Mastering Quadratic Functions | 一元二次函数精讲

📚 Mastering Quadratic Functions | 一元二次函数精讲

Quadratic functions are central to the Edexcel IGCSE Mathematics syllabus. This guide covers the essential techniques you need to solve, graph, and apply quadratic equations with confidence.
一元二次函数是 Edexcel IGCSE 数学考纲的核心内容。本指南将涵盖您需要掌握的关键技巧，帮助您自信地求解、作图并应用二次方程。

1. The General Form | 一般形式

A quadratic function can be written in the standard form f(x) = ax² + bx + c, where a, b, c are constants and a ≠ 0. The coefficient a controls the shape and direction of the parabola.
二次函数可以写成标准形式 f(x) = ax² + bx + c，其中 a、b、c 为常数，且 a ≠ 0。系数 a 控制抛物线的形状和开口方向。

When a is positive, the parabola opens upwards (U-shaped); when a is negative, it opens downwards (∩-shaped). The value of c gives the y-intercept.
当 a 为正数时，抛物线开口向上（U 形）；当 a 为负数时，开口向下（∩ 形）。常数 c 给出 y 轴截距。

The highest power of x is 2, so the graph has one turning point.
x 的最高次数是 2，因此图像只有一个转向点。
The graph of a quadratic is always a smooth curve called a parabola.
二次函数的图像总是一条平滑曲线，称为抛物线。

2. Solving by Factorisation | 因式分解法

To solve a quadratic equation ax² + bx + c = 0, first try to factorise it into two brackets. For example, x² + 5x + 6 = 0 can be written as (x + 2)(x + 3) = 0.
要求解一元二次方程 ax² + bx + c = 0，首先尝试将其因式分解为两个括号之积。例如，x² + 5x + 6 = 0 可写成 (x + 2)(x + 3) = 0。

Using the zero product property, if the product is zero, then at least one factor must be zero. So x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.
根据零乘积性质，若乘积为零，则至少有一个因式为零。所以 x + 2 = 0 或 x + 3 = 0，解得 x = -2 或 x = -3。

Always rearrange the equation so the right-hand side is zero before factorising.
在因式分解前，务必把方程整理为右边等于 0 的形式。
Common factors, difference of two squares, and perfect squares are useful patterns.
公因式、平方差公式和完全平方公式是常用的模式。

3. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. For x² + bx + c, we add and subtract (b/2)².
配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。对于 x² + bx + c，我们加上并减去 (b/2)²。

Example: x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4. The vertex is at (-3, -4).
例如：x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4。此时顶点坐标为 (-3, -4)。

x² + bx + c = (x + b/2)² – (b/2)² + c

This form is especially useful for finding the minimum or maximum value and for solving equations directly.
这种形式特别适用于求最小值或最大值，以及直接求解方程。

4. The Quadratic Formula | 二次公式

For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula:
对于任意一元二次方程 ax² + bx + c = 0，其解由二次公式给出：

x = (-b ± √(b² – 4ac)) / 2a

The formula works for all quadratic equations, including those that cannot be easily factorised. You must memorise it for the exam.
该公式适用于所有一元二次方程，包括不易因式分解的情况。您在考试中必须熟记。

The ± symbol means there are two solutions: one using addition and one using subtraction.
± 符号表示有两个解：一个用加法，一个用减法。
If the expression under the square root is negative, there are no real roots.
如果根号内的表达式为负数，则没有实数根。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: Δ = b² – 4ac. It determines the nature of the roots.
判别式是二次公式中根号内的部分：Δ = b² – 4ac。它决定根的性质。

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	一个相等的实数根（重根）
Δ < 0	No real roots	没有实数根

If Δ is a perfect square and a, b, c are rational, the equation can be factorised. Otherwise, use the formula or completing the square.
如果 Δ 是完全平方数且 a、b、c 为有理数，则方程可以因式分解。否则，请使用公式或配方法。

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

Every parabola has a vertical line of symmetry. The axis of symmetry is given by x = -b / 2a. The vertex lies on this line.
每条抛物线都有一条垂直对称轴。对称轴方程为 x = -b / 2a。顶点在这条直线上。

The x-coordinate of the vertex is -b / 2a. To find the y-coordinate, substitute this value back into the original equation.
顶点的 x 坐标为 -b / 2a。要求 y 坐标，只需将该值代回原方程。

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

When the parabola opens upwards, the vertex gives the minimum; when it opens downwards, the vertex gives the maximum.
当抛物线开口向上时，顶点给出最小值；当开口向下时，顶点给出最大值。

7. Sketching Graphs | 画抛物线图像

To sketch a quadratic graph, you need key points: the y-intercept, the x-intercepts (if any), and the vertex. The y-intercept is c if the equation is in standard form.
画二次函数图像需要关键点：y 轴截距、x 轴截距（如果有）和顶点。若方程为标准形式，y 轴截距为 c。

Solve f(x) = 0 to find the x-intercepts.
令 f(x) = 0 解方程得到 x 轴截距。
Find the vertex using -b/2a, then plot the point.
用 -b/2a 求顶点，并描出该点。
Use symmetry to draw the other half of the curve.
利用对称性画出曲线的另一半。

Mark the axis of symmetry and label the points clearly. The curve must be smooth, not straight lines between points.
标出对称轴并清晰标出各点。曲线必须平滑，不能在点之间画直线。

8. Quadratic Inequalities | 二次不等式

To solve a quadratic inequality such as x² – 5x + 6 < 0, first solve the corresponding equation x² – 5x + 6 = 0.
求解二次不等式，例如 x² – 5x + 6 < 0，首先解对应的方程 x² – 5x + 6 = 0。

The solutions are x = 2 and x = 3. Sketch the parabola or test intervals on a number line. Since the inequality is < 0, we want the part below the x-axis.
方程的解为 x = 2 和 x = 3。描画出抛物线或在数轴上测试区间。因为不等式是 < 0，我们要求图像在 x 轴下方的部分。

x² – 5x + 6 < 0 ⇒ 2 < x < 3

Remember to reverse the inequality sign when multiplying or dividing by a negative number.
记住，当乘以或除以一个负数时，不等号方向要改变。

9. Real-Life Applications | 实际应用

Quadratic equations model many real-world situations, such as projectile motion, area problems, and profit maximisation.
二次方程可以模拟许多现实情境，例如抛体运动、面积问题和利润最大化问题。

Example: A rectangle has a length that is 3 meters more than its width, and its area is 40 m². Find the width.
例如：一个矩形的长比宽多 3 米，面积为 40 平方米。求宽。

Let the width be x meters. Then x(x + 3) = 40, which gives x² + 3x – 40 = 0. Factorising gives (x + 8)(x – 5) = 0, so x = 5 (since x cannot be negative).
设宽为 x 米。则 x(x + 3) = 40，即 x² + 3x – 40 = 0。因式分解得 (x + 8)(x – 5) = 0，所以 x = 5（因为 x 不能为负数）。

Read the problem carefully and define a variable.
仔细阅读题目并定义一个变量。
Translate the information into a quadratic equation.
将信息转化为二次方程。
Reject any solution that does not make sense in the context.
舍弃在情境中没有意义的解。

10. Exam Tips | 考试技巧

In the Edexcel IGCSE exam, show all steps clearly and use the method that is most efficient for the given question.
在 Edexcel IGCSE 考试中，清晰地展示所有步骤，并针对给定题目选择最有效的方法。

Always check if the equation is set to zero before factorising.
因式分解前，务必检查方程是否已整理为等于 0。
If a = 1 and b is even, completing the square can be faster than the quadratic formula.
如果 a = 1 且 b 为偶数，配方法可能比二次公式更快。
Use a calculator to check your solutions, but do not rely on it for the working.
使用计算器检查答案，但不要依赖它来写过程。
Be careful with signs: x = (-b ± √(b² – 4ac)) / 2a involves several negatives and squares.
注意符号：x = (-b ± √(b² – 4ac)) / 2a 中涉及多个负号与平方。
Practise past paper questions to become familiar with common phrasing.
练习历年真题，熟悉常见问法。

Quadratic functions appear in many sections of the IGCSE syllabus, so mastering them is essential for a high grade.
二次函数出现在 IGCSE 考纲的许多章节中，因此掌握它们对于获得高分至关重要。

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Mastering Quadratic Functions | 一元二次函数精讲