📚 Quartic Graphs: Complete Revision Guide for A-Level Maths | 四次函数图像：A-Level 数学完整复习指南
Quartic graphs are the graphs of polynomial functions of degree four. In A-Level Mathematics, especially for Edexcel, you need to sketch these curves accurately, identify their key features, and understand how repeated factors change the shape of the curve near the x-axis. This article covers the essential ideas, the standard sketching method, and the exam techniques that turn a rough sketch into full marks.
四次函数图像是四次多项式函数的图像。在 A-Level 数学（尤其是 Edexcel 考试局）中，你需要会准确画出这些曲线、识别图像的关键特征，并理解重因式如何影响曲线在 x 轴附近的形状。本文涵盖四次函数图像的核心概念、标准作图步骤，以及能把草稿变成满分的考试技巧。
1. What is a Quartic Function? | 什么是四次函数
A quartic function is a polynomial whose highest power of x is 4. The general form is written below, where a, b, c, d and e are real constants and a ≠ 0. If a were zero, the x⁴ term would disappear and the function would drop back to a cubic, so the condition a ≠ 0 is essential.
四次函数是 x 的最高次数为 4 的多项式函数。其一般形式如下，其中 a、b、c、d、e 为实常数，且 a ≠ 0。如果 a 等于 0，x⁴ 项就会消失，函数就退化成三次函数，所以 a ≠ 0 这个条件必不可少。
y = ax⁴ + bx³ + cx² + dx + e, a ≠ 0
The coefficient a is called the leading coefficient, and it controls the overall direction of the curve. The names of polynomials follow the degree: degree 1 is linear, degree 2 is quadratic, degree 3 is cubic, degree 4 is quartic, and degree 5 is quintic. A quartic is therefore exactly one step above the cubic graphs you already know.
系数 a 称为首项系数，它决定了曲线整体的走向。多项式的命名与次数对应：1 次是线性，2 次是二次，3 次是三次，4 次是四次，5 次是五次。因此，四次函数正好是你在三次函数图像之上再进一步的内容。
sed form, such as y = (x + 1)²(x − 2)² or y = −(x − 3)(x − 1)²(x + 2). Recognising which form you have been given tells you immediately which features are quickest to find.
有些四次函数本质上是以 x² 为变量的二次函数，例如 y = x⁴ − 5x² + 4 可写成 y = (x² − 1)(x² − 4)。另一些则以完全因式分解形式给出，例如 y = (x + 1)²(x − 2)² 或 y = −(x − 3)(x − 1)²(x + 2)。识别出题目给的是哪种形式，就能立刻判断哪些特征最容易求。
2. Domain, Range and End Behaviour | 定义域、值域与两端趋势
The domain of every polynomial, including a quartic, is all real numbers, written as x ∈ ℝ no domain restrictions and no breaks: a quartic graph is continuous and smooth everywhere, so you never need to draw dashed lines as you would for a reciprocal or exponential graph.
包括四次函数在内的所有多项式函数，其定义域都是全体实数，记作 x ∈ ℝ 或 (−∞, +∞)。它没有渐近线、没有定义域限制、没有断点：四次函数图像处处连续且光滑，因此不需要像画倒数函数或指数函数那样画虚线渐近线。
pwards, so the curve has a global minimum value m and the range is [m, ∞). When a < 0, both ends point downwards, the curve has a global maximum value M, and the range is (−∞, M].
但值域取决于首项系数 a，因为它决定曲线两端的走向。当 a > 0 时，图像两端都向上，曲线存在全局最小值 m，值域为 [m, ∞)。当 a < 0 时，两端都向下，曲线存在全局最大值 M，值域为 (−∞, M]。
| Leading coefficient | As x → −∞ | As x → +∞ | Shape |
| a > 0 | y → +∞ | y → +∞ | W-like, opens upwards |
| a < 0 | y → −∞ | y → −∞ | M-like, opens downwards |
This is a key difference from a cubic: an odd-degree polynomial has ends pointing in opposite directions, while an even-degree polynomial has ends pointing the same way. If a sketch of a quartic has one end going up and the other going down, the sketch is wrong, no matter how good the turning points look.
这是与三次函数的关键区别：奇数次多项式两端方向相反，而偶数次多项式两端方向相同。如果一张四次函数草图出现一端向上、另一端向下，那么无论驻点画得多好，这张图都是错的。
3. Roots, Factors and x-axis Intercepts | 根、因式与 x 轴交点
y the factor theorem, if x = p is a root then (x − p) is a factor, so every root can be converted into a bracket and every bracket reveals a root.
函数的根就是使 y = 0 的 x 值，从图像上看，每个实根都对应曲线与 x 轴相交或相切的点。根据因式定理，若 x = p 是根，则 (x − p) 是因式，因此每个根都能写成一个括号，每个括号也能反过来读出根。
y = a(x − p)(x − q)(x − r)(x − s)
has exactly four roots in total: a repeated root such as x = p in (x − p)² counts twice. In an exam, roots are usually found by factorising, by the factor theorem with trial substitution, or by a substitution such as t = x².
由于四次函数的次数为 4，它最多有四个不同的线性因式，因此最多有四个实根。若按重数计算，它总共有四个根：例如 (x − p)² 中的根 x = p 要算两次。考试中，求根通常靠因式分解、用试值配合因式定理，或采用 t = x² 之类的换元法。
The number of real roots can be 0, 1, 2, 3 or 4, and the graph tells you which case you are in. Four roots means four crossings, three roots usually means one crossing plus one touching point, two roots may mean either two crossings or two touchings, and no real roots means the curve never reaches the x-axis at all. Always state the roots as coordinates, for example (−1, 0) and (2, 0), to gain method marks.
实根个数可以是 0、1、2、3 或 4，图像会告诉你属于哪种情况。四个根就是四个交点；三个根通常是一个交点加一个切点；两个根可能是两个交点，也可能是两个切点；没有实根则表示曲线完全不接触 x 轴。写答案时务必用坐标形式表示交点，例如 (−1, 0) 和 (2, 0)，以拿到过程分。
4. Repeated Factors and the Shape at Each Root | 重因式与各根处的形态
The most heavily examined idea in this topic is that the multiplicity of a factor decides how the curve behaves at the x-axis. A single factor means the curve crosses the axis at an angle, a squared factor means the curve touches the axis and turns back, and a cubed factor means the curve crosses the axis but flattens as it does so.
本专题最常考的思想是：因式的重数决定曲线在 x 轴处的表现。一次因式表示曲线斜穿 x 轴；平方因式表示曲线与 x 轴相切后折返；三次因式表示曲线穿过 x 轴但在穿过时变平。
| Factor | Multiplicity | Behaviour at (p, 0) |
| (x − p) | 1, simple | Crosses the axis |
| (x − p)² | 2, double | Touches the axis, turning point |
| (x − p)³ | 3, triple | Crosses with a flattened S-bend |
| (x − p)⁴ | 4, quadruple | Touches, very flat minimum or maximum |
rms this. If y = (x − p)²g(x), then differentiating gives dy/dx = 2(x − p)g(x) + (x − p)²g' lso a stationary point, and why the graph touches the axis rather than cutting through it.
一个简单的代数检验可以证实这一点。若 y = (x − p)²g(x)，求导得 dy/dx = 2(x − p)g(x) + (x − p)²g'(x)，只要 g(p) ≠ 0，就有 x = p 时 dy/dx = 0。这正是二重根同时是驻点的原因，也是图像与 x 轴相切而不穿过的原因。
A triple root behaves differently again: the derivative is zero there too, but the sign of the function still changes, so the curve crosses the axis while momentarily flattening. Candidates often draw a triple root as a simple crossing, which loses the shape mark even if every coordinate is correct.
三重根的表现又不一样：该处导数同样为零，但函数值的符号仍然改变，因此曲线在短暂变平的同时穿过 x 轴。考生常把三重根画成普通交点，这样即使所有坐标都对，也会丢掉形状分。
5. The y-intercept and Expanding Brackets | y 轴截距与展开括号
dx + e, the intercept is simply e. This is a common one-mark question, and it is also the anchor point that tells you whether your sketch sits above or below the axis at the centre of the page.
求 y 轴截距只需令 x = 0。在展开形式 y = ax⁴ + bx³ + cx² + dx + e 中，截距就是 e。这是一分的常见小题，同时也是关键锚点，能告诉你草图在页面中央是位于 x 轴上方还是下方。
ntributes a negative sign and four negatives multiply to a positive. For example y = (x + 1)²(x − 2)² gives y = (1)²(−2)² = 4 when x = 0, so the y-intercept is (0, 4).
在因式形式下计算同样快捷：若 y = a(x − p)(x − q)(x − r)(x − s)，代入 x = 0 得 y = a·p·q·r·s，因为每个括号都贡献一个负号，四个负号相乘得正。例如 y = (x + 1)²(x − 2)² 在 x = 0 时给出 y = (1)²(−2)² = 4，因此 y 轴截距是 (0, 4)。
Occasionally a question asks you to expand a quartic or to compare coefficients. Expanding (x + 1)²(x − 2)² step by step gives x⁴ − 2x³ − 3x² + 4x + 4, and you can check your answer by confirming that the constant term matches the y-intercept you already found. Comparing coefficients is also the standard way to find unknown constants when part of a factorisation is given.
有时题目要求你展开四次式或比较系数。逐步展开 (x + 1)²(x − 2)² 得 x⁴ − 2x³ − 3x² + 4x + 4，你可以通过核对常数项是否等于已求出的 y 轴截距来检验结果。当题目给出部分因式分解时，比较系数也是求未知常数的标准方法。
6. Stationary Points and the First Derivative | 驻点与一阶导数
Differentiating a quartic gives a cubic, because each term drops by one power. Stationary points therefore satisfy a cubic equation, and a cubic has at most three real solutions, so a quartic graph can have at most three stationary points and at most two points of inflection.
对四次函数求导得到三次函数，因为每一项的次数都减一。因此驻点满足一个三次方程，而三次方程最多有三个实根，所以四次函数图像最多有三个驻点、最多有两个拐点。
dy/dx = 4ax³ + 3bx² + 2cx + d = 0
nary point, use the second derivative d²y/dx² = 12ax² + 6bx + 2c: if d²y/dx² > 0 it is a minimum, if d²y/dx² < 0 it is a maximum, and if d²y/dx² = 0 the test fails and you must check the sign of the gradient on both sides.
四次函数必定至少有一个驻点，因为实系数三次方程至少有一个实根。判断每个驻点的类型要用二阶导数 d²y/dx² = 12ax² + 6bx + 2c：若 d²y/dx² > 0 则为极小值点，若 d²y/dx² < 0 则为极大值点，若 d²y/dx² = 0 则该判别法失效，必须检查该点两侧的梯度符号。
Remember the link with the previous section: every double root is automatically a stationary point that lies on the x-axis. This means that if a question tells you the curve touches the x-axis at two different points, you already know that two of the three stationary points are those touch points, and only one turning point remains to be found.
记住与上一节的联系：每个二重根自动成为位于 x 轴上的驻点。也就是说，如果题目告诉你曲线在两个不同点与 x 轴相切，你其实已经知道三个驻点中的两个就是这两个切点，只剩一个极值点需要另行求出。
7. W-shapes, M-shapes and Number of Turning Points | W 形、M 形与极值点个数
tive has only one real root, the graph looks like a very flat U with a single minimum, as in y = x⁴ or y = x⁴ − 2x² + 5 where the two turning points merge into a flatter structure.
当 a > 0 且导数有三个不同实根时，图像呈 W 形：两个极小值之间夹着一个极大值。当 a > 0 但导数只有一个实根时，图像看起来像一个非常平坦的 U 形，只有一个极小值，例如 y = x⁴ 或 y = x⁴ − 2x² + 5 这类结构。
When a < 0, the same logic applies upside down. The graph becomes an M shape with a local minimum between two local maxima, and again the number of turning points matches the number of real roots of the derivative. Being able to say "three stationary points, so a W shape" in one sentence saves time and shows the examiner you understand the structure.
当 a < 0 时，同样的逻辑上下颠倒。图像变成 M 形：两个极大值之间夹着一个极小值，极值点的个数同样等于导数的实根个数。能用一句话说出 "三个驻点，所以是 W 形"，既省时间，又向阅卷人展示你理解图像结构。
The number of real roots and the overall shape are also connected. A W-shaped quartic with a > 0 can have zero, two or four simple roots, or combinations involving repeated roots; it cannot have exactly one simple root, because the curve would have to cross an odd number of times, which is impossible when both ends point upwards.
实根个数与整体形状也有联系。a > 0 的 W 形四次函数可以有零个、两个或四个单根，也可以包含重根的混合情形；但它不可能只有一个单根，因为那样曲线需要穿过 x 轴奇数次，而两端同时向上时这是不可能的。
8. Step-by-step Curve Sketching Strategy | 逐步作图策略
For a typical Edexcel question worth four to six marks, follow the same fixed order every time. The order below is designed so that the easiest marks are secured first and the sketch improves as more information is added.
对于 Edexcel 中一道四到六分的典型题目，每次都用固定的顺序作答
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