📚 Quadratic Functions: IB & AQA Maths Key Points | 二次函数：IB与AQA考点精讲
Quadratic functions are a cornerstone of both IB and AQA A‑level Mathematics. Whether you are working with the standard form y = ax² + bx + c, completing the square, using the discriminant, or modelling real‑world scenarios, a deep understanding of quadratics is essential for success in papers 1, 2, and 3. This article consolidates the core concepts, techniques, and typical exam questions you will meet, with clear English–Chinese explanations to support bilingual learners.
二次函数是 IB 和 AQA A‑level 数学共同的基石。无论你是在处理标准式 y = ax² + bx + c、进行配平方、使用判别式，还是建立实际情景模型，扎实的二次函数理解对于在试卷 1、2 和 3 中取得成功都至关重要。本文凝练了核心概念、技巧和典型考题，并以清晰的中英双语解读帮助双语学习者掌握要点。
1. Standard Form and Key Features | 标准形式与关键特征
A quadratic function can be written in standard form: f(x) = ax² + bx + c, where a ≠ 0. The coefficient a determines the direction and width of the parabola; b influences the axis of symmetry; c is the y‑intercept.
二次函数可以写成标准形式：f(x) = ax² + bx + c，其中 a ≠ 0。系数 a 决定了抛物线的开口方向和宽窄；b 影响对称轴的位置；c 是 y 轴截距。
The graph of a quadratic is a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point. The y‑intercept is the point (0, c).
二次函数的图像是一条抛物线。若 a > 0，抛物线开口朝上，具有最小值点；若 a < 0，开口朝下，具有最大值点。y 轴截距为点 (0, c)。
The axis of symmetry is the vertical line x = –b / (2a). The vertex lies on this line, and its x‑coordinate is –b/(2a). Substituting this back into the function gives the corresponding y‑value of the vertex.
对称轴是直线 x = –b/(2a)。顶点位于该直线上，其 x 坐标为 –b/(2a)。将该 x 值代回函数即可得到顶点的 y 坐标。
2. Completing the Square and Vertex Form | 配平方与顶点式
Completing the square transforms a quadratic from standard form to vertex form: f(x) = a(x – h)² + k, where (h, k) is the vertex. This technique is vital for finding turning points without calculus.
配平方将二次函数从标准形式转化为顶点式：f(x) = a(x – h)² + k，其中 (h, k) 为顶点。这一技巧对于在不使用微积分的情况下求极值点至关重要。
For y = x² + bx + c, we add and subtract (b/2)²: x² + bx + (b/2)² – (b/2)² + c = (x + b/2)² + (c – b²/4). When a ≠ 1, first factor a from the x² and x terms, then complete the square inside the bracket, and finally expand carefully.
对于 y = x² + bx + c，我们加上并减去 (b/2)²：x² + bx + (b/2)² – (b/2)² + c = (x + b/2)² + (c – b²/4)。当 a ≠ 1 时，首先将 a 从 x² 和 x 项中提取出来，再在括号内配平方，最后仔细展开。
The vertex form immediately gives the coordinates of the turning point: (h, k). This is extremely useful in both IB and AQA exams, especially when answering questions on maximum/minimum values or sketching graphs.
顶点式直接给出极值点坐标 (h, k)。这在 IB 和 AQA 考试中极为有用，尤其是在回答有关最大值/最小值或草图绘制的问题时。
3. Factoring Quadratic Expressions | 因式分解二次式
Factoring (factorising) a quadratic involves writing it as a product of two linear factors. For x² + bx + c, we look for two numbers that multiply to c and add to b. For ax² + bx + c, the method includes splitting the middle term or using trial and error.
因式分解是将二次式写成两个一次因式的乘积。对于 x² + bx + c，我们需要找到两个数，它们的乘积为 c，和为 b。对于 ax² + bx + c，方法包括拆分中项或尝试法。
Once factorised, solving the equation becomes straightforward: set each factor to zero. This gives the x‑intercepts (roots) of the parabola. If the quadratic is already in completed square form, we can solve by rearranging and taking square roots.
一旦因式分解，解方程就变得简单：令每个因式等于零。这样可得到抛物线的 x 轴截距（根）。如果二次式已经是配平方形式，则可以通过移项和开方来求解。
In IB exams, factorisation is often the quickest way to find roots, but remember that not all quadratics factorise nicely over the integers. AQA papers equally test this skill, frequently leading into further work on inequalities or sketching.
在 IB 考试中，因式分解通常是求根的最快方法，但请记住，并非所有二次式都能在整数范围内因式分解。AQA 试卷同样考查这一技能，并经常引出后续的不等式或草图问题。
4. The Quadratic Formula and Discriminant | 求根公式与判别式
When factorisation is impossible or inconvenient, the quadratic formula gives the roots: x = [ –b ± √(b² – 4ac) ] / (2a). It works for all quadratic equations, provided we correctly identify a, b, and c from the standard form.
当因式分解不可能或不方便时，求根公式给出了根：x = [ –b ± √(b² – 4ac) ]/(2a)。只要我们能从标准式中正确识别出 a、b 和 c，该公式适用于所有二次方程。
The expression under the square root, Δ = b² – 4ac, is called the discriminant. It reveals the nature of the roots without solving the equation: if Δ > 0, two distinct real roots; Δ = 0, one repeated real root; Δ < 0, no real roots.
根号下的表达式 Δ = b² – 4ac 被称为判别式。它揭示了根的性质而无需解方程：Δ > 0 有两个不同实根；Δ = 0 有一个重实根；Δ < 0 无实根。
Both IB and AQA increasingly ask students to interpret the discriminant in modelling or geometrical contexts, for example determining when a line is tangent to a curve or predicting the number of solutions to a quadratic system.
IB 和 AQA 都越来越多地要求学生根据背景（如建模或几何）解读判别式，例如判断一条直线何时与曲线相切，或预测二次方程组的解的个数。
5. Graphs of Quadratic Functions | 二次函数的图像
A quadratic graph (parabola) is symmetric about its axis. Essential features to label are: the vertex, the axis of symmetry, the y‑intercept, and the roots (if they are real). Sketching these correctly can earn many marks even if the exact shape is not perfect.
二次函数图像（抛物线）关于其对称轴对称。需要标注的关键特征包括：顶点、对称轴、y 轴截距和根（若为实根）。即使形状不完全精准，正确地标出这些特征也可以获得很多分数。
When sketching, start by determining the direction (based on the sign of a), then find the vertex, then intercepts. An efficient approach in exams is to complete the square to get the vertex and use the quadratic formula or factoring for the x‑intercepts.
绘图时，首先根据 a 的符号确定开口方向，然后求顶点，最后求截距。考试中一个高效的方法是配平方得顶点，再用求根公式或因式分解找 x 截距。
Understanding how the graph shifts as parameters change is a key skill. For instance, compared to y = x², the graph of y = (x – 2)² + 3 is shifted right 2 units and up 3 units, with vertex (2, 3).
理解参数改变时图像如何移动是一项关键技能。例如，与 y = x² 相比，y = (x – 2)² + 3 的图像向右平移 2 个单位、向上平移 3 个单位，顶点为 (2, 3)。
6. Transformations of Parabolas | 抛物线的变换
Transformations include translations, stretches, and reflections. Given f(x) = x², we have: f(x) + k shifts vertically by k; f(x + h) shifts horizontally by –h; a f(x) stretches vertically by factor a; f(ax) stretches horizontally by factor 1/a.
变换包括平移、伸缩和反射。给定 f(x) = x²，我们有：f(x) + k 垂直平移 k；f(x + h) 水平平移 –h；a f(x) 垂直伸缩因子为 a；f(ax) 水平伸缩因子为 1/a。
An important transformation for quadratics is y = a(x – h)² + k, which combines a horizontal shift h, a vertical shift k, and a vertical stretch/reflection a. Mastering this form allows you to describe any parabola from its vertex.
对于二次函数，一个重要的变换是 y = a(x – h)² + k，它结合了水平平移 h、垂直平移 k 和垂直伸缩/对称 a。掌握这一形式便能从顶点出发描述任何抛物线。
In IB and AQA exams, you may be asked to map one parabola onto another, or to find the equation after a sequence of transformations. Always apply horizontal changes before vertical ones, and pay attention to the order of operations.
在 IB 和 AQA 考试中，可能要求你将一条抛物线映射到另一条，或求出一系列变换后的方程。始终先进行水平变化再进行垂直变化，并注意运算顺序。
7. Solving Quadratic Inequalities | 解二次不等式
A quadratic inequality such as ax² + bx + c > 0 is solved by first finding the roots of the corresponding equation, then testing intervals on the number line. The parabola’s shape (upward or downward) tells you where the function is positive or negative.
求解二次不等式 ax² + bx + c > 0，首先要找到对应方程的根，然后在数轴上检验区间。抛物线的形状（开口向上或向下）揭示了函数在何处为正或为负。
Sketching a quick graph is the safest method. Mark the x‑intercepts, draw the parabola’s orientation, and read off the intervals where y satisfies the inequality. Always check whether the inequality includes equality (≥) to decide if the endpoints are included.
快速绘制草图是最稳妥的方法。标出 x 截距，画出抛物线的开口方向，然后读出满足不等式的 y 值所对应的区间。始终检查不等式是否包含等号（≥），以决定是否包含端点。
AQA questions often ask for the solution set in set notation (e.g., {x : x < –1} ∪ {x : x > 3}), while IB may use both set notation and interval notation. Practice expressing answers clearly in the required format.
AQA 的题目通常要求用集合符号表示解集（如 {x : x < –1} ∪ {x : x > 3}），而 IB 可能会同时使用集合符号和区间符号。练习以所要求的格式清晰地表达答案。
8. Quadratic Systems and Simultaneous Equations | 二次方程组与联立方程
When a quadratic equation is combined with a linear equation, we can solve by substitution. For example, substituting y = mx + c into y = ax² + bx + d gives a quadratic in x. The discriminant of this resulting equation tells us how many points of intersection there are.
当二次方程与一次方程组合时，我们可以通过代入法求解。例如，将 y = mx + c 代入 y = ax² + bx + d 得到一个关于 x 的二次方程。该导出方程的判别式告诉我们有多少个交点。
If Δ > 0, the line intersects the parabola at two points; Δ = 0 exactly one point (tangent); Δ < 0 no intersection. This is a classic exam question, especially in AQA Core and IB Analysis & Approaches.
若 Δ > 0，直线与抛物线相交于两点；Δ = 0 一个交点（相切）；Δ < 0 无交点。这是一个经典的考题，尤其在 AQA Core 和 IB 分析与方法中。
For two quadratics, eliminate one variable to obtain a single quadratic in the other. Solving gives the intersection points. Consistent practice with algebraic manipulation is essential to avoid sign errors.
对于两个二次式，消去一个变量可得到另一个变量的二次方程。求解即得交点坐标。不断练习代数操作对于避免正负号错误至关重要。
9. Applications and Modelling | 应用与建模
Quadratic functions model many real‑world contexts: projectile motion, area problems, profit maximisation, and more. Typically, you will be given a quadratic expression and asked to find a maximum or minimum value, or to solve for a specific condition using the discriminant.
二次函数可用于建立许多实际情景的模型：抛体运动、面积问题、利润最大化等。通常会给出一个二次式，要求求出最大值或最小值，或者使用判别式求解特定条件。
To maximise or minimise a quadratic, you can complete the square or use the vertex formula x = –b/(2a). In context, remember to interpret your mathematical solution in terms of the original problem and check for realistic constraints (e.g., time or length cannot be negative).
要对二次式求最值，可以通过配平方或使用顶点公式 x = –b/(2a)。在应用背景中，请记住将数学解代入原问题中进行解读，并检查实际约束条件（例如时间或长度不能为负）。
IB examinations often include an extended modelling problem involving a quadratic function. You might need to derive the equation from data, predict the maximum height, or find when a moving object hits the ground. AQA also features such questions in applied units.
IB 考试经常包含涉及二次函数的扩展建模问题。你可能需要根据数据推导方程、预测最大高度，或求出一个运动物体何时落地。AQA 在其应用单元中也有类似题目。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
1. Sign errors when completing the square: When you expand (x + b/2)², it gives x² + bx + (b/2)². The correction term is –(b/2)² added to c. Always double‑check the arithmetic.
1. 配平方时的正负号错误：展开 (x + b/2)² 得到 x² + bx + (b/2)²。修正项是 –(b/2)² 加到 c 上。务必仔细检查算术计算。
2. Misidentifying a, b, c in the quadratic formula: Make sure the equation is set to zero and in standard form before extracting coefficients. Watch for negative c values.
2. 在求根公式中错误识别 a、b、c：确保方程已化为标准形式并等于零再提取系数。注意负的 c 值。
3. Forgetting the discriminant’s interpretation: Δ > 0 does not just mean ‘two roots’ — it means two distinct real roots. For a repeated root (tangency), you need Δ = 0 explicitly.
3. 忘记判别式的解读：Δ > 0 不仅仅意味着“两个根”——而是两个不同的实根。对于重根（相切），你需要的是 Δ = 0 的精确条件。
4. Not drawing a sketch for inequalities: Always do a quick graph. It prevents you from picking the wrong intervals and losing easy marks.
4. 解不等式时不画草图：始终快速画一张草图。它能避免选择错误区间而白白丢分。
5. Units and contextual answers: In modelling questions, present your answer with correct units and relate it back to the problem statement. IB and AQA mark schemes award marks for interpretation.
5. 单位与上下文的答案：在建模题中，用正确的单位呈现答案，并将其与问题陈述相关联。IB 和 AQA 的评分方案都为解读步骤给分。
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