📚 Exercise 4E: Mastering Quadratic Functions and the Discriminant | 练习4E：掌握二次函数与判别式
Welcome to Exercise 4E, your focused guide to quadratic functions and the discriminant — a cornerstone topic in IB Mathematics. Understanding how the discriminant governs the nature of roots will transform your ability to analyze equations, sketch graphs, and tackle exam problems with confidence. This article walks you through every key concept, provides worked examples, and highlights common pitfalls, giving you a complete revision resource.
欢迎来到练习4E，这是你深入学习二次函数与判别式的专题指南——IB数学中的基石主题。理解判别式如何决定根的性质，将大大提升你分析方程、绘制图像和应对考试的能力。本文带你梳理每一个关键概念，提供典型例题，并指出常见误区，为你提供一份完整的复习资料。
1. What is a Quadratic Function? | 什么是二次函数？
c, where a, b, and c are real constants and a ≠ 0. The graph of a quadratic function is a parabola. It opens upward if a > 0 and downward if a < 0. The y-intercept is always (0, c).
二次函数是次数为2的多项式函数，通常写成标准形式 f(x) = ax² + bx + c，其中 a、b、c 为实常数且 a ≠ 0。二次函数的图像是一条抛物线。当 a > 0 时开口向上，a < 0 时开口向下。y轴截距总是 (0, c)。
2. The Discriminant (Δ) and Why It Matters | 判别式Δ及其重要性
lly. The value of Δ determines whether the roots are real or non-real, and whether they are distinct or repeated.
判别式用希腊大写字母 Δ 表示，是二次求根公式中平方根号下的表达式：Δ = b² − 4ac。它能让我们在不必完全求解方程 ax² + bx + c = 0 的情况下，就知道根的信息。Δ 的值决定了根是实数还是非实数，是相异还是重根。
x = (−b ± √Δ) / (2a)
The quadratic formula itself is x = (−b ± √b² − 4ac) / (2a), which you can rewrite using the discriminant for efficiency.
二次求根公式本身为 x = (−b ± √b² − 4ac) / (2a)，你可以使用判别式将它改写，以提升计算效率。
3. Determining the Nature of Roots Using Δ | 利用Δ判断根的性质
By examining the sign of the discriminant, you can classify the roots without calculating them. The following table summarizes the relationship between Δ, the nature of roots, and the behaviour of the graph.
通过考察判别式的符号，你无需计算根就能对其进行分类。下表总结了 Δ、根的性质以及图像行为之间的关系。
| Δ = b² − 4ac | Nature of Roots | 根的性质 | Graph Behavior | 图像行为 |
|---|---|---|---|---|
| Δ > 0 | Two distinct real roots | 两个不同的实根 | Parabola cuts x-axis at two points | 抛物线与x轴交于两点 |
| Δ = 0 | One real repeated root (double root) | 一个实重根（二重根） | Parabola touches x-axis at the vertex | 抛物线与x轴相切于顶点 |
| Δ < 0 | No real roots (two complex conjugate roots) | 无实根（一对共轭复根） | Parabola does not intersect x-axis | 抛物线不与x轴相交 |
For the IB, when Δ < 0 you can simply say "no real roots". At HL level, you may also be asked to state that the roots are complex conjugates. Always give the reasoning: "since Δ < 0, the equation has no real solutions".
在IB中，当 Δ < 0 时，你可以简单地说“无实根”。在HL层次，你可能还需要说明根是一对共轭复数。始终给出推理：“因为 Δ < 0，所以方程无实数解”。
4. The Quadratic Formula and Finding Exact Roots | 求根公式与精确求根
6. Then Δ = (−4)² − 4(2)(−6) = 16 + 48 = 64. √Δ = 8, so x = (4 ± 8) / 4 → x = 3 or x = −1.
当判别式非负时，使用 x = (−b ± √Δ) / (2a) 求精确根。除非要求小数近似值，否则应将答案保留为最简根式形式。例如：解 2x² − 4x − 6 = 0。确定 a = 2, b = −4, c = −6。则 Δ = (−4)² − 4(2)(−6) = 16 + 48 = 64。√Δ = 8，因此 x = (4 ± 8)/4，得 x = 3 或 x = −1。
5. Vertex Form: Completing the Square | 顶点式：配方法
(x + 3)² − 4. So the vertex is (−3, −4). Pay attention to the sign: (x + 3)² means h = −3.
要将 f(x)
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