📚 Edexcel Maths: Quadratic Functions | 二次函数考点精讲
Quadratic functions form a core topic in the Edexcel Maths specification, appearing extensively in both Pure and Applied sections. This guide covers every essential concept — from algebraic manipulation to graphical interpretation — and provides clear, exam‑focused explanations. Master these ideas to confidently tackle equation solving, curve sketching, optimisation, and inequality problems.
二次函数是 Edexcel 数学考纲中的核心主题，广泛出现在纯数及应用模块中。本文涵盖每一个重要概念——从代数变形到图像解读——并提供清晰、贴合考试的讲解。掌握这些内容，你将能自信地解决方程求解、曲线草图、最优化及不等式问题。
Below we break down the topic into 12 focused sections, each pairing English explanations with Chinese translations. Work through them systematically, practice Edexcel‑style questions, and you’ll be well prepared for any quadratic‑related task on your exam.
下面我们将主题细分为12个重点小节，每个小节都提供中英文解释配对。请系统学习并练习 Edexcel 风格的题目，这样你就能为考试中任何与二次函数相关的题目做好充分准备。
1. Quadratic Functions: Definition and Standard Form | 二次函数：定义与标准形式
A quadratic function is a polynomial of degree two, written as f(x) = ax² + bx + c, where a, b, c are real constants and a ≠ 0. This is called the standard form. The highest power of x is 2, which gives the graph its characteristic parabolic shape.
二次函数是一个二次多项式，写作 f(x) = ax² + bx + c，其中 a、b、c 为实常数且 a ≠ 0。这种形式被称为标准形式。x 的最高次幂为2，使得图像呈抛物线特征。
The coefficient a controls the opening direction and width of the parabola. If a > 0, the parabola opens upwards (∪-shaped); if a < 0, it opens downwards (∩-shaped). The constant term c gives the y‑intercept, because f(0) = c. In Edexcel exams, you often need to identify a, b, c and use them to determine key features.
系数 a 控制抛物线的开口方向和宽度。若 a > 0，开口向上（∪形）；若 a < 0，开口向下（∩形）。常数项 c 给出 y 轴截距，因为 f(0) = c。在 Edexcel 考试中，经常需要识别 a、b、c 并用它们确定关键特征。
2. The Discriminant and Nature of Roots | 判别式与根的情况
The discriminant of a quadratic ax² + bx + c = 0 is Δ = b² – 4ac. It reveals the number and type of roots without solving the equation: Δ > 0 ⇒ two distinct real roots; Δ = 0 ⇒ one repeated real root (or two equal real roots); Δ < 0 ⇒ no real roots (the roots are complex conjugates).
二次方程 ax² + bx + c = 0 的判别式为 Δ = b² – 4ac。它无需解方程就能揭示根的数量和类型：Δ > 0 ⇒ 两个不等实根；Δ = 0 ⇒ 一个重根（两个相等实根）；Δ < 0 ⇒ 无实根（根为共轭复数）。
In Edexcel pure maths, discriminant questions often link to intersections of a quadratic curve with a line or another curve. For example, setting the functions equal and examining the resulting quadratic’s Δ tells you if they meet (Δ ≥ 0) or not (Δ < 0). This is a powerful tool for coordinate geometry and modelling.
在 Edexcel 纯数中，判别式问题常与二次曲线同直线或其他曲线的交点相联系。例如，令函数相等并根据所得二次方程的 Δ 判断是否相交（Δ ≥ 0 有交点，Δ < 0 无交点）。这是坐标几何和数学建模中的有力工具。
3. Factorising Quadratics | 因式分解二次式
Factorising converts a quadratic expression ax² + bx + c into a product of two linear factors. For simple quadratics with a = 1, find two numbers that multiply to c and add to b. For a ≠ 1, use the ‘ac method’: multiply a by c, find two numbers whose product is ac and sum is b, then split the middle term and factor by grouping.
因式分解将二次式 ax² + bx + c 转化为两个一次因式的乘积。对于 a = 1 的简单二次式，找出两个乘积为 c 且和为 b 的数。当 a ≠ 1 时，使用“ac 法”：将 a 与 c 相乘，找两数使其积为 ac 且和为 b，再拆中项并分组分解。
Factorisation is the fastest way to solve a quadratic equation if the coefficients are nice. Once you have (px + q)(rx + s) = 0, apply the null factor law: px + q = 0 or rx + s = 0, giving the roots x = –q/p and x = –s/r. Always check by expanding to verify.
若系数合适，因式分解是解二次方程最快的方法。一旦得到 (px + q)(rx + s) = 0，应用零因子律：px + q = 0 或 rx + s = 0，得到根 x = –q/p 和 x = –s/r。务必展开验证。
4. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. The steps: factor a from the x² and x terms; inside the bracket, add and subtract (b/(2a))²; then simplify to get a(x + b/(2a))² + (c – b²/(4a)). This is a core algebraic skill tested in Edexcel papers.
配方法将 ax² + bx + c 写成 a(x + p)² + q 的形式。步骤为：从 x² 和 x 项中提取 a；在括号内加上并减去 (b/(2a))²；化简得到 a(x + b/(2a))² + (c – b²/(4a))。这是 Edexcel 试卷考查的核心代数技能。
This form immediately gives the vertex coordinates: (–b/(2a), c – b²/(4a)). Completing the square is also essential for deriving the quadratic formula, for integrating certain rational functions, and for solving equations where factorisation is messy. Practise with negative a values and fractional b.
该形式直接给出顶点坐标：(–b/(2a), c – b²/(4a))。配方法对推导求根公式、积分某些有理函数、以及在不易因式分解时解方程都至关重要。请练习 a 为负值和 b 为分数的情况。
5. Quadratic Formula Derivation and Use | 求根公式推导与使用
Starting from ax² + bx + c = 0 (a ≠ 0), complete the square: x² + (b/a)x = –c/a. Add (b/(2a))² to both sides, giving (x + b/(2a))² = (b² – 4ac)/(4a²). Taking the square root yields x + b/(2a) = ±√(b² – 4ac)/(2a), and isolating x gives the formula: x = [–b ± √(b² – 4ac)] / (2a).
从 ax² + bx + c = 0（a ≠ 0）出发，配方：x² + (b/a)x = –c/a。两边加上 (b/(2a))²，得 (x + b/(2a))² = (b² – 4ac)/(4a²)。开平方得 x + b/(2a) = ±√(b² – 4ac)/(2a)，移项得到公式：x = [–b ± √(b² – 4ac)] / (2a)。
The quadratic formula always works, even when factorisation is difficult or the roots are irrational. In Edexcel exams, you are expected to memorise it and to substitute values carefully, simplifying surds where necessary. Always check the discriminant first to anticipate the nature of the roots.
求根公式总有效，即使因式分解困难或根为无理数时也可用。Edexcel 考试要求你熟记公式并仔细代入数值，按需化简根式。务必先算判别式，以预判根的情况。
6. Graphs of Quadratic Functions (Parabolas) | 二次函数图像（抛物线）
The graph of y = ax² + bx + c is a smooth, symmetric U‑ or ∩‑shaped curve. It has a turning point (vertex) that is either a minimum (a > 0) or a maximum (a < 0). The axis of symmetry is a vertical line passing through the vertex, with equation x = –b/(2a).
y = ax² + bx + c 的图像是一条平滑、对称的 U 形或 ∩ 形曲线。它有一个转折点（顶点），为最小值点（a > 0）或最大值点（a < 0）。对称轴是通过顶点的竖直线，方程为 x = –b/(2a)。
To sketch a parabola, plot the vertex, the y‑intercept (0, c), and the x‑intercepts (if real). The x‑intercepts are the solutions of ax² + bx + c = 0. If Δ < 0, the graph lies entirely above or below the x‑axis. A quick sketch often helps solve inequalities and optimisation problems.
画抛物线草图时，需标出顶点、y 截距 (0, c) 以及 x 截距（若存在实根）。x 截距是 ax² + bx + c = 0 的解。若 Δ < 0，图像完全在 x 轴上方或下方。快速草图常有助于解不等式和最优化问题。
7. Vertex and Axis of Symmetry | 顶点与对称轴
For the standard form, the vertex x‑coordinate is –b/(2a). Substitute this back into the function to find the y‑coordinate. In vertex form y = a(x – h)² + k, the vertex is simply (h, k). The axis of symmetry is x = h (which equals –b/(2a) in standard form).
对于标准形式，顶点横坐标为 –b/(2a)。将其代回函数即可求得纵坐标。在顶点式 y = a(x – h)² + k 中，顶点为 (h, k)。对称轴为 x = h（在标准形式中等于 –b/(2a)）。
Edexcel questions often give a quadratic in vertex form and ask for the line of symmetry or the turning point. Conversely, you may need to transform standard to vertex form by completing the square to extract these features. Remember that the sign of h in y = a(x – h)² + k is the x‑coordinate of the vertex, so (x + 3)² means h = –3.
Edexcel 题目常给出顶点式的二次函数，要求写出对称轴或转折点。反之，你可能需要通过配方法将标准形式化为顶点式，以提取这些特征。记住 y = a(x – h)² + k 中 h 的符号就是顶点的横坐标，因此 (x + 3)² 对应 h = –3。
8. Transformations of Quadratic Graphs | 二次图像变换
Applying the usual graph transformations to a quadratic function is a common Edexcel topic. For y = f(x) = ax²: f(x) + k shifts the graph up (k > 0) or down; f(x + k) shifts left (k > 0) or right; –f(x) reflects in the x‑axis; af(x) with |a| > 1 stretches vertically, 0 < |a| < 1 compresses vertically.
对二次函数应用常规图像变换是 Edexcel 的常见考点。对于 y = f(x) = ax²：f(x) + k 使图像上/下平移（k > 0 上移）；f(x + k) 左/右平移（k > 0 左移）；–f(x) 关于 x 轴反射；af(x) 中 |a| > 1 纵向拉伸，0 < |a| < 1 纵向压缩。
Using vertex form y = a(x – h)² + k ties transformations directly to the graph’s features: h is the horizontal shift, k the vertical shift, and a determines the stretch and reflection. For example, y = –2(x + 1)² + 5 is a parabola opening downwards, stretched by factor 2, shifted 1 unit left and 5 units up.
使用顶点式 y = a(x – h)² + k 可将变换与图像特征直接关联：h 为水平平移量，k 为垂直平移量，a 决定拉伸和反射。例如 y = –2(x + 1)² + 5 表示抛物线开口向下，拉伸因子为2，左移1个单位、上移5个单位。
9. Applications: Maxima and Minima Problems | 应用：最大值与最小值问题
Since the vertex of a quadratic gives the maximum or minimum value of the function, modelling scenarios often lead to quadratic functions. Typical Edexcel optimisation problems involve finding the maximum area of a rectangle, maximum height of a projectile, or minimum cost by completing the square to locate the vertex.
由于二次函数的顶点给出最大值或最小值，许多建模场景会导出二次函数。典型的 Edexcel 优化问题包括求矩形最大面积、抛射体最大高度或最小成本，通过配方法找到顶点。
Worked example: A ball is thrown upwards, its height h (metres) after t seconds being h(t) = –5t² + 20t + 2. Find its maximum height. Complete the square: h(t) = –5(t² – 4t) + 2 = –5[(t – 2)² – 4] + 2 = –5(t – 2)² + 22. Maximum height 22 m occurs at t = 2 s. Clearly state units and interpret the vertex in context.
例题：一球向上抛出，t 秒后的高度 h（米）为 h(t) = –5t² + 20t + 2。求最大高度。配方得：h(t) = –5(t – 2)² + 22。最大高度 22 米在 t = 2 秒时达到。务必写明单位并在情境中解读顶点的意义。
10. Quadratic Inequalities | 二次不等式
To solve ax² + bx + c > 0 or < 0, first find the roots of ax² + bx + c = 0. Sketch the parabola quickly: if a > 0, it opens upwards; if a < 0, downwards. The solution to > 0 is the x‑values where the curve lies above the x‑axis; for < 0, where it lies below. Write intervals using union (U) and strictly avoid including the roots if the inequality is strict.
解 ax² + bx + c > 0 或 < 0 时，先求出 ax² + bx + c = 0 的根。快速画出抛物线：a > 0 开口向上，a < 0 开口向下。> 0 的解为曲线在 x 轴上方的 x 值区间；< 0 为下方区间。严格不等式时用并集 U 表示，且不包含根点。
Edexcel marks are awarded for the correct use of the discriminant, the sign of a, and proper set notation. For example, solve x² – 5x + 6 > 0. Roots are 2 and 3; a > 0, so the parabola opens up; it is > 0 for x < 2 or x > 3, written as {x | x < 2} ∪ {x | x > 3}. Always draw a quick sign diagram.
Edexcel 评卷时要求正确使用判别式、a 的符号以及恰当的集合符号。例如解 x² – 5x + 6 > 0，根为 2 和 3，a > 0，抛物线开口向上，解为 x < 2 或 x > 3，记作 {x | x < 2} ∪ {x | x > 3}。始终画一个简明的符号图。
11. Solving Quadratic Equations by Graphing | 图像法解二次方程
Graphing devices or careful hand sketches allow you to solve quadratic equations graphically. The solutions to ax² + bx + c = 0 are the x‑intercepts (where the graph crosses the x‑axis). If the parabola touches the axis at exactly one point, the equation has a repeated root; if it does not intersect, it has no real roots.
通过绘图工具或精细手绘草图可直观解二次方程。ax² + bx + c = 0 的解为 x 截距（图像与 x 轴的交点）。若抛物线恰与轴相切于一点，则方程有重根；若不相交，则无实根。
Edexcel often asks you to use a given graph to estimate roots or to solve simultaneous equations such as a quadratic and a linear function. Read coordinates accurately from the graph and check you understand how the intersection of two graphs relates to the equation formed by setting their expressions equal.
Edexcel 常要求利用给定图像估算根，或解如一个二次函数与一个一次函数的联立方程。准确读取坐标，并理解两个图像的交点如何对应令表达式相等后所形成的方程。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Edexcel-specific advice: Always check that a ≠ 0 before using quadratic techniques; otherwise, the function is linear. When factorising, pull out any common factors first. With the discriminant, remember it sits inside the square root — if Δ is negative, there are no real solutions, but in some contexts you may still give complex roots using i = √(–1) if the question allows.
Edexcel 特别建议：使用二次函数方法前务必检查 a ≠ 0，否则函数为线性。因式分解时先提取公因式。处理判别式时，记住它在平方根内——若 Δ 为负，则无实数解，但在某些允许的语境下你可能需要用 i = √(–1) 给出复数根。
Common mistakes: confusing the sign of h when converting vertex form, missing the sign of a when sketching inequalities, and forgetting to divide both sides by a before completing the square. For optimisation, students often state x or t but forget the corresponding maximum/minimum value; always compute both coordinates of the vertex and interpret them in context. Practise past Edexcel papers under timed conditions to build speed and accuracy.
常见错误：转换顶点式时混淆 h 的符号，画不等式草图时弄错 a 的符号，配方法前忘记两边同时除以 a。在优化问题中，学生常只给出 x 或 t 而忘记对应的最大/最小值；务必计算顶点的两个坐标并结合背景解读。限时练习 Edexcel 历年真题，以提高解题速度与准确度。
Finally, never leave a quadratic answer untested — plug it back into the original equation. This quick check saves marks and reinforces your understanding.
最后，切勿跳过一个二次方程答案的检验——将其代回原方程。这种快速检查能守住分数并加深理解。
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