轴对称翻转。
Stretching: Increasing |a| makes the parabola narrower; decreasing |a| makes it wider. | 伸缩：|a| 增大使抛物线变窄；|a| 减小使抛物线变宽。

For example, the graph of y = (x − 2)² + 3 is the graph of y = x² shifted 2 units to the right and 3 units upward.

例如，y = (x − 2)² + 3 的图像是 y = x² 的图像向右平移 2 个单位、再向上平移 3 个单位得到的结果。

8. Quadratic Inequalities | 二次不等式

Solving quadratic inequalities is an extension of solving quadratic equations. The method involves finding the roots and then determining the sign of the quadratic expression in each interval.

解二次不等式是解二次方程的延伸。方法是先求根，再判断每个区间内二次表达式的正负。

Example | 示例: Solve x² − 3x − 4 < 0.

Step 1: Factorise: (x − 4)(x + 1) = 0, so x = 4 or x = −1.

步骤 1：因式分解：(x − 4)(x + 1) = 0，因此 x = 4 或 x = −1。

Step 2: Since a > 0, the parabola opens upward. The expression is negative between the roots. Therefore, x² − 3x − 4 < 0 when −1 < x < 4.

步骤 2：因为 a > 0，抛物线开口向上。表达式在两个根之间为负。因此，当 −1 < x < 4 时，x² − 3x − 4 < 0。

Note that if the inequality were ≤, we would write −1 ≤ x ≤ 4, including the endpoints.

注意，如果不等式为 ≤，则答案写为 −1 ≤ x ≤ 4，包含端点。

A useful visual aid is a number line with open (unfilled) circles for strict inequalities and closed (filled) circles for inclusive inequalities.

一个有用的工具是数轴：严格不等式使用空心圆圈，含等号的不等式使用实心圆圈。

9. Word Problems | 应用题

Quadratic functions frequently appear in real-world problems. The most common types include projectile motion, area optimisation, and revenue maximisation.

二次函数频繁出现在现实应用题中。最常见的类型包括抛体运动、面积优化和收益最大化。

Example | 示例: A rectangle has a perimeter of 40 m. Find the maximum possible area.

示例：一个矩形的周长为 40 米。求最大可能面积。

Let the width be x. Then the length is (40 − 2x) ÷ 2 = 20 − x. The area A is given by:

设宽为 x，则长为 (40 − 2x) ÷ 2 = 20 − x。面积 A 为：

A = x(20 − x) = 20x − x²

This is a quadratic function with a = −1 < 0, so it opens downwards and has a maximum. The maximum occurs at x = −b ÷ (2a) = −20 ÷ (−2) = 10. Therefore, the dimensions are 10 m × 10 m and the maximum area is 100 m².

这是一个二次函数，a = −1 < 0，开口向下，因此有最大值。最大值出现在 x = −b ÷ (2a) = −20 ÷ (−2) = 10 处。因此，矩形为 10 米 × 10 米，最大面积为 100 平方米。

When solving word problems, always define your variables clearly, translate the conditions into a quadratic equation or function, and then interpret the mathematical solution in the context of the original problem.

解应用题时，务必明确定义变量，将条件转化为二次方程或函数，然后结合题目情境解释数学解。

10. Sketching Quadratic Graphs | 绘制二次函数图像

A common IGCSE question asks students to sketch the graph of a quadratic function. A complete sketch must show all key features:

IGCSE 常见考题要求学生画出二次函数的示意图。完整的示意图必须标出所有关键特征：

The y-intercept (where x = 0). | y 截距（x = 0 处）。
The x-intercepts/roots (where y = 0). | x 截距/根（y = 0 处）。
The vertex (turning point). | 顶点（转折点）。
The axis of symmetry. | 对称轴。

Example | 示例: Sketch y = x² − 2x − 3.

Step 1: y-intercept: c = −3, so the graph crosses the y-axis at (0, −3).

步骤 1：y 截距：c = −3，因此图像与 y 轴交于 (0, −3)。

Step 2: Roots: x² − 2x − 3 = (x − 3)(x + 1) = 0, so x = 3 or x = −1.

步骤 2：根：x² − 2x − 3 = (x − 3)(x + 1) = 0，因此 x = 3 或 x = −1。

Step 3: Vertex: x = −b ÷ (2a) = 2 ÷ 2 = 1, and y = 1 − 2 − 3 = −4. Vertex is (1, −4).

步骤 3：顶点：x = −b ÷ (2a) = 2 ÷ 2 = 1，y = 1 − 2 − 3 = −4，顶点为 (1, −4)。

Since a = 1 > 0, the parabola opens upward. Draw the axis of symmetry x = 1 as a dashed vertical line. Finally, connect the points with a smooth U-shaped curve.

因为 a = 1 > 0，抛物线开口向上。用垂直虚线标出对称轴 x = 1，最后用光滑的 U 形曲线连接各点。

11. Common Exam Pitfalls | 常见考试误区

Avoiding common errors is just as important as knowing the correct methods. Here are typical mistakes students make:

避免常见错误与掌握正确方法同样重要。以下是学生常犯的典型错误：

Forgetting a ≠ 0: A function with a = 0 is linear, not quadratic. | 忘记 a ≠ 0：a = 0 时函数为一次函数，而非二次函数。
Incorrect factorisation signs: Always double-check the signs in your factors. | 因式分解符号错误：务必仔细检查因子中的符号。
Misapplying the quadratic formula: Remember the entire formula, and be especially careful with the sign of b when b is negative. | 求根公式使用错误：记住完整公式，特别注意 b 为负数时的符号。
Confusing the roots with the vertex: The roots are where the graph crosses the x-axis; the vertex is the turning point. They are different objects. | 混淆根与顶点：根是图像与 x 轴的交点，顶点是转折点，两者完全不同。
Forgetting to state solutions as x = …: Always present your final answer clearly. | 忘记以 x = … 的形式写出解：始终清晰呈现最终答案。

As you practice, keep a checklist of these common mistakes and verify each solution step by step.

练习时，保留这份常见错误清单，逐步检查每一步解答。

12. Practice Checklist | 练习清单

To master quadratic functions, work through the following checklist to ensure you have covered all essential skills:

要掌握二次函数，请对照以下清单逐一检查自己是否掌握了所有关键技能：

✓ Solve quadratic equations using all three methods. | ✓ 用三种方法解二次方程。
✓ Determine the nature of roots using the discriminant. | ✓ 用判别式判断根的性质。
✓ Find the vertex and axis of symmetry. | ✓ 求顶点和对称轴。
✓ Convert between standard form and completed square form. | ✓ 在标准式和顶点式之间进行转换。
✓ Sketch graphs showing all key features. | ✓ 绘制包含所有关键特征的图像。
✓ Solve quadratic inequalities. | ✓ 解二次不等式。
✓ Apply quadratic functions to word problems. | ✓ 将二次函数应用于应用题。
✓ Describe graph transformations. | ✓ 描述图像变换。

Once you can perform each skill confidently, try mixed-topic past paper questions that combine quadratics with other areas like coordinate geometry, sequences, or basic calculus (for Additional Mathematics students).

当你能够自信地完成每项技能后，尝试将二次函数与坐标几何、数列或基础微积分（针对 Additional Mathematics 学生）结合的综合题。

Remember: consistent daily practice is the key to transforming quadratic functions from a challenging topic into one of your strongest areas on the IGCSE exam.

请记住：每天坚持练习，是把二次函数这一难点转化为 IGCSE 考试中你最擅长领域的关键。

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Mastering Quadratic Functions | 精通二次函数