📚 Constructing and Analysing Quadratic Models in IB Mathematics | IB数学：二次模型的构建与分析
Quadratic models are one of the most versatile tools in IB Mathematics. They appear in topics ranging from functions and equations to calculus and statistics. Understanding how to construct and analyse these models is essential for both Analysis and Approaches (AA) and Applications and Interpretation (AI) students.
二次模型是IB数学中最通用的工具之一。它们出现在从函数与方程到微积分与统计的各个主题中。理解如何构建和分析这些模型，对学习分析与方法（AA）以及应用与解释（AI）的学生都至关重要。
1. The Standard Form and Graph Features | 二次函数的标准形式与图像特征
The standard form of a quadratic function is f(x) = ax² + bx + c, where a, b and c are real constants and a ≠ 0. The graph of such a function is a parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
二次函数的标准形式为 f(x) = ax² + bx + c，其中 a、b、c 为实数常数，且 a ≠ 0。这种函数的图像是一条抛物线。若 a > 0，抛物线开口向上；若 a < 0，抛物线开口向下。
The y-intercept is simply the point (0, c). The axis of symmetry is the vertical line x = -b/(2a), and the vertex lies on this axis. These features provide a quick sketch of the graph and help in interpreting the model in context.
与 y 轴的交点就是点 (0, c)。对称轴是竖直线 x = -b/(2a)，顶点位于该轴上。这些特征有助于快速绘制图像，并帮助我们在实际情境中解释模型。
2. Vertex Form and Factored Form | 顶点式与交点式
The vertex form of a quadratic function is f(x) = a(x – h)² + k, where (h, k) is the vertex of the parabola. This form is especially useful when a maximum or minimum value is known, for example when modelling profit or area.
二次函数的顶点式为 f(x) = a(x – h)² + k，其中 (h, k) 是抛物线的顶点。这种形式在已知最大值或最小值时特别有用，例如在模拟利润或面积时。
The factored form is f(x) = a(x – r₁)(x – r₂), where r₁ and r₂ are the roots (x-intercepts). This form directly reveals the solutions of the equation f(x) = 0. Converting between forms is a key algebraic skill in IB.
交点式为 f(x) = a(x – r₁)(x – r₂)，其中 r₁ 和 r₂ 是根（与 x 轴的交点）。这种形式直接给出方程 f(x) = 0 的解。在不同形式之间进行转换是IB中的关键代数技能。
f(x) = ax² + bx + c = a(x – h)² + k = a(x – r₁)(x – r₂)
3. Determining a Quadratic Model from Three Points | 通过三点确定二次模型
Since a quadratic function has three unknown parameters (a, b and c), three distinct points on the parabola are generally sufficient to determine its equation. Substitute each point into y = ax² + bx + c to obtain a system of three linear equations.
由于二次函数有三个未知参数（a、b、c），抛物线上三个不同的点通常足以确定其方程。将每个点代入 y = ax² + bx + c，可以得到一个包含三个线性方程的方程组。
Alternatively, if one point is the vertex (h, k), use the vertex form and one other point to solve for a. If two points are x-intercepts, use the factored form and one other point to solve for a. Choosing the most efficient form saves time in exam conditions.
或者，如果已知一个点是顶点 (h, k)，可使用顶点式并代入另一个点求解 a。如果已知两个点是 x 轴交点，则使用交点式并代入另一个点求解 a。在考试中选择最有效的形式可以节省时间。
4. Discriminant and Nature of Roots | 判别式与根的性质
The discriminant Δ = b² – 4ac determines the number and type of roots of the quadratic equation ax² + bx + c = 0. If Δ > 0, there are two distinct real roots; if Δ = 0, there is one repeated real root; if Δ < 0, there are no real roots (two complex conjugate roots).
判别式 Δ = b² – 4ac 决定了二次方程 ax² + bx + c = 0 的根的数量和类型。若 Δ > 0，则有两个不同的实数根；若 Δ = 0，则有一个重根；若 Δ < 0，则没有实数根（有两个共轭复数根）。
In modelling, the discriminant can indicate whether the model predicts real solutions. For example, in a profit model, Δ < 0 might mean the business never breaks even, while Δ ≥ 0 gives the break-even quantities.
在建模中，判别式可以指示模型是否预测出实数解。例如，在利润模型中，Δ < 0 可能意味着企业永远无法盈亏平衡，而 Δ ≥ 0 则给出盈亏平衡时的产量。
5. Practical Modelling with Quadratics | 二次模型的实际建模
Many real-world situations can be approximated by quadratic functions. Typical IB examples include the area of a rectangular enclosure, revenue as a function of price change, the height of a projectile, and the shape of a suspension cable.
许多现实情境可以用二次函数近似描述。IB中常见的例子包括矩形围栏的面积、收入作为价格变化的函数、抛射物的高度以及悬索的形状。
When constructing a model, define the variables clearly, identify the domain that makes sense in context, and check whether the quadratic behaviour is justified by the situation. A quadratic model is often valid only over a restricted interval.
在构建模型时，要明确定义变量，确定在情境中有意义的定义域，并检查二次行为是否合理。二次模型通常只在有限的区间内有效。
6. Optimisation: Maximum and Minimum Values | 最优化：最大最小值
For a quadratic function, the maximum or minimum value occurs at the vertex. If a > 0, the vertex gives the minimum; if a < 0, it gives the maximum. The x-coordinate of the vertex is x = -b/(2a), and the corresponding value is f(-b/(2a)).
对于二次函数，最大值或最小值出现在顶点处。若 a > 0，顶点给出最小值；若 a < 0，顶点给出最大值。顶点的 x 坐标为 x = -b/(2a)，对应的函数值为 f(-b/(2a))。
In IB problems, optimisation often involves constraints from geometry or business. For example, if a farmer has 100 m of fencing and wants to maximise the rectangular area, the area is A(x) = x(50 – x), with maximum at x = 25 m.
在IB问题中，最优化通常涉及几何或商业约束。例如，若一位农民有100米围栏，想要最大化矩形面积，则面积为 A(x) = x(50 – x)，在 x = 25 米时取得最大值。
7. Projectile Motion and the Parabola | 抛物运动与抛物线
In kinematics, the height of a projectile under constant gravity is modelled by h(t) = -½gt² + v₀t + h₀, where g is gravitational acceleration, v₀ is initial velocity and h₀ is initial height. This is a quadratic model in time t.
在运动学中，恒重力作用下抛射物的高度可以用 h(t) = -½gt² + v₀t + h₀ 来模拟，其中 g 是重力加速度，v₀ 是初速度，h₀ 是初始高度。这是关于时间 t 的二次模型。
Key questions include finding the maximum height (vertex), the time when the object hits the ground (positive root), and the time when the object reaches a certain height (solving a quadratic equation). Interpreting the context correctly is as important as the algebra.
常见问题包括求最大高度（顶点）、物体落地的时间（正根），以及物体达到某一高度的时间（解二次方程）。正确理解情境与代数运算同样重要。
8. Fitting Quadratic Models to Data | 数据拟合与二次回归
In Applications and Interpretation, students often use technology to fit a quadratic regression model to given data. The calculator or software outputs an equation of the form y = ax² + bx + c together with the correlation coefficient r² (coefficient of determination).
在应用与解释课程中，学生常使用技术手段将二次回归模型拟合到给定数据。计算器或软件会输出形如 y = ax² + bx + c 的方程，以及决定系数 r²。
A value of r² close to 1 indicates that the quadratic model fits the data well. However, students should be cautious about extrapolating beyond the data range, as quadratic behaviour may not continue in the real world.
r² 的值接近1表明二次模型对数据拟合得很好。然而，学生应警惕在数据范围之外进行外推，因为在现实世界中二次行为可能不会继续成立。
9. Common Exam Pitfalls | 常见考点与易错点
One common mistake is confusing the sign of a when interpreting the graph: a positive coefficient means a minimum point, not a maximum. Another is using the wrong vertex formula: the x-coordinate is -b/(2a), not -b/a.
一个常见的错误是解释图像时混淆 a 的符号：a 为正意味着有最小值，而不是最大值。另一个错误是使用错误的顶点公式：x 坐标应为 -b/(2a)，而不是 -b/a。
Students also forget to state the domain of a real-world model. For example, time cannot be negative, so the domain should be restricted to t ≥ 0. Always check whether your answers make sense in the given context, not just mathematically.
学生还经常忘记说明现实模型的定义域。例如，时间不能为负，因此定义域应限制为 t ≥ 0。始终检查你的答案在给定情境中是否有意义，而不仅仅是在数学上成立。
10. Summary and Revision Tips | 小结与复习建议
To master quadratic models, practise converting between the three forms, sketching graphs accurately, and solving application problems. Pay attention to the meaning of the parameters in context and to the domain restrictions.
要掌握二次模型，需要练习在三种形式之间转换，准确地绘制图像，并解决应用问题。注意参数在情境中的意义以及定义域的限制。
In exams, show all steps when finding the equation of a quadratic from given points. If you use a calculator for regression, state the equation clearly and justify why a quadratic model is appropriate. A solid understanding of quadratics will support many other topics in IB Mathematics.
在考试中，从已知点求二次方程时要展示所有步骤。如果使用计算器进行回归，请清晰地写出方程，并说明为什么二次模型是合适的。扎实理解二次函数将为IB数学中的许多其他主题提供支持。
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