📚 13B Quadratic Models | 13B 二次模型
cribes countless natural and designed phenomena. In the IB Diploma, quadratic models are not just about solving equations; they are about interpreting vertexes, intercepts and curvature in context, and learning to move fluently between algebraic forms to answer real questions.
二次模型是应用代数的基石，它让我们能够捕捉变化率本身以恒定速率变化的关系。从足球的飞行弧线到企业利润的最大化，简单的方程 y = ax² + bx + c 描述了无数自然和人为现象。在 IB 文凭课程中，二次模型不仅仅是解方程；更重要的是结合情境解读顶点、截距和曲率，并学会在不同代数形式之间灵活转换，以回答真实世界中的问题。
1. Introduction to Quadratic Models | 1. 二次模型导论
A quadratic model expresses a dependent variable as a quadratic function of an independent variable. The hallmark of such a relationship is a constant second difference. In table form, if the first differences of y-values change by a fixed amount each time x increases by 1, then a quadratic is the natural choice. This makes quadratics ideal for modelling acceleration, area optimisation, and any scenario involving a turning point.
二次模型将因变量表示为自变量的二次函数。这种关系的标志是恒定的二阶差分。在表格形式中，如果 y 值的一阶差分在 x 每次增加 1 时按固定量变化，那么二次函数就是自然的选择。这使得二次函数非常适合为加速度、面积优化以及任何涉及转折点的情境建模。
= −4.9t² + 20t + 1.5 is a demonstration of how mathematics formalises physical intuition.
在 IB 课程中，要求你能够识别数据中的二次模式、使用技术工具拟合二次回归，然后在真实情境中解释系数的含义。从散点图到 y = −4.9t² + 20t + 1.5 的过程，展示了数学如何将物理直觉形式化。
2. The Three Algebraic Forms and Their Uses | 2. 三种代数形式及其用途
A quadratic function can be written in standard form, vertex form and factored form, each offering unique insights. Understanding when to switch between them is essential for efficient modelling.
二次函数可以写成标准式、顶点式和因式分解式，每种形式都提供独特的见解。理解何时在它们之间切换对于高效建模至关重要。
Standard form: y = ax² + bx + c
标准式: y = ax² + bx + c
Standard form reveals the y-intercept immediately — it is simply c. The sign of a tells us whether the parabola opens upward (a > 0) or downward (a < 0). This form is convenient for input into the quadratic formula and for identifying the initial value in a time-based model, such as initial height in projectile motion.
标准式立即显露出 y 轴截距——它就是 c。a 的符号告诉我们抛物线是向上开口（a > 0）还是向下开口（a < 0）。这种形式便于代入二次公式求解，也可识别基于时间模型中的初始值，比如抛体运动中的初始高度。
Vertex form: y = a(x − h)² + k
顶点式: y = a(x − h)² + k
imum point of the model. In optimisation problems, this form directly gives the best outcome and the value of x that achieves it. For example, if a business' an instantly read off that selling 15 units yields a maximum profit of 450 currency units.
顶点 (h, k) 是模型的最大值或最小值点。在优化问题中，这种形式直接给出了最佳结果以及达到该结果所需的 x 值。例如，如果某企业的利润 P(x) = −2(x − 15)² + 450，我们可以立即读出卖出 15 单位产品可获得 450 货币单位的最大利润。
Factored form: y = a(x − p)(x − q)
因式分解式: y = a(x − p)(x − q)
Here p and q are the x-intercepts, or roots. This form is indispensable when modelling situations where the output returns to zero, such as a projectile landing on the ground. It also links beautifully to the null factor law, allowing you to solve for when a quantity becomes zero without expanding first.
这里 p 和 q 是 x 轴截距，即根。当建模输出归零的情形时，例如抛体落回地面，这种形式必不可少。它还与零因子律完美衔接，使你能在不展开的前提下求解一个量何时为零。
3. Vertex and Axis of Symmetry in Context | 3. 顶点与对称轴的情境解读
sses through the vertex. In a real-world model, this line often represents the midpoint of the action — ll reaches its peak or the production level exactly halfway between two break-even points.
每个二次模型都有一条对称轴 x = −b/(2a)，它穿过顶点。在现实模型中，这条线通常代表动作的中点——球达到最高点的时刻，或者恰好处于两个盈亏平衡点中间的生产水平。
The y-coordinate of the vertex is found by substituting the axis of symmetry back into the function. If a model is given in vertex form, both coordinates are explicit. Knowing the vertex allows you to describe the range of possible outputs, which is critical when assessing the feasibility of a solution. For a downward-opening model, the maximum value must appear as a constraint in the final answer.
顶点的 y 坐标通过将对称轴代入原函数求得。如果模型以顶点式给出，两个坐标都清晰显现。掌握顶点让你能够描述可能输出的值域，这在评估解决方案的可行性时至关重要。对于向下开口的模型，最大值必须作为约束条件出现在最终答案中。
In IB exams, a typical question might provide a quadratic formula for the height of a firework and ask for the maximum height and the time it occurs. Recognising that the vertex answers both questions instantly saves time and reduces algebraic errors.
在 IB 考试中，一道典型题目可能给出烟花高度的二次公式，并要求最大高度及其发生时间。识别出顶点可以同时回答这两个问题，从而节省时间并减少代数错误。
4. Modelling Projectile Motion | 4. 抛体运动建模
+ v₀t + h₀, where g is the acceleration due to gravity (9.8 m/s² or 32 ft/s²), v₀ is the initial vertical velocity, and h₀ is the initial height.
最经典的二次模型之一是在均匀重力下（忽略空气阻力）抛体的轨迹。一般形式为 h(t) = −½gt² + v₀t + h₀，其中 g 是重力加速度（9.8 m/s² 或 32 ft/s²），v₀ 是初始垂直速度，h₀ 是初始高度。
s at ground level. Students should practise writing h(t) = −4.9(t − 2)² + 21.6 and reading off the maximum height of 21.6 m at t = 2 s.
通过这个函数，顶点给出最大高度，正根给出飞行时间。在形式之间转换变得非常强大：标准式直接与物理参数相关，顶点式突显峰值，因式分解式显示物体何时位于地面。学生应当练习写出 h(t) = −4.9(t − 2)² + 21.6，并直接读出 t = 2 s 时最大高度为 21.6 m。
e the projectile hits the ground. Extrapolating beyond the domain produces meaningless results and is a common pitfall in modelling questions.
请记住，此类模型的定义域被限制在 t ≥ 0 直至物体落地的时间。超出定义域进行外推会产生无意义的结果，这是建模题中常见的陷阱。
5. Revenue, Cost and Profit Models | 5. 收入、成本与利润模型
+ mq, a quadratic. Subtracting a linear cost function yields a quadratic profit function.
当价格与销量呈线性关系时，二次方程自然出现在经济学中。如果每件商品的价格为 p = m − nq，那么收入 R(q) = q × p = −nq² + mq，就是一个二次函数。减去线性成本函数后得到二次利润函数。
Maximising profit becomes a matter of finding the vertex of the profit parabola. The break-even points — where profit is zero — are the x-intercepts of the profit function. This dual interpretation is extremely rich: the vertex tells you the optimal quantity, while the intercepts inform you about the viable production range.
利润最大化就变成了求利润抛物线顶点的问题。盈亏平衡点——利润为零的点——就是利润函数的 x 轴截距。这种双重解读极为丰富：顶点告诉你最优生产数量，截距则揭示了可行的生产范围。
Consider a café a maximum profit of 500. The roots, found via factoring or the quadratic formula, indicate that selling fewer than 10 or more than 70 meals results in a loss. Such a model guides both ambition and caution.
设想一家咖啡馆发现其日利润为 P(x) = −0.5x² + 40x − 300，其中 x 是售出的餐食数量。顶点在 x = 40 处，得到最大利润 500。通过因式分解或二次公式求得的根表明，售出少于 10 份或多于 70 份餐食将导致亏损。这样的模型既指导进取又提醒谨慎。
6. Area Maximisation and Geometric Problems | 6. 面积最大化与几何问题
x) = −2x² + Lx, where L is the total fencing material. The maximum area occurs at x = L/4.
经典的优化问题涉及利用固定长度的围栏靠墙围一个矩形场地。设 x 为垂直于墙的边长，则面积 A = x(L − 2x) = −2x² + Lx，其中 L 是围栏总长。最大面积出现在 x = L/4 处。
In these problems, the domain is constrained by x > 0 and L − 2x > 0, giving 0 < x < L/2. The vertex lies exactly midway within this interval, illustrating the symmetry of optimisation. Asking students to explore how the maximum changes when the wall is replaced by a river that also requires no fencing reinforces the concept of variable constraints.
在这些问题中，定义域受限于 x > 0 和 L − 2x > 0，即 0 < x < L/2。顶点恰好位于该区间的中点，说明优化的对称性。要求学生探讨当墙壁换成同样无需围栏的河流时最大值如何变化，可以强化可变约束的概念。
L/4. Encourage connecting the algebraic representation to the physical shape: a width-to-length ratio emerges, proving that the optimal rectangle is twice as long as it is wide.
将面积函数写成顶点式 A = −2(x − L/4)² + L²/8，可以明显看出当 x = L/4 时最大面积为 L²/8。鼓励学生将代数表征与几何形状联系起来：宽长比随之浮现，证明最优矩形长为宽的两倍。
7. Quadratic Regression and Modelling from Data | 7. 二次回归与基于数据的建模
bx + c, along with the coefficient of determination R², which indicates the goodness of fit.
当真实数据呈现单次转向的曲线模式时，便可使用 GDC 或统计软件进行二次回归。输出结果为最佳拟合二次函数 y = ax² + bx + c 提供系数 a、b 和 c，同时给出决定系数 R²，用以指示拟合优度。
In IB assessment, you might be given a table of stopping distances against speed for a car. The strong quadratic relationship can be modelled, and then the model used to predict a stopping distance for an untested speed. However, caution must be taken: prediction outside the range of data (extrapolation) can be unreliable because the physical law may change.
在 IB 评估中，你可能会面对一个汽车制动距离随速度变化的表格。强大的二次关系可以被建模，然后用该模型预测未经测试的速度下的制动距离。然而必须谨慎：超出数据范围的预测（外推）可能不可靠，因为物理规律可能改变。
A good modelling report should comment on the appropriateness of the quadratic, compare it with a linear model if relevant, and discuss the meaning of the coefficients. For instance, in a stopping-distance model, the a-coefficient relates to deceleration characteristics, while c might indicate a reaction-time offset.
一份好的建模报告应评论二次模型是否恰当，必要时与线性模型进行比较，并讨论系数的含义。例如，在制动距离模型中，系数 a 与减速特性相关，而 c 可能表示反应时间的偏移。
8. The Discriminant and Modeling Reality | 8. 判别式与现实的联系
ow many real solutions a quadratic model has. In modelling terms, this corresponds to how many times a certain y-value is attained. If a projectile' iminant of the resulting equation determines whether it ever reaches that height, touches it once, or crosses it twice.
判别式 Δ = b² − 4ac 告诉我们二次模型有多少个实数解。在建模语境中，这对应于某个特定的 y 值被达到多少次。如果将射弹的高度设为 10 米，所得方程的判别式决定了它是从未达到该高度、恰好达到一次，还是穿过两次。
ht equals the maximum height; the vertex just touches that level. When Δ < 0, the goal is physically unattainable under the given parameters. This is a powerful tool for establishing feasibility without fully solving the equation.
当 Δ > 0 时，物体在上行和下行过程中各穿过一次目标高度。当 Δ = 0 时，目标高度等于最大高度；顶点刚好触及该高度。当 Δ < 0 时，给定参数下该目标在物理上无法达到。这是无需完全解方程即可判定可行性的强大工具。
In an optimisation problem where profit must reach a certain benchmark, the discriminant can quickly reveal whether the target is ever achievable, and if so, between which two sales volumes. This makes the discriminant a decision-making aid, not just a symbolic exercise.
在利润须达到某一基准水平的优化问题中，判别式可以迅速揭示该目标是否可能实现，以及如果可能，在哪两个销售区间内可以实现。这使得判别式成为决策辅助工具，而不仅仅是符号练习。
9. Transforming Quadratics to Reveal the Model’s Story | 9. 转换二次函数以揭示模型的故事
+ b/2)² − (b/2)² unlocks the coordinates of the optimum and makes graphing vastly simpler.
配方法将标准式转换为顶点式，这一步骤讲述了二次模型的完整故事。过程 (x² + bx) → (x + b/2)² − (b/2)² 解锁了最优点的坐标，并使作图大为简化。
rofit model, that narrowing effect might indicate high sensitivity to changes in production levels.
以模型 y = 3x² − 12x + 7 为例。配方法得到 y = 3(x − 2)² − 5。我们立即看到当 x = 2 时最小值为 −5。首项系数 a = 3 告诉我们该抛物线比基础的 y = x² 更窄。在利润模型中，这种变窄的效应可能表明对生产水平变化的灵敏度很高。
Moving between factored form and vertex form can answer both “when does it hit zero?” and “what is the peak?” without solving two separate equations. Encouraging fluency in these transformations produces students who see the entire shape of a model, not just isolated points.
在因式分解式和顶点式之间切换，可以回答“何时归零？”和“峰值是什么？”而无需求解两个独立方程。鼓励学生熟练掌握这些变换，使他们能够看到模型的整个形状，而不仅仅是孤立的数据点。
10. Solving Quadratic Inequalities in Context | 10. 在情境中解二次不等式
oblems demand more than a single answer; they ask for a range: When does the profit exceed £500? During what time interval is the ball above 3 m? + 20t + 1 ≥ 3 involves finding the roots of the associated equation and testing intervals.
许多 IB 题目要求的不仅仅是一个答案，而是一个范围：利润何时超过 500 英镑？在哪个时间间隔内球的高度超过 3 米？求解二次不等式如 −4.9t² + 20t + 1 ≥ 3 涉及到找到对应方程的根并检验区间。
A clear sketch of the parabola makes the inequality's solution visible. For a downward-opening parabola, the y-values are above a certain threshold between the two roots. In a business model, the interval between the break-even points is the profitable zone. Accurate inequality solutions ensure the model's output is useful for strategic decisions.
清晰地画出抛物线的草图可以使不等式的解一目了然。对于向下开口的抛物线，在两个根之间的 y 值高于特定阈值。在商业模型中，两个盈亏平衡点之间的区间就是盈利区间。准确的不等式解答确保了模型输出对战略决策有价值。
Technology can solve inequalities directly, but the IB values the reasoning: plotting the horizontal line representing the threshold, finding intersection points, and shading the region where the quadratic is above or below the line. This visual method reinforces the connection between algebraic and graphic representations.
技术工具可以直接求解不等式，但 IB 重视推理过程：绘制表示阈值的水平线，寻找交点，并将二次函数高于或低于该线的区域涂上阴影。这种视觉方法强化了代数表征与图形表征之间的联系。
11. Real-World Constructions: Arches, Bridges and Suspension | 11. 现实世界的构造：拱门、桥梁与悬索
The shape of a suspension bridge' parabola. The equation y = a x² + b gives the height of the cable above the lowest point.
悬索桥主缆的形状通常可以用二次函数来模拟，特别是当重量沿跨度均匀分布时。虽然自由悬挂的链条呈悬链线形状，但在桥面负载下曲线近似于抛物线。方程 y = a x² + b 给出了缆索相对于最低点的高度。
k can be built by setting the vertex at the highest point and finding a such that the arch meets the ground at the endpoints. This is a direct application of modelling with boundary conditions.
拱桥出于结构效率常常采用抛物线拱形。给定跨度和最大净空，可以将顶点设在最高点，并确定 a 值使拱在端点处触地，从而建立顶点式 y = a(x − h)² + k。这是利用边界条件直接进行建模的应用。
When architects want a fountain's water stream to land exactly in a basin, a quadratic trajectory is designed. By placing the coordinate axes appropriately, the jet's nozzle at (0, h) becomes the y-intercept, and the landing point gives an x-intercept. The design of the nozzle angle and pressure determines the quadratic that connects them.
当建筑师希望喷泉的水流精确落入水池时，就会设计一条二次轨迹。通过适当放置坐标轴，喷嘴在 (0, h) 处成为 y 轴截距，而落点给出 x 轴截距。喷嘴角度和压力的设计决定了连接它们的二次函数。
12. Summary and Strategic Tips for IB Success | 12. 总结与 IB 解题策略提示
Quadratic models unite algebra, geometry and practical reasoning. Always begin by identifying what the variables represent and which quantity is being maximised or minimised. Choose the most revealing form of the quadratic: vertex form for optimum, factored form for roots, standard form for initial values and discriminant analysis.
二次模型将代数、几何和实践推理融为一体。始终从识别变量含义、明确哪个量被最大化或最小化开始。选择最能揭示信息的二次函数形式：顶点式用于最值，因式分解式用于根，标准式用于初始值及判别式分析。
When confronted with a modelling question, sketch a quick graph—even a rough one. Note axis intercepts, direction of opening, and the vertex. State any domain restrictions explicitly. Verify whether your answers make sense in context: a negative time or a length beyond available fencing must be discarded, always tying back to the real-world scenario.
遇到建模问题时，迅速画出草图——哪怕很粗略。标注坐标轴截距、开口方向和顶点。明确陈述任何定义域限制。验证答案在语境中是否合理：负时间或超出可用围栏的长度必须舍弃，始终与真实情境相结合。
Finally, articulate your findings clearly. The IB values communication: use phrases like “the model predicts a maximum revenue of... when...” or “according to the discriminant, the target is only achievable between... and...”. This turns mathematical working into a convincing, evidence-based argument.
最后，清晰地表述你的发现。IB 重视交流：使用诸如“模型预测最大收入为...，当...时”或“根据判别式，目标仅在...到...之间可以实现”这样的措辞。这将数学过程转化为令人信服、基于证据的论证。
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