Cubic Functions: Shape, Roots and Points of Inflection | 三次函数图像：形状、零点与拐点

📚 Cubic Functions: Shape, Roots and Points of Inflection | 三次函数图像：形状、零点与拐点

Cubic functions are among the most important polynomial functions studied in A-level mathematics. Their distinctive S-shaped curves, varying numbers of real roots, and the presence of inflection points make them a rich topic for both pure mathematics and applications. This article provides a systematic treatment of cubic function graphs, focusing on their shape, roots, and points of inflection.

三次函数是A-level数学中最重要的多项式函数之一。其独特的”S”形曲线、实数根数量的变化，以及拐点的存在，使其成为纯数学与应用数学中的丰富课题。本文系统讲解三次函数图像，重点围绕形状、零点与拐点展开。

1. General Form and Basic Shape | 一般形式与基本形状

A cubic function is a polynomial of degree three, which can be written in the general form f(x) = ax³ + bx² + cx + d, where a ≠ 0. The domain and range of any cubic function are both the set of all real numbers, meaning the graph extends infinitely in both the positive and negative x- and y-directions.

三次函数是三次多项式，其一般形式为 f(x) = ax³ + bx² + cx + d，其中 a ≠ 0。任何三次函数的定义域与值域均为全体实数，这意味着图像在x轴与y轴的正负方向上均无限延伸。

Unlike quadratic functions, which are symmetric about a vertical line, cubic functions do not possess a simple axis of symmetry in general. Instead, their graphs exhibit a characteristic “S” shape (or reversed “S”), with one end of the curve rising to positive infinity and the other falling to negative infinity.

与关于某条竖直线对称的二次函数不同，三次函数一般不具有简单的对称轴。相反，其图像呈现出特征性的”S”形（或反”S”形），曲线一端趋于正无穷，另一端趋于负无穷。

2. The Role of the Leading Coefficient a | 首项系数a的作用

The sign of the leading coefficient a determines the overall orientation of the cubic curve. When a > 0, as x → +∞, f(x) → +∞, and as x → −∞, f(x) → −∞. This produces a curve that rises from the bottom left to the top right, often described as a “positive cubic”.

首项系数a的正负决定了三次曲线的总体朝向。当 a > 0 时，随着 x → +∞，f(x) → +∞；随着 x → −∞，f(x) → −∞。这产生一条从左下方向右上方上升的曲线，通常称为”正三次曲线”。

When a < 0, the behaviour is reversed: as x → +∞, f(x) → −∞, and as x → −∞, f(x) → +∞. The curve then falls from top left to bottom right, forming a "negative cubic". This distinction is crucial when sketching graphs quickly and when analysing end behaviour in exam questions.

当 a < 0 时，趋势相反：随着 x → +∞，f(x) → −∞；随着 x → −∞，f(x) → +∞。曲线从左上方降至右下方，形成"负三次曲线"。这一区别在快速作图以及分析考试题中的端部行为时至关重要。

3. Derivatives and Critical Points | 导数与临界点

The first derivative of a cubic function is f′(x) = 3ax² + 2bx + c, which is a quadratic function. Setting f′(x) = 0 gives the x-coordinates of the stationary points (turning points) of the cubic. Since a quadratic equation has at most two real solutions, a cubic function has at most two stationary points.

三次函数的一阶导数为 f′(x) = 3ax² + 2bx + c，这是一个二次函数。令 f′(x) = 0，可得三次函数驻点（转折点）的横坐标。由于二次方程至多有两个实数解，因此三次函数至多有两个驻点。

The discriminant of this quadratic, namely Δ = (2b)² − 4(3a)(c) = 4b² − 12ac, determines the number of stationary points:

该二次方程的判别式为 Δ = (2b)² − 4(3a)(c) = 4b² − 12ac，它决定了驻点的数量：

判别式 Δ	驻点数量	图像特征
Δ > 0	两个驻点（一个极大值，一个极小值）	典型的波浪形”S”曲线
Δ = 0	一个驻点（水平拐点）	曲线在某点处切线水平但无局部极值
Δ < 0	无驻点	单调递增或单调递减，无转折

4. The Point of Inflection | 拐点

The second derivative of a cubic function is f″(x) = 6ax + 2b, which is a linear function. Setting f″(x) = 0 yields the unique solution x = −b/(3a). This x-coordinate gives the point of inflection, where the curvature of the graph changes sign.

三次函数的二阶导数为 f″(x) = 6ax + 2b，这是一个线性函数。令 f″(x) = 0，得唯一解 x = −b/(3a)。该横坐标对应拐点，即图像曲率改变符号的位置。

At the point of inflection, the graph changes from being concave down to concave up, or vice versa. The y-coordinate is obtained by substituting x = −b/(3a) back into f(x). Since a cubic’s second derivative is linear, it crosses zero exactly once, so every cubic function has exactly one point of inflection.

在拐点处，图像由向下凹变为向上凹，或反之。将 x = −b/(3a) 代回 f(x)，即可得到拐点的纵坐标。由于三次函数的二阶导数为线性函数，它恰好穿过零一次，因此每个三次函数都恰好有一个拐点。

The point of inflection is a central point of the cubic in a certain sense: it is the centre of symmetry when the cubic has a symmetric form relative to that point. Indeed, every cubic function is symmetric about its point of inflection in the sense that the graph is invariant under a 180° rotation about that point.

在某种意义上，拐点是三次函数的中心点：当三次函数关于该点具有对称形式时，它就是对称中心。事实上，每个三次函数都关于其拐点对称，即图像绕该点旋转180°后保持不变。

5. Symmetry of Cubic Curves | 三次曲线的对称性

The rotational symmetry of a cubic about its inflection point can be verified algebraically. Let the inflection point be (h, k), where h = −b/(3a). If we translate the axes so that the origin moves to (h, k), the translated function becomes a pure cubic of the form y = aX³ + mX, where X = x − h and m is a constant. This form is an odd function in X, confirming rotational symmetry.

三次函数关于其拐点的旋转对称性可以通过代数方法验证。设拐点为 (h, k)，其中 h = −b/(3a)。若将坐标轴平移，使原点移到 (h, k)，则平移后的函数化为纯三次形式 y = aX³ + mX，其中 X = x − h，m 为常数。该形式关于 X 是奇函数，从而证实了旋转对称性。

This symmetry property is often used to locate the inflection point quickly and to reason about the relative positions of roots. For example, if a cubic has three real roots that are equally spaced in some sense, the middle root lies at the inflection point only in special cases; generally, the inflection point lies at the average coordinate of the three roots when the cubic is written with equal spacing.

这种对称性常用于快速定位拐点，以及推理零点之间的相对位置。例如，若一个三次函数有三个实根，在某种含义下等距排列，则中间根仅在特殊情形下位于拐点；一般来说，当三根等距时，拐点位于三根横坐标的算术平均处。

6. Roots and the Factor Theorem | 零点与因式定理

A root of a cubic equation f(x) = 0 is an x-value at which the graph crosses or touches the x-axis. The Factor Theorem states that if f(r) = 0, then (x − r) is a factor of f(x). Thus, if a cubic has roots r₁, r₂, and r₃ (allowing repeated roots), the function can be factorised as f(x) = a(x − r₁)(x − r₂)(x − r₃).

三次方程 f(x) = 0 的零点是指图像与x轴相交或相切的横坐标。因式定理指出：若 f(r) = 0，则 (x − r) 是 f(x) 的一个因式。因此，若一个三次函数有根 r₁、r₂、r₃（允许重根），则该函数可分解为 f(x) = a(x − r₁)(x − r₂)(x − r₃)。

Every cubic equation with real coefficients has at least one real root, because as x → −∞ the function tends to ∓∞ and as x → +∞ it tends to ±∞, so by the Intermediate Value Theorem the graph must cross the x-axis at least once.

每个实系数三次方程至少有一个实根，因为当 x → −∞ 时函数趋于 ∓∞，而当 x → +∞ 时趋于 ±∞，根据介值定理，图像至少穿过x轴一次。

Multiplying out the factor form gives the useful relationships:

展开因式形式，可得如下重要关系：

r₁ + r₂ + r₃ = −b/a, r₁r₂ + r₂r₃ + r₃r₁ = c/a, r₁r₂r₃ = −d/a

These are particular cases of Vieta’s formulas, often used to solve problems relating roots to coefficients without explicit factorisation.

这些是韦达定理的特例，常用于在无需显式因式分解的情况下，建立根与系数之间的关系。

7. Number and Types of Real Roots | 实数零点的数量与类型

A cubic equation with real coefficients can have either one real root or three real roots (counting multiplicities). It cannot have exactly two real roots unless one of them is repeated, in which case it has a double root plus a simple root, giving three real roots in total when counted with multiplicity.

实系数三次方程要么有一个实根，要么有三个实根（按重数计）。它不可能恰好有两个不同的实根，除非其中一个为重根——此时它有一个二重根和一个单根，按重数计算仍为三个实根。

The three possible configurations of roots produce distinct shapes:

三种不同的零点配置会产生不同的图像形状：

Three distinct real roots: the graph crosses the x-axis at three separate points, producing two turning points between them. This occurs when the cubic has a local maximum and a local minimum on opposite sides of the x-axis.

三个互异实根：图像在三个不同的点穿过x轴，其间产生两个转折点。这发生在三次函数有一个局部最大值和一个局部最小值，且它们分别位于x轴的两侧时。
One real root and two complex roots: the graph crosses the x-axis only once, and the two stationary points (if they exist) are both entirely above or entirely below the x-axis.

一个实根与两个复根：图像仅穿过x轴一次，两个驻点（若存在）均完全位于x轴上方或完全位于x轴下方。
A double root and a simple root: the graph touches the x-axis at the double root (where it has a stationary point) and crosses at the simple root.

一个二重根与一个单根：图像在二重根处与x轴相切（该处为驻点），在单根处穿过x轴。
A triple root: f(x) = a(x − r)³, the graph crosses the x-axis with a horizontal tangent at r, producing no separate turning points.

一个三重根：f(x) = a(x − r)³，图像在r处以水平切线穿过x轴，没有独立的转折点。

8. Discriminant of a Cubic | 三次函数的判别式

For a cubic equation ax³ + bx² + cx + d = 0, the discriminant Δ₃ provides information about the nature of the roots. Its formula is:

对于三次方程 ax³ + bx² + cx + d = 0，判别式 Δ₃ 提供了关于根的性质的信息。其公式为：

Δ₃ = b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd

If Δ₃ > 0, the cubic has three distinct real roots; if Δ₃ = 0, it has repeated roots (a double root or a triple root); if Δ₃ < 0, it has one real root and two complex conjugate roots.

若 Δ₃ > 0，则三次方程有三个互异实根；若 Δ₃ = 0，则有重根（二重根或三重根）；若 Δ₃ < 0，则有一个实根和一对共轭复根。

While this discriminant is not always required at A-level, understanding it deepens one’s grasp of the relationship between the coefficients and the graph’s intersections with the x-axis.

虽然A-level不一定要求掌握此判别式，但理解它有助于加深对系数与图像x轴交点之间关系的认识。

9. Sketching Cubic Graphs | 绘制三次函数图像

To sketch a cubic graph quickly and accurately, follow a systematic sequence. First, determine the sign of a to establish the end behaviour. Second, find the y-intercept by evaluating f(0) = d. Third, factorise or use numerical methods to find the x-intercepts (roots). Fourth, differentiate to locate stationary points and determine their nature using the second derivative test.

要快速而准确地绘制三次函数图像，可按系统步骤进行。首先确定a的符号以判断端部趋势。其次，计算 f(0) = d 得到y轴截距。第三，通过因式分解或数值方法求x轴截距（零点）。第四，通过求导定位驻点，并利用二阶导数判断其性质。

Finally, locate the inflection point at x = −b/(3a) and confirm the overall shape. Plot the intercepts, turning points, and inflection point, then join them smoothly with a curve that respects the end behaviour and the concavity on either side of the inflection point.

最后，在 x = −b/(3a) 处定位拐点并确认整体形状。在坐标系中标出截距、转折点和拐点，然后以平滑曲线连接，确保符合端部趋势以及拐点两侧的凹凸性。

When the cubic cannot be factorised easily, one can use a numerical method such as the Newton–Raphson method to approximate a real root, then divide through by the corresponding linear factor to obtain a quadratic for the remaining roots.

当三次函数不易因式分解时，可用牛顿-拉弗森法等数值方法近似求出一个实根，再除以相应的线性因式得到二次方程，进而求出其余根。

10. Worked Example 1: Three Distinct Roots | 例1：三个互异实根

Consider f(x) = x³ − 3x² − x + 3. Factorise: f(x) = x²(x − 3) − 1(x − 3) = (x² − 1)(x − 3) = (x − 1)(x + 1)(x − 3). Hence the roots are x = −1, 1, 3, all distinct and real.

考察 f(x) = x³ − 3x² − x + 3。因式分解：f(x) = x²(x − 3) − 1(x − 3) = (x² − 1)(x − 3) = (x − 1)(x + 1)(x − 3)。因此零点为 x = −1、1、3，均为互异实根。

The derivative is f′(x) = 3x² − 6x − 1. Setting it to zero gives the stationary points x = 1 ± (2√3)/3. Substituting these back into f(x) gives the local maximum at approximately (−0.155, 3.079) and the local minimum at approximately (2.155, −3.079). Observe that the y-coordinates are opposite in sign and equal in magnitude, reflecting the fact that the inflection point lies at (1, 0), the midpoint of the turning points and the average of the roots.

其导数为 f′(x) = 3x² − 6x − 1。令其为零，得驻点 x = 1 ± (2√3)/3。代回 f(x) 得局部极大值约为 (−0.155, 3.079)，局部极小值约为 (2.155, −3.079)。注意，两个纵坐标互为相反数且绝对值相等，这反映了拐点位于 (1, 0)，即转折点连线的中点，也是三个零点横坐标的平均值。

11. Worked Example 2: One Real Root | 例2：一个实根

Consider f(x) = x³ − 3x + 3. The derivative is f′(x) = 3x² − 3 = 3(x − 1)(x + 1), giving stationary points at x = ±1. Evaluating f(1) = 1 − 3 + 3 = 1 > 0 and f(−1) = −1 + 3 + 3 = 5 > 0, both stationary points lie above the x-axis.

考察 f(x) = x³ − 3x + 3。其导数为 f′(x) = 3x² − 3 = 3(x − 1)(x + 1)，驻点为 x = ±1。计算 f(1) = 1 − 3 + 3 = 1 > 0，f(−1) = −1 + 3 + 3 = 5 > 0，两个驻点均位于x轴上方。

Since both turning points are above the x-axis and the curve descends to −∞ as x → −∞, the graph crosses the x-axis exactly once, to the left of x = −1. Numerically, this root is approximately x ≈ −2.532, confirming that the cubic has one real root and a pair of complex conjugate roots.

由于两个转折点均在x轴上方，且曲线在 x → −∞ 时降至 −∞，图像在 x = −1 左侧恰好穿过x轴一次。数值计算表明该根约为 x ≈ −2.532，从而证实该三次函数有一个实根和一对共轭复根。

12. Summary of Key Properties | 关键性质总结

Every cubic function f(x) = ax³ + bx² + cx + d has exactly one inflection point, located at x = −b/(3a). It has at most two stationary points, determined by the sign of Δ = 4b² − 12ac. The graph crosses the x-axis at least once, with either one distinct root or three distinct roots; repeated roots produce tangency instead of crossing at certain points.

每个三次函数 f(x) = ax³ + bx² + cx + d 都恰好有一个拐点，位于 x = −b/(3a)。它至多有两个驻点，由 Δ = 4b² − 12ac 的符号决定。图像至少穿过x轴一次，要么有一个不同实根，要么有三个不同实根；重根在特定点处产生相切而非穿越。

When sketching cubic graphs, always check the end behaviour from the sign of a, mark the y-intercept, find all real roots, locate stationary points and the inflection point, and finally draw a smooth curve that respects concavity and symmetry.

绘制三次函数图像时，务必先从a的符号判断端部趋势，标出y轴截距，求出所有实根，定位驻点与拐点，最后绘制一条符合凹凸性与对称性的平滑曲线。

Mastering these properties enables students to handle a wide range of examination questions, from factorisation and sketching to optimisation and applied modelling problems involving cubic relationships.

掌握这些性质，学生即可从容应对各类考试题型，从因式分解与作图，到最优化问题以及涉及三次关系的应用建模题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Cubic Functions: Shape, Roots and Points of Inflection | 三次函数图像：形状、零点与拐点