📚 Exercise 1F: Quadratic Functions and Equations | 练习1F：二次函数与方程
Quadratic functions are the cornerstone of algebraic manipulation in IB Mathematics, bridging coordinate geometry, equation solving and real-world modelling. Exercise 1F commonly consolidates skills such as completing the square, interpreting the discriminant, sketching parabolas, and applying quadratic forms to optimisation problems. This article unpacks all essential techniques and theory that students encounter in this typical exercise set, ensuring you can navigate quadratic questions with confidence in both Analysis & Approaches and Applications & Interpretation.
二次函数是 IB 数学代数运算的基石，将坐标几何、方程求解与现实建模连接在一起。练习 1F 通常旨在强化配方法、判别式解读、抛物线草图绘制以及二次型在最优化问题中的应用等技能。本文梳理了这类典型练习中涉及的所有核心技巧与理论，帮助你在分析与方法和应用与解释课程中都能自信地应对二次函数题目。
1. The Standard Form and Graph Shape | 标准形式与图形形状
are real constants and a ≠ 0. The graph of f is a parabola. If a > 0, the parabola opens upward and has a minimum point; if a < 0, it opens downward and has a maximum point. The coefficient |a| controls the 'steepness' of the curve — larger |a| gives a narrower parabola.
二次函数通常写作 f(x) = ax² + bx + c，其中 a、b 和 c 为实数常数且 a ≠ 0。其图像是一条抛物线。若 a > 0，抛物线开口向上并具有最低点；若 a < 0，开口向下并具有最高点。|a| 的大小控制曲线的 '陡峭' 程度 — |a| 越大，抛物线越窄。
f(x) = ax² + bx + c, a ≠ 0
arabola may cross the x-axis twice, touch it once, or not cross it at all, depending on the discriminant.
y 轴截距为 (0, c)，因为 f(0) = c。x 轴截距（根）通过求解 ax² + bx + c = 0 得到。抛物线可能与 x 轴相交两次、相切一次，或者完全不相交，这取决于判别式。
2. Vertex Form and Completing the Square | 顶点式与配方法
x − h)² + k, where (h, k) is the vertex. Completing the square is the algebraic technique that achieves this conversion:
将一般式化为顶点式可以直接得到转折点。顶点式为 f(x) = a(x − h)² + k，其中 (h, k) 是顶点。配方法正是实现这一转换的代数技巧：
ax² + bx + c = a(x² + (b/a)x) + c = a[(x + b/(2a))² − (b/(2a))²] + c
crucial for optimisation problems and for sketching the graph accurately. Exercise 1F often asks you to write a quadratic in vertex form and then state the coordinates of the maximum or minimum.
化简后可得 h = −b/(2a)，k = c − b²/(4a)。了解顶点坐标对于最优化问题和精确绘制图像至关重要。练习 1F 经常要求将二次函数写成顶点式，然后写出最大值或最小值的坐标。
3. Factored Form and Roots | 因式分解式与根
(x − p)(x − q). Here p and q are the x-intercepts (roots) of the parabola. The factored form instantly shows the zeros and helps in sketching sign diagrams. If the quadratic does not factorise neatly, we resort to the quadratic formula or completing the square.
当二次函数可以在整数范围内因式分解时，我们将其写成 f(x) = a(x − p)(x − q)。此时 p 和 q 为抛物线的 x 轴截距（根）。因式分解式能立刻显示零点，并有助于绘制符号图。如果二次式无法整齐地分解，则需要借助求根公式或配方法。
ams, you may be asked to factorise and then solve or determine the sign of the expression.
请记住，并非所有二次函数都有实数根。只有当判别式 Δ = b² − 4ac ≥ 0 时，才存在实数范围内的因式分解式。IB 考试可能会要求你先进行因式分解，然后求解或判断表达式的符号。
4. The Discriminant and Nature of Roots | 判别式与根的性质
es how many real roots a quadratic equation has. This concept appears in nearly every IB Exercise 1F and is a favourite exam topic.
判别式 Δ = b² − 4ac 决定了一个二次方程有多少个实数根。这一概念几乎出现在每份 IB 练习 1F 中，也是考试的热门考点。
| Condition | Nature of Roots |
|---|---|
| Δ > 0 | Two distinct real roots (graph crosses x-axis twice) |
| Δ = 0 | One repeated real root (graph touches x-axis) |
| Δ < 0 | No real roots (graph does not intersect x-axis) |
(set Δ = 0), or show that a quadratic is always positive by proving Δ < 0 and a > 0.
在 IB 问题中，你可能需要求未知系数，使得方程恰好有一个实数根（令 Δ = 0），或者通过证明 Δ < 0 且 a > 0 来说明一个二次式恒正。
5. Solving Quadratic Equations | 解二次方程
orising, completing the square, and the quadratic formula. The quadratic formula x = [−b ± √b² − 4ac] / (2a) works in every case and is derived from completing the square.
求解 ax² + bx + c = 0 有三种主要的代数方法：因式分解法、配方法和求根公式法。求根公式 x = [−b ± √b² − 4ac] / (2a) 适用于所有情形，并由配方法推导而来。
x = (−b ± √Δ) / (2a)
In Exercise 1F you may also encounter equations that are 'hidden' quadratics, such as exponential equations of the form 2²ˣ − 5·2ˣ + 6 = 0, which can be reduced by substitution t = 2ˣ. Recognising these disguised quadratics is a valuable skill.
在练习 1F 中，你还可能遇到 '隐藏' 二次方程，例如形如 2²ˣ − 5·2ˣ + 6 = 0 的指数方程，可通过令 t = 2ˣ 来转化为二次方程。识别这类伪装二次式是一项重要的能力。
6. Quadratic Inequalities | 二次不等式
Solving a quadratic inequality like ax² + bx + c > 0 requires a sign diagram. First find the real roots (if any), then sketch the parabola or use a number line to determine where the expression is positive or negative.
求解如 ax² + bx + c > 0 的二次不等式需要借助符号图。首先找出实数根（若有），然后绘制抛物线草图或使用数轴来判断表达式在哪些区间为正或为负。
- For a > 0: the expression is positive outside the roots and negative between them.
- For a < 0: the expression is negative outside the roots and positive between them.
- 当 a > 0 时：表达式在两根外侧为正，在两根之间为负。
- 当 a < 0 时：表达式在两根外侧为负，在两根之间为正。
sign of a. Pay close attention to strict (>, <) versus inclusive (≥, ≤) boundaries.
若 Δ ≤ 0，二次式符号不变，因此不等式要么对所有实数 x 成立，要么无解，这取决于 a 的符号。要特别注意严格不等号（>、<）与包含等号（≥、≤）的边界处理。
7. The Axis of Symmetry and Vertex | 对称轴与顶点
y transformed, the axis of symmetry remains x = h in the vertex form f(x) = a(x − h)² + k.
每条抛物线都有一条垂直对称轴 x = h，其中 h = −b/(2a)。这条线将图像分成对称的两半。顶点 (h, k) 是对称轴与抛物线的交点。即使图像经历了大幅度变换，在顶点式 f(x) = a(x − h)² + k 中对称轴始终为 x = h。
r₂ = 2h − r₁. This symmetry property is frequently tested in IB multiple-choice questions.
了解对称轴有助于在已知一个根的情况下求出另一个根：若一个 x 轴截距为 r₁，对称轴为 x = h，则另一个截距为 r₂ = 2h − r₁。这种对称性在 IB 选择题中经常出现。
8. Transformations of y = x² | y = x² 的图像变换
y = x². A function in vertex form f(x) = a(x − h)² + k illustrates three transformations:
所有二次函数图像都可以通过变换母函数 y = x² 得到。顶点式 f(x) = a(x − h)² + k 体现了三种变换：
- Vertical stretch/compression and possible reflection by factor a: if |a| > 1 the graph narrows; if 0 < |a| < 1 it widens; if a is negative it reflects vertically.
- Horizontal translation by h: shift right if h > 0, left if h < 0.
- Vertical translation by k: shift upward if k > 0, downward if k < 0.
- 由系数 a 引起的垂直伸缩/压缩以及可能的反射：若 |a| > 1 图像变窄；若 0 < |a| < 1 图像变宽；若 a 为负则关于 x 轴反射。
- 水平平移 h：若 h > 0 向右平移，若 h < 0 向左平移。
- 垂直平移 k：若 k > 0 向上平移，若 k < 0 向下平移。
Understanding the order of transformations matters: stretching is applied before translations when the function is written as a f(x − h) + k. Exercise 1F often asks you to describe the sequence of transformations mapping one quadratic onto another.
理解变换的顺序非常重要：当函数写成 a f(x − h) + k 的形式时，应先进行伸缩再进行平移。练习 1F 经常要求你描述将一个二次函数映射为另一个的变换序列。
9. Applications: Maximum and Minimum Problems | 应用：最大值与最小值问题
Quadratic functions model many optimisation scenarios in IB: projectile motion, profit/revenue functions, areas of enclosures, and more. The maximum or minimum value of a quadratic occurs at its vertex.
二次函数可以模拟 IB 中的许多最优化场景：抛体运动、利润/收益函数、围栏面积等等。二次函数的最大值或最小值出现在其顶点处。
When solving word problems, first define variables, write an expression for the quantity to be optimised, reduce it to a quadratic form, and then find the vertex. Always check that the vertex lies within the domain dictated by the real-world context (e.g. time cannot be negative).
在解决应用题时，首先定义变量，写出需要优化的量的表达式，将其化为二次形式，然后找到顶点。始终要检查顶点是否处于实际情况确定的定义域内（例如时间不能为负）。
Max/Min value = k, occurring at x = h (vertex form), or at x = −b/(2a) (standard form).
Exercise 1F may include a question like 'Find the maximum area of a rectangular enclosure with a fixed perimeter', which reduces to a quadratic in width or length.
练习 1F 可能包含类似 '求固定周长下矩形围栏的最大面积' 的题目，它最终归结为一个关于宽或长的二次函数。
10. Quadratic Systems and Intersections | 二次方程组与交点
Finding the intersection points of a line and a parabola, or of two parabolas, means solving a system of equations. Substitute the linear equation into the quadratic to obtain a single equation in one variable. This resulting quadratic equation's discriminant determines the number of intersection points:
- Δ > 0: two distinct intersections;
- Δ = 0: the line is tangent to the parabola (one intersection);
- Δ < 0: no intersection.
求直线与抛物线或两条抛物线的交点，就是求解方程组。将直线方程代入二次函数，得到关于一个变量的一元方程。该二次方程的判别式决定了交点个数：
- Δ > 0：两个不同的交点；
- Δ = 0：直线与抛物线相切（一个交点）；
- Δ < 0：无交点。
In some IB questions, you are asked to find the value of a parameter such that a line just touches the curve — this is a tangent condition requiring the discriminant of the combined equation to be set to zero.
在某些 IB 题目中，会要求你求出使得直线恰好与曲线相切的参数值 —— 此时需要令组合方程的判别式为零，即相切条件。
11. The Sum and Product of Roots | 根的和与积
x), the sum α + β = −b/a and the product αβ = c/a. These relationships, derived from Vieta' torisations and for solving problems that involve the roots without actually finding them.
对于有根 α 和 β（实数或复数）的二次方程 ax² + bx + c = 0，根的和 α + β = −b/a，根的积 αβ = c/a。这些关系源于韦达定理，在检验因式分解以及在不需要实际求出根的情况下解决涉及根的问题时非常有用。
Exercise 1F may present a problem like 'If the sum of the squares of the roots is 5, find k' parameter k. Here, using α² + β² = (α + β)² − 2αβ avoids solving for the roots explicitly.
练习 1F 可能出现这样的题目：对于一个含有参数 k 的二次方程，'若两根平方和为 5，求 k'。此时使用 α² + β² = (α + β)² − 2αβ，可避免直接求根。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Quadratic questions are often labelled as 'easy' but lose candidates marks through careless errors. Always check the sign of a before sketching or interpreting inequalities. Remember that completing the square requires factoring a from the x² and x terms before inserting the bracket square.
二次函数的题目常被认为是 '简单题'，但考生常因粗心而失分。在画草图和解读不等式之前，一定要检查 a 的符号。请记住，配方法需要先将 x² 和 x 项的系数 a 提取出来，再进行括号内平方的处理。
Common pitfalls: confusing the discriminant formula (it is b² − 4ac, not b² + 4ac); forgetting to consider the domain in optimisation; mixing up the signs when writing the vertex from completed square form (x − h)²; and misapplying the axis of symmetry when roots are complex.
常见陷阱：混淆判别式公式（是 b² − 4ac，而不是 b² + 4ac）；在最优化问题中忘记考虑定义域；由完全平方形式 (x − h)² 写顶点时弄错符号；在根为复数时误用对称轴性质。
Finally, practice rewriting the same quadratic in all three forms — standard, vertex and factored — fluently. This flexibility is the key to mastering Exercise 1F and performing well in Paper 1 and Paper 2.
最后，请熟练地将同一个二次式分别写成标准式、顶点式和因式分解式。这种灵活转换的能力是掌握练习 1F 并在试卷一和试卷二中取得好成绩的关键。
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