📚 Quartic Graphs Investigation | 四次函数图像探究
A quartic function can produce some of the most varied and interesting graphs in the whole of algebra. In this investigation, we explore the key features of quartic graphs: their possible shapes, symmetry, roots, turning points, points of inflection, end behaviour, and the transformations that connect them — all within the IB Mathematics framework.
四次函数能够生成代数中最丰富多彩、最有意思的图像。在这项探究中，我们将逐一剖析四次函数图像的关键特征：可能的形状、对称性、根、驻点、拐点、无穷远行为以及图像之间的变换——内容完全契合 IB 数学课程框架。
1. What Is a Quartic Function? | 什么是四次函数？
ere a, b, c, d, e are real constants and a ≠ 0. The term ax⁴ dominates the shape for large |x|, while the lower-degree terms control the wiggles in the middle.
四次函数是四次多项式。它的一般形式为 f(x) = ax⁴ + bx³ + cx² + dx + e，其中 a, b, c, d, e 为实数系数且 a ≠ 0。当 |x| 很大时，ax⁴ 项决定图像的走向，而低次项则控制图像中段的起伏与弯曲。
- Leading coefficient (a): Determines the vertical stretch and end behaviour.
- 主导系数 (a)：决定纵向拉伸程度和无穷远行为。
- Lower-degree terms: Create local maxima, minima and asymmetry.
- 低次项：产生局部极大值、极小值及非对称性。
- Constant term (e): Gives the y‑intercept.
- 常数项 (e)：给出图像在 y 轴上的截距。
2. The Overall Shape of a Quartic Graph | 四次函数图像的整体形状
Quartic graphs generally belong to one of four broad shape families: a 'W' shape, an 'M' shape, a flattened parabola with one local extremum, or a simple steep bowl. If a > 0, the graph eventually rises on both sides; if a < 0, it falls on both sides. The number of 'turns' depends on the first derivative.
四次函数图像大致可归为四类：“W 形”、“M 形”、仅有一个极值点的扁平抛物线，以及简单陡峭的碗形。若 a > 0，图像两端最终都向上延伸；若 a < 0，图像两端向下。转弯的次数取决于一阶导数的根。
Canonical examples:
| Function | Shape when a > 0 |
|---|---|
| y = x⁴ | Symmetric bowl, one minimum |
| y = x⁴ − 4x² | W‑shaped, two minima and one local maximum |
| y = −x⁴ + 5x² | M‑shaped, two maxima and one local minimum |
典型例子：
| 函数 | a > 0 时的形状 |
|---|---|
| y = x⁴ | 对称碗形，有一个极小值 |
| y = x⁴ − 4x² | W 形，两个极小值和一个局部极大值 |
| y = −x⁴ + 5x² | M 形，两个极大值和一个局部极小值 |
3. Symmetry in Quartic Functions | 四次函数的对称性
If a quartic contains only even powers of x (the terms ax⁴, cx², e), then the function is even and its graph is symmetric about the y‑axis. Any odd‑power term (bx³, dx) breaks that symmetry. Investigation often begins by comparing pure‑odd quartic variants with their even counterparts.
如果四次函数只含 x 的偶次幂（ax⁴、cx²、e 三项），则函数为偶函数，图像关于 y 轴对称。任何奇次幂项（bx³、dx）都会打破这种对称性。探究时通常会从纯偶四次函数与其含奇次项的变形开始对比分析。
- Even quartic: f(−x) = f(x). Graph mirrors across the y‑axis.
- 偶四次函数：满足 f(−x) = f(x)，图像关于 y 轴镜像对称。
- Odd terms present: Translates or skews the graph, losing reflectional symmetry.
- 存在奇次项时：图像发生平移或倾斜，失去轴对称性。
4. Roots and x‑intercepts of Quartics | 四次函数的根与 x 轴截距
to four real roots, depending on the discriminant and the nature of the polynomial. Repeated roots (double, triple, or quadruple) create points where the graph touches or flattens against the x‑axis rather than crossing it.
四次方程 f(x) = 0 最多可以有四个实根，具体取决于判别式和多项式的根的性质。重根（二重、三重或四重）会使图像在 x 轴上相切或压平，而不是直接穿过。
Example: f(x) = (x+1)²(x−2)² — touches the x‑axis at x = −1 and x = 2 (double roots).
例子：f(x) = (x+1)²(x−2)² — 在 x = −1 和 x = 2 处与 x 轴相切（二重根）。
Complex roots occur in conjugate pairs, so a quartic can have 0, 2 or 4 real roots. Investigators are encouraged to factorise quartics into quadratics and linear factors to fully interpret the graph.
复根以共轭对形式出现，因此四次函数可以有 0、2 或 4 个实根。鼓励探究者将四次多项式因式分解为二次式与一次式的乘积，从而完满解读图像。
5. Stationary Points and Turning Points | 驻点与转折点
The first derivative f ' ch can have one, two or three real zeros. Therefore a quartic graph can possess at most three stationary points. Depending on the signs of the second derivative at these points, they may be local minima, local maxima, or stationary points of inflection.
一阶导数 f '(x) = 4ax³ + 3bx² + 2cx + d 是一个三次函数，它可以有一个、两个或三个实零点。因此四次函数图像最多可有三个驻点。根据二阶导数在这些点处的符号，驻点可能是局部极小值、局部极大值或平稳拐点。
- 3 turning points: Typical of the full 'W' or 'M' family.
- 3 个转折点：是完整的“W 形”或“M 形”族的典型特征。
- 1 turning point: Pure x⁴ or heavily biased quartic.
- 1 个转折点：纯 x⁴ 或严重偏斜的四次函数。
- Stationary inflection: Rare, when f '(x) has a repeated root matching f ''(x) = 0.
- 平稳拐点：很少见，当 f '(x) 有重根且 f ''(x) 在该点为零时出现。
6. Points of Inflection | 拐点
A quartic's second derivative f '' can have zero, one or two real solutions, giving a maximum of two points of inflection. At a point of inflection, the concavity changes from upward to downward (or vice versa).
四次函数的二阶导数 f ''(x) = 12ax² + 6bx + 2c 是一个二次函数。它可以有零个、一个或两个实解，因此最多拥有两个拐点。在拐点处，图像的凹凸性发生改变。
For y = x⁴, f ''(x) = 12x², which is positive except at x = 0, so there is no genuine inflection — the concavity is upward everywhere. In contrast, y = x⁴ − 6x² has f ''(x) = 12x² − 12, giving two inflections at x = ±1.
对于 y = x⁴，f ''(x) = 12x²，除 x = 0 外始终为正，因此没有真正的拐点——所有地方都是上凹的。而 y = x⁴ − 6x² 的二阶导数为 f ''(x) = 12x² − 12，在 x = ±1 处有两个拐点。
7. End Behaviour – As x → ±∞ | 无穷远处的行为
Because the degree of a quartic is even and the leading coefficient a determines sign, the end behaviour is uniform on both sides (unlike odd‑degree polynomials). The left tail and the right tail either both point upward (a > 0) or both point downward (a < 0).
由于四次函数为偶次多项式，且主导系数 a 决定符号，无穷远处的行为在两侧是一致的（与奇次多项式不同）。左尾和右尾要么同时指向上方（a > 0），要么同时指向下方（a < 0）。
If a > 0: as x → −∞, f(x) → +∞; as x → +∞, f(x) → +∞.
若 a > 0：当 x → −∞ 时，f(x) → +∞；当 x → +∞ 时，f(x) → +∞。
The steepness of the tails is amplified by |a|, while the lower terms merely adjust the graph's interior.
|a| 越大，尾部越陡峭，而低次项仅调整图像中段的形态。
8. Transformations of Quartic Functions | 四次函数的图像变换
Standard transformations apply to quartic graphs just as to other polynomials. The ability to rewrite a quartic in completed form or in factorised form helps predict the effect of each parameter.
图像变换的基本规则同样适用于四次函数。将四次函数写成完全平方形式或因式分解形式，有助于预测每个参数的影响。
| Transformation | Effect on y = f(x) |
|---|---|
| f(x) + k | Vertical translation by k |
| f(x + h) | Horizontal translation by −h |
| a f(x) with |a| > 1 | Vertical stretch; if a < 0, reflection in x‑axis |
| f(bx) with |b| > 1 | Horizontal compression; if b < 0, reflection in y‑axis |
变换 / 对 y = f(x) 的影响
f(x) + k: 上下平移 k 个单位；f(x + h): 左右平移 −h 个单位；a f(x) (|a|>1): 纵向拉伸，a<0 时关于 x 轴反射；f(bx) (|b|>1): 横向压缩，b<0 时关于 y 轴反射。
Combining these transforms lets you sketch any quartic from a known parent graph.
综合运用这些变换就能从已知的母函数图像出发绘制任意四次函数草图。
9. Special Families of Quartic Graphs | 特殊的四次函数图像族
Several families are particularly useful for investigation due to their clean algebraic properties.
以下几类函数因其清晰的代数性质，特别适合用于探究。
- Perfect square quartics: y = (x² + p)², which is always non‑negative and has a single minimum if p > 0, or a W‑shaped double minimum if p < 0.
- 完全平方四次函数：y = (x² + p)²，始终非负；p > 0 时只有一个极小值，p < 0 时呈现 W 形双极小值。
- Difference of squares quartics: y = x⁴ − a² = (x² − a)(x² + a). Real roots exist only if a ≥ 0.
- 平方差四次函数：y = x⁴ − a² = (x² − a)(x² + a)，仅当 a ≥ 0 时有实根。
- Repeated quadratic factor: y = (x² + bx + c)², which always touches the x‑axis at the roots of the inner quadratic (if real).
- 二次因子重复四次函数: y = (x² + bx + c)²，如果内部的二次式有实根，图像总是在那些根处与 x 轴相切。
10. Using Calculus to Sketch Quartics | 利用微积分绘制四次函数草图
A reliable sketching procedure for IB investigation combines algebraic and calculus steps: first find intercepts and factorise if possible, then compute f '(x) to locate critical points and f ''(x) to classify them and find inflections. Finally, confirm end behaviour.
IB 探究中，一套可靠的作图程序融合了代数与微积分步骤：先找出截距并尽可能因式分解，然后求 f '(x) 定位临界点，再借助 f ''(x) 对它们进行分类并寻找拐点，最后验证无穷远行为。
Step‑by‑step for f(x) = x⁴ − 5x² + 4
1. Factorise: f(x) = (x² − 1)(x² − 4) = (x−1)(x+1)(x−2)(x+2). Roots at x = ±1, ±2.
2. f '(x) = 4x³ − 10x = 2x(2x² − 5) → critical at x = 0, ±√5/2.
3. f ''(x) = 12x² − 10 → inflections at x = ±√5/6.
逐步绘制 f(x) = x⁴ − 5x² + 4 的草图
1. 因式分解：f(x) = (x²−1)(x²−4) = (x−1)(x+1)(x−2)(x+2)，根在 x = ±1, ±2。
2. f '(x) = 4x³ − 10x = 2x(2x² −5) → 临界点在 x = 0, ±√5/2。
3. f ''(x) = 12x² − 10 → 拐点在 x = ±√5/6。
Such systematic analysis reveals a clear W‑shaped graph with correct turning points.
这种系统分析能够清晰地展示出一个具有正确转折点的 W 形图像。
11. Real‑World Applications of Quartics | 四次函数的实际应用
Quartic models appear in physics and engineering, often when a system has two stable equilibria. Examples include the elastic potential energy of a double‑well system, the track shape of a roller‑coaster segment, or the profile of certain suspension bridge cables approximated by polynomial curves.
四次函数模型时常出现在物理和工程领域，尤其是当系统存在两个稳定平衡点时。例如双势阱系统的弹性势能、过山车轨道的某段形状，或某些悬索桥缆索的多项式近似曲线。
In optimisation, quartic objective functions may represent a trade‑off with multiple competing goals. Investigating their graphs helps students interpret real critical points and choose realistic solutions.
在最优化问题中，四次目标函数可以表示多个竞争目标之间的权衡取舍。探究图像能帮助学生解读真实临界点并选出符合实际的解。
12. Investigation Tips and Summary | 探究技巧与总结
When answering an IB investigation on quartic graphs, always connect algebraic expressions to geometric features. Use technology (GDC or graphing software) to check your work, but also explain how calculus confirms the picture. Discuss the effect of varying each coefficient one at a time, and look for patterns like the number of turning points and the sign of the discriminant.
在回答 IB 四次函数图像探究题时，务必把代数表达式与几何特征联系起来。可以借助技术工具（图形计算器或绘图软件）验证结果，同时也要阐述微积分如何佐证图像。逐一改变各系数的值，观察其影响，寻找转折点个数和判别式符号等方面的规律。
Ultimately, quartic graphs showcase the rich behaviour of higher‑degree polynomials. Mastering them deepens your overall understanding of function analysis and equips you with sophisticated sketching and modelling skills.
归根结底，四次函数图像展示了高次多项式丰富多样的行为。掌握它们能加深你对函数分析的整体理解，并为你提供更高级的作图和建模技巧。
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