📚 Investigation 2 - Cubic Graphs | 探究二：三次函数图像
, where a ≠ 0. The graph of a cubic function is a smooth, continuous curve with distinctive shapes. By examining the role of each coefficient and the function’s factorised form, we can sketch accurate graphs, locate key points such as intercepts, turning points, and points of inflection, and understand how transformations affect the curve. This exploration is essential for IB Mathematics: Analysis and Approaches, where cubic models appear in algebra, calculus, and real-world contexts.
在本次探究中，我们将研究三次函数的性质及其图像特征。三次函数是一个三次多项式，通常写作 f(x) = ax³ + bx² + cx + d，其中 a ≠ 0。三次函数的图像是一条光滑、连续的曲线，具有独特的形状。通过分析各项系数的作用以及函数的因式分解形式，我们可以准确绘制图像，确定截距、转折点和拐点等关键特征，并理解图像变换对曲线的影响。这一探究对于 IB 数学（分析与方法）至关重要，三次函数模型常出现在代数、微积分和实际应用中。
1. Understanding Cubic Functions | 理解三次函数
osite infinities. As x → +∞, the sign of the leading term ax³ determines whether y → +∞ or –∞, and as x → – f any cubic function are all real numbers. Understanding this fundamental behaviour is the first step in graphing cubic functions.
三次函数是指任何形如 f(x) = ax³ + bx² + cx + d 的多项式，其中 a、b、c、d 为实数常数且 a ≠ 0。x 的最高次数为 3，这意味着图像最多有两个转折点，并且两端总会趋向于相反方向的无穷大。当 x → +∞ 时，首项 ax³ 的符号决定了 y → +∞ 还是 –∞；当 x → –∞ 时，变化方向则相反。与二次函数不同，三次函数图像没有全局最大值或最小值，它是无界的。任何三次函数的定义域和值域都是全体实数。理解这一基本行为是绘制三次函数图像的第一步。
2. Standard Form and Parameters | 标准形式与参数
fficient b affects the curvature near the y-axis and the position of the inflection point. The coefficient c influences the slope at the y-intercept, and d is the y-intercept itself, since f(0) = d. By adjusting these parameters, one can produce a family of cubic graphs with varying shapes. In an investigation, it is instructive to vary one parameter while holding others fixed to observe the changing graph dynamics.
一般方程 f(x) = ax³ + bx² + cx + d 包含四个参数，每个参数对图像的影响各不相同。系数 a 控制整体的陡峭程度和两端趋势：若 a > 0，图像右端上升、左端下降；若 a < 0，则右端下降、左端上升。系数 b 影响 y 轴附近的弯曲程度以及拐点的位置。系数 c 影响图像在 y 截距处的斜率，而 d 本身就是 y 截距，因为 f(0) = d。通过调整这些参数，我们可以得到一系列形状各异的三次函数图像。在探究中，固定其他参数而单独改变一个参数，有助于观察图像动态变化，这是一种很有启发性的做法。
3. Factored Form and Roots | 因式分解形式与根
If a cubic function can be factorised, it is often written as f(x) = a(x – r₁)(x – r₂)(x – tinct real roots. The multiplicity of a root affects how the graph crosses or touches the x-axis. A simple root (multiplicity 1) causes the graph to cross the axis. A double root (multiplicity 2) makes the graph touch the axis and turn around, appearing as a local extremum on the axis. A triple root (multiplicity 3) results in a flattening at the axis combined with a point of inflection. Recognising the factored form allows for quick sketching and solving cubic equations.
如果三次函数可以因式分解，则常写作 f(x) = a(x – r₁)(x – r₂)(x – r₃)，其中 r₁, r₂, r₃ 为实根（即 x 轴截距）。三次函数至少有一个实根，但可能有一个、两个（其中一个是重根）或三个互异的实根。根的重数影响图像在 x 轴处的穿越或接触方式。单根（重数 1）使图像直接穿过 x 轴。二重根（重数 2）会使图像碰触 x 轴并折返，在轴上呈现一个局部极值。三重根（重数 3）则导致图像在 x 轴处变平，并带有一个拐点。识别因式分解形式有助于快速绘制草图和求解三次方程。
4. Y-intercept and Symmetry | y 轴截距与对称性
rder 2 about the origin. In general, however, the presence of an x² term (b ≠ 0) or a constant term breaks this symmetry. The point of inflection itself is a centre of rotational symmetry for the graph only when the cubic is expressed in the form y = a(x – int (h, k). This property is useful in determining the shape without plotting many points.
任何三次函数的 y 轴截距就是常数项 d，因为代入 x = 0 得到 f(0) = d。这一点为图像提供了一个定位点。在对称性方面，只有当 b = 0 且 d = 0 时，三次函数才是奇函数，形式为 f(x) = ax³ + cx，此时图像关于原点具有二阶旋转对称性。然而，一般情况下，若存在 x² 项（b ≠ 0）或常数项，这种对称性就会被打破。只有当三次函数表示为 y = a(x – h)³ + k 时，其拐点本身才是图像的旋转对称中心，此时图像关于点 (h, k) 对称。这一性质有助于无需绘制大量点即可确定图形形状。
5. Shape and Leading Coefficient | 图像形状与首项系数
basic orientation of the cubic graph. For a > 0, the graph starts low on the left (as x → –∞, y → – posite is true: the graph starts high on the left and ends low on the right. The magnitude |a| affects the steepness; a larger |a| gentler slopes. Investigations often compare graphs of y = x³, y = 2x³, y = 0.5x³, and y = – his effect. Understanding this primary shape is the basis for all further sketching steps.
首项系数 a 的符号从根本上决定了三次函数图像的基本朝向。当 a > 0 时，图像左端向下延伸（当 x → –∞ 时，y → –∞），右端向上延伸（当 x → +∞ 时，y → +∞）。当 a < 0 时，则正好相反：图像左端向上、右端向下。a 的绝对值 |a| 影响图像的陡峭程度；|a| 越大，图像越陡；而当 |a| 接近零时，曲线趋于平缓，斜率更温和。探究中常常通过比较 y = x³、y = 2x³、y = 0.5x³ 和 y = –x³ 的图像来直观感受这一影响。理解这一基本形状是所有后续绘图步骤的基础。
6. Stationary Points and Turning Points | 驻点与转折点
nts (where the derivative f′(x) = 0). Since f′(x) is a quadratic, its discriminant D = 4b² – ph is strictly monotonic (always increasing or always decreasing) with no bends. The coordinates of turning points are found by solving f′(x) = 0 and substituting back into f(x). These points are crucial for accurately sketching the curve.
三次函数可以没有驻点、有一个或有两个驻点（即导数 f′(x) = 0 的点）。因为 f′(x) 是一个二次函数，其判别式 D = 4b² – 12ac 决定了实驻点的个数。若 D > 0，则存在两个不同的转折点：一个局部极大值和一个局部极小值。若 D = 0，则只有一个驻点，该点也是具有水平切线的拐点。若 D < 0，则没有驻点，图像严格单调（始终递增或始终递减），没有弯曲。转折点的坐标通过求解 f′(x) = 0 并代回 f(x) 求得。这些点对于准确绘制曲线至关重要。
7. Point of Inflection | 拐点
f(x) = ax³ + bx² + cx + d, f″(x) = 6ax + 2b, so the inflection point has x-coordinate x = – the function is shifted to eliminate the quadratic term (by writing it in the form y = a(x – y a stationary point unless the tangent is horizontal, which happens when f′(x) = 0 at x = –b/(3a), i.e. when b² = 3ac.
每一个三次函数的图像恰好有一个拐点，在该点处曲线的凹凸性发生改变。从数学上讲，拐点出现在二阶导数 f″(x) = 0 处。对于 f(x) = ax³ + bx² + cx + d，有 f″(x) = 6ax + 2b，所以拐点的 x 坐标为 x = –b/(3a)。将其代回原函数即可得到 y 坐标。有趣的是，当通过配方消除二次项（即将函数写成 y = a(x – h)³ + k 的形式）时，该拐点就是图像的旋转对称中心。拐点不一定是一个驻点，除非其切线是水平的，即当在 x = –b/(3a) 处 f′(x) = 0 ，也就是 b² = 3ac 时。
8. Using Derivatives to Sketch Graphs | 利用导数绘制图像
t or a sign chart. Third, find the point of inflection by setting f″(x) = 0. Fourth, evaluate the end behaviour using the leading coefficient. Finally, plot these key points and connect them with a smooth curve that respects the monotonic intervals and curvature. This derivative-based method ensures an accurate graph without plotting an excessive number of points. In IB investigations, students often use technology to check their hand-drawn sketches.
绘制三次函数图像的系统化方法包括以下步骤：首先，确定 y 截距（d），并尽可能进行因式分解以找到 x 截距。第二，计算 f′(x) 并解方程 f′(x) = 0，以定位驻点，并通过二阶导数检验或符号表判断其性质。第三，通过解 f″(x) = 0 找到拐点。第四，根据首项系数判断两端的变化趋势。最后，标出这些关键点，并用一条光滑的曲线将它们连接起来，确保曲线符合单调区间和凹凸性。这种基于导数的方法可以确保图像的准确性，而无需绘制过多的点。在 IB 探究中，学生常使用技术工具来核对手绘图。
9. Transformations of Cubic Graphs | 三次函数图像的变换
f(x) + k, shifting the whole graph up or down. A horizontal translation by h units is f(x – A vertical stretch or compression by factor v is v·f(x). A horizontal stretch by factor 1/|p| is f(px). A reflection in the x-axis is –f(x), and in the y-axis is f(–x). The most powerful form for transformations is the completed cubic form: y = a(x – This form highlights the symmetry and makes graphing trivial once (h, k) and a are known.
三次函数图像的变换与其他函数类似。沿 y 轴垂直平移 k 个单位表示为 f(x) + k，可将整个图像向上或向下移动。沿 x 轴水平平移 h 个单位表示为 f(x – h)，可使图像向右移动（当 h 为正时）。垂直方向拉伸或压缩 v 倍表示为 v·f(x)。水平方向拉伸因子为 1/|p| 表示为 f(px)。关于 x 轴的反射为 –f(x)，关于 y 轴的反射为 f(–x)。对于变换来说，最有用的形式是完全三次形式：y = a(x – h)³ + k，其中 (h, k) 为拐点，a 控制拉伸和朝向。这一形式凸显了几何对称性，并且一旦知道 (h, k) 和 a，绘图就变得非常简单。
10. Solving Cubic Equations Graphically | 图像法解三次方程
solutions. This method is particularly useful for equations that cannot be easily factorised or solved algebraically. In IB coursework, students might be asked to find approximate roots using a graph on their GDC (graphic display calculator) and then refine them using numerical methods. Visualising cubic equations graphically reinforces the connection between algebraic and geometric representations and is a key investigative skill.
方程 ax³ + bx² + cx + d = 0 的解对应于函数 y = f(x) 图像上的 x 截距。如果方程形式为 ax³ + bx² + cx + d = k，则可以绘制 y = f(x) 和水平线 y = k，它们的交点即为方程的解。这一方法对于不易因式分解或代数求解的方程尤为有用。在 IB 课程作业中，学生可能被要求使用图形计算器（GDC）通过图像找到近似根，然后利用数值方法进行精确化。通过图像法解三次方程，能够强化代数表示与几何表示之间的联系，是一项关键的探究技能。
11. Discriminant and Nature of Roots | 判别式与根的性质
For a cubic in the form ax³ + bx² + cx + d = 0, the discriminant Δ is given by Δ = 18abcd – 4b³d + b²c² – 4ac³ – or three equal). If Δ < 0, the cubic has one real root and two non-real complex conjugate roots. While the IB may not require memorising this formula, understanding its significance helps in predicting graph intersections with the x-axis without plotting.
与二次方程类似，三次方程也有一个判别式，用于揭示根的性质。对于形如 ax³ + bx² + cx + d = 0 的三次方程，其判别式 Δ 为 Δ = 18abcd – 4b³d + b²c² – 4ac³ – 27a²d²。尽管表达式较为复杂，但它能表明根是互异实根，还是包含重根或复根。若 Δ > 0，则三次方程有三个互异的实根。若 Δ = 0，则存在重根（两个或三个相等）。若 Δ < 0，则三次方程有一个实根和两个非实的共轭复根。虽然 IB 可能不要求记住这个公式，但理解其意义有助于无需绘图即可预测图像与 x 轴的交点情况。
12. Real-World Applications and Modelling | 实际应用与建模
Cubic functions frequently appear in real-life contexts such as physics, engineering, and economics. For example, the volume of a box made from a rectangular sheet with cut-out corners is often modelled by a cubic function in terms of the cut size. The relationship between the side length of a cube and its volume is cubic. In economics, cubic cost functions can represent increasing, then decreasing, then increasing marginal costs. In IB internal assessments, students may collect data and fit a cubic model, investigating properties like maximum volume or minimum cost using calculus. Such investigations demonstrate the power of cubic graphs in solving optimisation and design problems.
三次函数经常出现在物理、工程和经济学等实际情境中。例如，用一块矩形板材通过切角制作盒子的体积，往往表示为关于切角尺寸的三次函数。立方体的边长与其体积之间的关系就是三次的。在经济学中，三次成本函数可以表示边际成本先增后减再增的情况。在 IB 内部评估中，学生可能会收集数据并拟合三次模型，利用微积分探究最大体积或最小成本等性质。这类探究展示了三次函数图像在解决优化与设计问题中的强大功能。
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