📚 Quadratic Functions and Equations | 二次函数与方程
Quadratic functions are a central topic in the IB Mathematics curriculum, bridging algebraic manipulation with graphical interpretation and real-world problem-solving. A solid grasp of quadratics is essential not only for exam success but also for understanding advanced concepts in calculus and physics. This article provides a comprehensive walkthrough of Exercise B.4, covering key techniques from factoring to the quadratic formula, alongside typical IB-style problem-solving strategies.
二次函数是IB数学课程的核心主题之一，它将代数运算与图像解读以及实际问题的解决紧密相连。扎实掌握二次函数不仅对考试成功至关重要，也是理解微积分和物理中高级概念的基础。本文全面解析练习B.4，涵盖从因式分解到求根公式的关键技巧，以及典型的IB风格解题策略。
1. Definition and Standard Form | 定义与标准形式
the linear term, and c is the constant term, which also gives the y-intercept when x = 0.
二次函数是形如 f(x) = ax² + bx + c 的函数，其中 a、b、c 为实数且 a ≠ 0。二次函数的图像是一条抛物线，当 a > 0 时开口向上，当 a < 0 时开口向下。ax² 项为二次项，bx 为一次项，c 为常数项，当 x = 0 时 c 也代表 y 轴截距。
2. Vertex Form and Completing the Square | 顶点形式与配方法
tor a from the first two terms, then add and subtract (b/(2a))² inside the parentheses. The vertex represents the maximum or minimum point of the parabola, depending on the sign of a.
顶点形式 f(x) = a(x - h)² + k 直接揭示了顶点 (h, k)。要从标准形式转换，我们使用配方法：取 f(x) = ax² + bx + c，从前两项中提取 a，然后在括号内加上并减去 (b/(2a))²。顶点代表抛物线的最大值或最小值点，具体取决于 a 的符号。
3. Axis of Symmetry | 对称轴
e divides the parabola into two mirror-image halves. In IB problems, you may be asked to find the axis of symmetry to solve optimisation questions or to sketch the graph accurately.
对称轴是穿过顶点的垂直线，由标准形式给出 x = -b/(2a)，或由顶点形式给出 x = h。这条线将抛物线分成两个镜像对称的部分。在IB问题中，你可能需要找出对称轴以解决优化问题或准确绘制函数图像。
4. Factoring Quadratics | 二次式的因式分解
o c and add to b. Always check for a common factor first, as IB exam questions often include coefficients that can be simplified.
因式分解将 ax² + bx + c 表示为两个二项式 (px + q)(rx + s) 的乘积，其中 p×r = a，q×s = c，且交叉项之和等于 b。对于首项系数为1的二次式 (a = 1)，找出两个数，其乘积为 c 且和为 b。务必首先检查是否有公因子，因为IB考试题常常包含可以化简的系数。
5. The Quadratic Formula | 求根公式
The quadratic formula x = [-b ± √b² - 4ac leting the square and provides the x-intercepts (roots) of the parabola. In IB exams, you must memorise this formula and apply it accurately, paying close attention to signs and the order of operations under the radical.
求根公式 x = [-b ± √b² - 4ac] / (2a) 可解任何二次方程 ax² + bx + c = 0。它由配方法推导而来，并给出抛物线的 x 轴截距（根）。在IB考试中，你必须熟记此公式并准确应用，特别注意符号和根号下的运算顺序。
6. The Discriminant and Nature of Roots | 判别式与根的性质
, one repeated real root (the vertex touches the x-axis); if Δ < 0, no real roots (the parabola does not cross the x-axis). IB questions frequently ask you to find the range of a parameter for a certain number of roots.
判别式 Δ = b² - 4ac 无需解方程即可确定根的性质和数量。若 Δ > 0，有两个不同的实根；若 Δ = 0，有一个重根（顶点与 x 轴相切）；若 Δ < 0，无实根（抛物线不与 x 轴相交）。IB题目经常要求你找出使方程具有一定数量根的参数范围。
7. Sum and Product of Roots | 根的和与积
and β, the sum α + β = -b/a, and the product αβ = c/a. These relationships, known as Vieta' without calculating the roots individually. They appear regularly in IB AA and AI papers.
对于二次方程 ax² + bx + c = 0，其根为 α 和 β，则和 α + β = -b/a，积 αβ = c/a。这些关系被称为韦达定理，对于根据给定根构造方程或在不逐个计算根的情况下化简对称表达式极为有用。它们经常出现在IB AA和AI试卷中。
8. Quadratic Inequalities | 二次不等式
Solving quadratic inequalities like ax² + bx + c > 0 involves finding the roots and determining intervals where the parabola lies above or below the x-axis. Sketch a quick sign diagram or use the shape of the parabola based on the sign of a. In IB, always present your final answer using interval notation or set notation, and pay attention to whether the inequality is strict or inclusive.
解二次不等式如 ax² + bx + c > 0 需要求出根并确定抛物线在 x 轴上方或下方所处的区间。根据 a 的符号快速绘制符号图或利用抛物线的形状进行判断。在IB考试中，始终使用区间符号或集合符号呈现最终答案，并注意不等号是严格的还是包含等号的。
9. Quadratic Modelling and Optimisation | 二次函数建模与优化
Many IB problems ask you to model a real-life scenario with a quadratic function: projectile motion, area maximisation, revenue and profit functions. Identify the variable, write the quadratic expression, then find the vertex to determine the maximum or minimum value. The domain often restricts the meaningful part of the parabola, so always interpret your answer in context.
许多IB问题要求你用二次函数对现实场景进行建模：抛体运动、面积最大化、收益与利润函数。确定变量，写出二次表达式，然后求出顶点以确定最大值或最小值。定义域通常会限制抛物线的有效部分，因此始终要在具体情境中解读你的答案。
10. Graphs and Transformations | 图像与变换
rmations: horizontal translation by h, vertical translation by k, and vertical stretch by factor |a| f transformations is crucial in IB, especially when deducing the equation from a described sequence of shifts and stretches.
函数 f(x) = a(x - h)² + k 的图像可通过变换从 y = x² 得到：水平平移 h，垂直平移 k，以 |a| 为因子进行垂直拉伸，若 a 为负则关于 x 轴反射。在IB中理解变换的顺序至关重要，尤其是在根据所描述的一系列平移和拉伸推导方程时。
11. Common IB Exam Pitfalls | IB考试常见易错点
Students often forget to divide the entire equation by the coefficient of x² before completing the square when a ≠ 1. Another frequent error is misapplying the quadratic formula, particularly mishandling negative signs or forgetting the '2a' in the denominator. Always double-check your factoring by expanding your binomials, and remember that the discriminant only applies to the standard form, not to factored or vertex forms.
学生常常在 a ≠ 1 时忘记在配方法之前将整个方程除以 x² 的系数。另一个常见错误是错误应用求根公式，尤其是错误处理负号或忘记分母中的 '2a'。务必通过展开二项式来复核因式分解，并记住判别式仅适用于标准形式，而非因式分解形式或顶点形式。
12. Practice Problem from Exercise B.4 | 来自练习B.4的典型题目
axis x = 2. Discriminant: Δ = (-8)² - 4(2)(5) = 64 - 40 = 24, so two real roots: x = (8 ± √24)/4 = 2 ± √6/2. For f(x) ≥ -1: 2(x - 2)² - 3 ≥ -1 → 2(x - 2)² ≥ 2 → (x - 2)² ≥ 1 → x ≤ 1 or x ≥ 3.
考虑函数 f(x) = 2x² - 8x + 5。求其顶点、对称轴、判别式和 x 轴截距。然后解不等式 f(x) ≥ -1。首先配方：f(x) = 2(x² - 4x) + 5 = 2[(x - 2)² - 4] + 5 = 2(x - 2)² - 3。顶点为 (2, -3)，对称轴为 x = 2。判别式：Δ = (-8)² - 4(2)(5) = 64 - 40 = 24，因此有两个实根：x = (8 ± √24)/4 = 2 ± √6/2。对于 f(x) ≥ -1：2(x - 2)² - 3 ≥ -1 → 2(x - 2)² ≥ 2 → (x - 2)² ≥ 1 → x ≤ 1 或 x ≥ 3。
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