Quartic Function Graphs: Shapes and Key Features | 四次函数图像的形状与关键特征

📚 Quartic Function Graphs: Shapes and Key Features | 四次函数图像的形状与关键特征

A quartic function is a polynomial of degree four, typically written in the form f(x) = ax⁴ + bx³ + cx² + dx + e, where a ≠ 0. Its graph is a smooth continuous curve that can display up to three turning points and as many as four real roots. In this article, we explore the possible shapes of quartic graphs and the key features needed to sketch them accurately.

四次函数是次数为四的多项式函数，通常写作 f(x) = ax⁴ + bx³ + cx² + dx + e，其中 a ≠ 0。它的图像是一条光滑连续的曲线，最多可以有三个转向点，最多可以有四个实数根。在本文中，我们将探究四次函数图像的可能形状，以及准确绘制图像所需的关键特征。

1. The General Form and End Behaviour | 一般形式与端部行为

Every quartic function has the standard form f(x) = ax⁴ + bx³ + cx² + dx + e. Because the leading term is x⁴, the end behaviour of the graph depends only on the sign of a. When a > 0, both ends of the graph rise toward +∞ as x → ±∞. When a < 0, both ends fall toward −∞ as x → ±∞. This "same-end" behaviour distinguishes quartics from cubics, which have opposite ends.

每一个四次函数都有标准形式 f(x) = ax⁴ + bx³ + cx² + dx + e。由于最高次项是 x⁴，图像的端部行为只取决于 a 的符号。当 a > 0 时，图像两端在 x → ±∞ 时都趋向 +∞。当 a < 0 时，图像两端在 x → ±∞ 时都趋向 −∞。这种“两端同向”的行为将四次函数与三次函数区分开来，三次函数的两端方向相反。

If a > 0: f(x) → +∞ as x → ±∞
If a < 0: f(x) → −∞ as x → ±∞

2. Possible Shapes: W, M, and Beyond | 可能的形状：W 形、M 形及其他

A quartic graph can take several distinct shapes. When a > 0, the graph often resembles the letter W, with two minima and one maximum between them. When a < 0, the graph resembles the letter M, with two maxima and one minimum. However, not every quartic has three turning points. If the derivative equation f′(x) = 0 has fewer than three real solutions, the graph may have only one turning point or two turning points.

四次函数的图像可以呈现几种不同的形状。当 a > 0 时，图像常像字母 W，有两个极小值点和它们之间的一个极大值点。当 a < 0 时，图像像字母 M，有两个极大值点和一个极小值点。但并非每个四次函数都有三个转向点。如果导数方程 f′(x) = 0 的实数解少于三个，图像可能只有一个或两个转向点。

Number of real roots of f′(x) = 0	Number of turning points	Typical shape
3	3	W (a > 0) or M (a < 0)
2 (one repeated)	2	Single valley or hill with a plateau
1	1	U-like or inverted U-like

3. The Derivative and Critical Points | 导数与临界点

The first derivative of a quartic is a cubic: f′(x) = 4ax³ + 3bx² + 2cx + d. Setting f′(x) = 0 gives the x-coordinates of all turning points. A cubic equation always has at least one real root, but it may have two or three real roots depending on its discriminant. The second derivative f″(x) = 12ax² + 6bx + 2c is a quadratic, and its roots locate the points of inflection.

四次函数的一阶导数是三次函数：f′(x) = 4ax³ + 3bx² + 2cx + d。令 f′(x) = 0 可得所有转向点的 x 坐标。三次方程至少有一个实数根，但根据判别式，它可能有两个或三个实数根。二阶导数 f″(x) = 12ax² + 6bx + 2c 是二次函数，其根对应拐点的位置。

f′(x) = 4ax³ + 3bx² + 2cx + d = 0
f″(x) = 12ax² + 6bx + 2c = 0

4. Roots and Multiplicity | 根及其重数

A quartic equation can have 0, 2, or 4 real roots (counting multiplicities). Complex roots always occur in conjugate pairs, so the number of real roots must be even. A root of odd multiplicity means the graph crosses the x-axis at that point. A root of even multiplicity means the graph touches the x-axis and turns around without crossing.

四次方程可以有 0、2 或 4 个实数根（计重数）。复根总是成共轭对出现，因此实数根的个数必须为偶数。奇数重数的根意味着图像在该点穿过 x 轴；偶数重数的根意味着图像在该点接触 x 轴并折返，而不穿过。

Multiplicity 1: the graph crosses the axis with a slope.

重数 1：图像以一定斜率穿过 x 轴。
Multiplicity 2: the graph touches the axis and turns; this creates a turning point on the axis.

重数 2：图像接触 x 轴并折返；这会在轴上形成一个转向点。
Multiplicity 4: the graph is tangent to the axis and both sides stay on the same side.

重数 4：图像与 x 轴相切，两侧保持在轴的同一侧。

5. The Role of the Constant Term e | 常数项 e 的作用

The constant term e gives the y-intercept. Since f(0) = e, the graph always passes through the point (0, e). Changing e shifts the entire graph vertically without altering its shape or the x-coordinates of its turning points. This vertical shift is one of the simplest transformations of a quartic function.

常数项 e 给出 y 轴截距。因为 f(0) = e，图像总是经过点 (0, e)。改变 e 会使整个图像垂直平移，而不改变其形状或转向点的 x 坐标。这种垂直平移是四次函数最简单的变换之一。

6. Symmetry and Special Forms | 对称性与特殊形式

When all odd-power coefficients are zero, the quartic is even: f(x) = ax⁴ + cx² + e. Its graph is symmetric about the y-axis, meaning f(−x) = f(x). This creates a graph with either one turning point at x = 0 or three turning points arranged symmetrically. When all even-power coefficients except a are zero, the quartic is odd: f(x) = ax⁴ + bx³, but this is not a true odd function because the x⁴ term remains.

当所有奇次幂系数为零时，四次函数为偶函数：f(x) = ax⁴ + cx² + e。其图像关于 y 轴对称，即 f(−x) = f(x)。这会形成在 x = 0 处有一个转向点或三个对称排列转向点的图像。当除 a 外所有偶次幂系数为零时，形式为 f(x) = ax⁴ + bx³，但这并非真正的奇函数，因为 x⁴ 项仍然存在。

Even quartic: f(x) = ax⁴ + cx² + e, symmetric about the y-axis

7. The Discriminant and Number of Turning Points | 判别式与转向点个数

The derivative f′(x) = 4ax³ + 3bx² + 2cx + d is a cubic. The number of distinct real roots of this cubic determines the number of turning points of the quartic. A cubic has three distinct real roots when its discriminant is positive, one real root when the discriminant is negative, and repeated roots when the discriminant is zero. Thus the quartic has three turning points only when the cubic discriminant is positive.

导数 f′(x) = 4ax³ + 3bx² + 2cx + d 是三次函数。该三次函数不同实数根的个数决定了四次函数转向点的个数。当三次函数的判别式为正时，它有三个不同的实数根；为负时，有一个实数根；为零时有重根。因此，只有当三次判别式为正时，四次函数才有三个转向点。

8. Points of Inflection | 拐点

A point of inflection is a point where the concavity changes. For a quartic, the second derivative f″(x) = 12ax² + 6bx + 2c is a quadratic, which can have zero, one, or two real roots. If the quadratic has two distinct real roots, the quartic has two points of inflection. If it has one repeated root, there is one point of inflection. If it has no real roots, the concavity never changes, although the graph still curves.

拐点是凹凸性发生改变的点。对于四次函数，二阶导数 f″(x) = 12ax² + 6bx + 2c 是二次函数，它可能有零个、一个或两个实数根。若该二次函数有两个不同的实数根，则四次函数有两个拐点；若有一个重根，则有一个拐点；若无实数根，则凹凸性从不改变，尽管图像仍有弯曲。

f″(x) > 0: the graph is concave up (holds water).

f″(x) > 0：图像凹向上（开口朝上）。
f″(x) < 0: the graph is concave down (spills water).

f″(x) < 0：图像凹向下（开口朝下）。

9. Sketching a Quartic Step by Step | 分步绘制四次函数图像

To sketch the graph of a quartic function accurately, follow these steps. First, determine the sign of a to know the end behaviour. Second, find the y-intercept e. Third, solve f′(x) = 0 to locate turning points and evaluate f at each critical x. Fourth, solve f″(x) = 0 to locate points of inflection. Fifth, find the real roots of f(x) = 0 if possible, using factoring or numerical methods. Finally, plot all key points and connect them with a smooth curve respecting the end behaviour.

为了准确绘制四次函数的图像，请按照以下步骤操作。首先，判断 a 的符号以确定端部行为。其次，找出 y 轴截距 e。第三，解 f′(x) = 0 以定位转向点，并在每个临界 x 处计算 f 的值。第四，解 f″(x) = 0 以定位拐点。第五，尽可能通过因式分解或数值方法求出 f(x) = 0 的实数根。最后，标出所有关键点，并用一条平滑曲线连接它们，同时满足端部行为。

10. Worked Example: f(x) = x⁴ − 4x² + 3 | 例题：f(x) = x⁴ − 4x² + 3

Consider the even quartic f(x) = x⁴ − 4x² + 3. Here a = 1 > 0, so both ends rise. The y-intercept is f(0) = 3. Factor the expression: f(x) = (x² − 1)(x² − 3) = (x − 1)(x + 1)(x − √3)(x + √3). Thus the real roots are x = ±1 and x = ±√3, all of multiplicity 1, so the graph crosses the x-axis at all four points.

考虑偶四次函数 f(x) = x⁴ − 4x² + 3。这里 a = 1 > 0，所以两端都上升。y 轴截距为 f(0) = 3。因式分解得：f(x) = (x² − 1)(x² − 3) = (x − 1)(x + 1)(x − √3)(x + √3)。因此实数根为 x = ±1 和 x = ±√3，重数均为 1，所以图像在这四个点都穿过 x 轴。

Now compute the derivative: f′(x) = 4x³ − 8x = 4x(x² − 2). Setting f′(x) = 0 gives x = 0, x = √2, and x = −√2. Evaluating f at these points: f(0) = 3, f(±√2) = 4 − 8 + 3 = −1. So there are turning points at (0, 3), (√2, −1), and (−√2, −1). Since a > 0, the two outer turning points are minima and the middle one is a maximum.

现在计算导数：f′(x) = 4x³ − 8x = 4x(x² − 2)。令 f′(x) = 0 得 x = 0、x = √2 和 x = −√2。计算这些点的函数值：f(0) = 3，f(±√2) = 4 − 8 + 3 = −1。因此转向点为 (0, 3)、(√2, −1) 和 (−√2, −1)。由于 a > 0，外侧两个转向点为极小值点，中间的为极大值点。

11. Common Mistakes and Exam Tips | 常见错误与考试提示

One common mistake is assuming every quartic has three turning points. Always check the derivative equation first. Another error is confusing the number of real roots with the number of turning points; a quartic can have four real roots but only one turning point, as in f(x) = (x − 1)⁴. In exams, clearly state the end behaviour, label all intercepts, and mark the coordinates of every turning point and inflection point you find.

一个常见错误是假设每个四次函数都有三个转向点。务必先检查导数方程。另一个错误是混淆实数根的个数与转向点的个数；四次函数可以有四个实数根但只有一个转向点，例如 f(x) = (x − 1)⁴。在考试中，要清晰说明端部行为，标出所有截距，并标出你求得的每个转向点和拐点的坐标。

12. Summary | 总结

The graph of a quartic function is fully determined by its leading coefficient a, its derivative, and its roots. The sign of a controls the end behaviour; the number of real solutions of f′(x) = 0 controls the number of turning points; and the roots, with their multiplicities, control where the graph touches or crosses the x-axis. Mastering these features allows you to sketch any quartic quickly and accurately.

四次函数图像完全由首项系数 a、导数以及根决定。a 的符号控制端部行为；f′(x) = 0 的实数解个数控制转向点个数；根及其重数控制图像在 x 轴上接触或穿过的位置。掌握这些特征，你就能快速而准确地绘制任意四次函数的图像。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Quartic Function Graphs: Shapes and Key Features | 四次函数图像的形状与关键特征