📚 Exercise 4D.1: Quadratic Functions – Vertex, Roots, and Discriminant | 练习 4D.1：二次函数 – 顶点、根与判别式
Exercise 4D.1 in the IB Mathematics course focuses on key skills for working with quadratic functions: converting between standard and vertex form by completing the square, using the discriminant to analyse the nature of roots, finding intercepts, and sketching graphs. Mastering these techniques is essential for success in both Analysis and Approaches and Applications and Interpretation courses.
IB 数学课程中的练习 4D.1 重点训练二次函数的核心技能：通过配方法在标准式与顶点式之间转换，运用判别式分析根的性质，求交点并绘制图像。熟练掌握这些技巧对于分析与方法和应用与解释两门课程的成功都至关重要。
1. Standard and Vertex Forms | 标准式与顶点式
etry x = h. Converting between these forms is a fundamental skill tested in Exercise 4D.1.
二次函数可以写成标准式 f(x) = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。顶点式 f(x) = a(x − h)² + k 则直接呈现顶点 (h, k) 和对称轴 x = h。在练习 4D.1 中，两种形式之间的转换是一项基本技能。
2. Completing the Square Step by Step | 逐步配方法
f(x) = a(x + b/(2a))² + (c − b²/(4a)). The vertex is therefore at (−b/(2a), c − b²/(4a)).
要将 f(x) = ax² + bx + c 转换为顶点式，先从首两项中提取 a：f(x) = a[x² + (b/a)x] + c。然后在括号内加上并减去 (b/(2a))²：f(x) = a[(x + b/(2a))² − (b/(2a))²] + c。化简得 f(x) = a(x + b/(2a))² + (c − b²/(4a))，顶点坐标为 (−b/(2a), c − b²/(4a))。
4: f(x) = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 8 + 5 = 2(x + 2)² − 3. The vertex is (−2, −3).
例如，对于 f(x) = 2x² + 8x + 5，提取 2：f(x) = 2(x² + 4x) + 5。4 的一半是 2，2² = 4。加减 4 得到 f(x) = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 8 + 5 = 2(x + 2)² − 3。顶点为 (−2, −3)。
3. Identifying the Vertex and Axis of Symmetry | 确定顶点与对称轴
s are (−b/(2a), f(−b/(2a))). Exercise 4D.1 often asks you to state these without fully graphing the function.
从顶点式 f(x) = a(x − h)² + k 可知，顶点为 (h, k)，对称轴为直线 x = h。在标准式中，对称轴为 x = −b/(2a)，顶点坐标为 (−b/(2a), f(−b/(2a)))。练习 4D.1 经常要求在不完整绘图的情况下说出这些信息。
4. Direction of Opening and the Sign of a | 开口方向与 a 的符号
The coefficient a determines the parabola's opening direction: if a > 0, the parabola opens upward and the vertex is a minimum point; if a < 0, it opens downward and the vertex is a maximum. The magnitude |a| also affects the graph's width — a larger |a| produces a narrower parabola, while a smaller |a| makes it wider.
系数 a 决定了抛物线的开口方向：若 a > 0，开口向上，顶点为最小值点；若 a < 0，开口向下，顶点为最大值点。|a| 的大小也影响图形宽度 —— |a| 越大，抛物线越窄；|a| 越小，抛物线越宽。
5. The Discriminant and Root Analysis | 判别式与根的分析
tells us how many real roots the quadratic has without solving the equation. The following table summarises the three possible cases that frequently appear in Exercise 4D.1.
对于二次方程 ax² + bx + c = 0，判别式为 Δ = b
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