📚 Cubic Functions: Graphs and Properties | 三次函数图像与性质
A cubic function is a polynomial of degree three, expressed generally as f(x) = ax³ + bx² + cx + d, where a ≠ 0. It is one of the most frequently tested topics in A-Level Mathematics, particularly in the Pure Mathematics component, where students are expected to analyse its graph, locate stationary points and understand the discriminant conditions that determine the number of real roots.
三次函数是最高次数为三的多项式函数，其一般形式为 f(x) = ax³ + bx² + cx + d，其中 a ≠ 0。这是 A-Level 数学，尤其是纯数学部分的高频考点。考生需要掌握其图像特征、驻点求解以及判别式条件，从而判断函数实根的个数。
1. Standard Form and End Behavior | 标准形式与端行为
The standard form of a cubic function is f(x) = ax³ + bx² + cx + d. The leading coefficient a determines the end behavior of the graph. When a > 0, as x → +∞, f(x) → +∞, and as x → −∞, f(x) → −∞. This gives a rising curve from lower-left to upper-right. When a < 0, the direction reverses: the graph rises from upper-left and falls to lower-right.
三次函数的标准形式为 f(x) = ax³ + bx² + cx + d。首项系数 a 决定了图像的端行为。当 a > 0 时，x 趋向正无穷，f(x) 趋向正无穷；x 趋向负无穷，f(x) 趋向负无穷，图像从左下方延伸到右上方。当 a < 0 时，方向相反，图像从左上方向右下方延伸。
Unlike quadratic functions, a cubic always has at least one real root. This is guaranteed by the Intermediate Value Theorem, as the function takes both positive and negative values for sufficiently large and small x.
与二次函数不同，三次函数至少有一个实根。这由介值定理保证——由于函数在 x 足够大和足够小时分别取正值和负值，其图像必然穿过 x 轴至少一次。
2. First Derivative and Stationary Points | 一阶导数与驻点
To locate stationary points, we differentiate: f'(x) = 3ax² + 2bx + c. Setting f'(x) = 0 gives a quadratic equation whose discriminant Δ = (2b)² − 4(3a)(c) = 4b² − 12ac = 4(b² − 3ac). Since the factor 4 does not affect the sign, the reduced discriminant Δ’ = b² − 3ac determines the number of stationary points.
为了求驻点，对 f(x) 求导：f'(x) = 3ax² + 2bx + c。令 f'(x) = 0 得到一个二次方程，其判别式为 Δ = (2b)² − 4(3a)(c) = 4b² − 12ac = 4(b² − 3ac)。由于因子 4 不影响符号，简化后的判别式 Δ’ = b² − 3ac 就决定了驻点的数量。
Δ’ = b² − 3ac
If Δ’ > 0, the equation f'(x) = 0 has two distinct real solutions, so the cubic has one local maximum and one local minimum. If Δ’ = 0, the two stationary points coincide, creating a stationary point of inflection. If Δ’ < 0, there are no real stationary points, and the graph is strictly monotonic.
若 Δ’ > 0，方程 f'(x) = 0 有两个不同实根，三次函数有一个极大值点和一个极小值点。若 Δ’ = 0，两个驻点重合为一个拐点。若 Δ’ < 0，则不存在实驻点，图像严格单调。
3. Second Derivative and Point of Inflection | 二阶导数与拐点
The second derivative is f”(x) = 6ax + 2b. Setting f”(x) = 0 gives x = −b/(3a), the unique point of inflection. At this point, the curvature of the graph changes from concave down to concave up (or vice versa), and the graph is symmetric about this point for the cubic function.
三次函数的二阶导数为 f”(x) = 6ax + 2b。令 f”(x) = 0，得 x = −b/(3a)，这是唯一的拐点。在拐点处，图像的凹凸性发生改变（由凹向上变为凹向下，或反之），且三次函数的图像关于该拐点对称。
x = −b/(3a)
To determine whether a stationary point is a maximum or a minimum, substitute the x-coordinate into f”(x). If f”(x) < 0, it is a local maximum; if f''(x) > 0, it is a local minimum. For an inflection point involving a stationary point (Δ’ = 0), f”(x) = 0 as well.
要判断驻点是极大值还是极小值，可将驻点 x 坐标代入 f”(x)。若 f”(x) < 0，则为极大值；若 f''(x) > 0，则为极小值。当 Δ’ = 0 时，驻点与拐点重合，此时 f”(x) 也为零。
4. The Discriminant and Number of Real Roots | 判别式与实根个数
For a cubic equation ax³ + bx² + cx + d = 0, the number of real roots depends on the relative positions of the local maximum and minimum relative to the x-axis. The cubic discriminant is given by Δ₃ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d². When Δ₃ > 0, the equation has three distinct real roots; when Δ₃ = 0, there is a repeated root; when Δ₃ < 0, there is exactly one real root.
对于三次方程 ax³ + bx² + cx + d = 0，实根的个数取决于极大值与极小值相对于 x 轴的位置。三次方程的判别式为 Δ₃ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d²。当 Δ₃ > 0 时，方程有三个互不相同的实根；当 Δ₃ = 0 时，存在重根；当 Δ₃ < 0 时，只有一个实根。
In A-Level exams, students are not usually required to memorise this formula. Instead, they are expected to use calculus to analyse the graph. A common method is to find the local maximum and minimum values; if they have opposite signs, there are three real roots; if the maximum or minimum equals zero, there is a repeated root; otherwise, one real root exists.
在 A-Level 考试中，考生通常不需要记忆这个复杂公式，而是通过微积分分析图形。常用的方法是求出极大值与极小值；若二者异号，则存在三个实根；若极大值或极小值恰为零，则存在重根；否则只有一个实根。
5. Graph Shapes: Three Cases | 三种图像类型
When b² − 3ac > 0 and the maximum and minimum values lie on opposite sides of the x-axis, the graph crosses the x-axis three times. This is the classic S-shaped cubic curve with two turning points. It appears when the cubic has three distinct real roots.
当 b² − 3ac > 0 且极大值与极小值分别位于 x 轴两侧时，图像与 x 轴相交三次。这就是经典的 S 形三次曲线，具有两个转折点，表示该三次方程有三个不同的实根。
When the local maximum or minimum just touches the x-axis, the graph intersects it at two points, one of which is a repeated root (tangent to the axis). In this case, the equation is of the form f(x) = (x − α)²(x − β).
当极大值或极小值恰好与 x 轴相切时，图像与 x 轴有两个交点，其一为重根（切点）。此时三次方程可以写成 f(x) = (x − α)²(x − β) 的形式。
When b² − 3ac < 0, the cubic has no stationary points and increases or decreases monotonically throughout. The graph crosses the x-axis exactly once, corresponding to one real root. Additionally, even when Δ' > 0, if both extrema lie entirely above or below the x-axis, the curve still crosses the axis only once — the so-called one-root, two-turning-point shape.
当 b² − 3ac < 0 时，三次函数没有驻点，整个函数单调递增或递减，图像与 x 轴仅有一个交点。即使 Δ' > 0，若两个极值均位于 x 轴同侧（全部为正或全部为负），图像也仅与 x 轴相交一次，这种类型称为”单根双驻点”形态。
6. Roots and Coefficients: Vieta’s Formulas | 根与系数：韦达定理
Suppose the roots of ax³ + bx² + cx + d = 0 are α, β and γ. Then the following relations hold: α + β + γ = −b/a, αβ + βγ + γα = c/a, and αβγ = −d/a. These are Vieta’s formulas for cubic equations and are extremely useful for solving problems where the roots are related in a certain way.
设三次方程 ax³ + bx² + cx + d = 0 的三个根为 α、β、γ，则有以下关系成立：α + β + γ = −b/a，αβ + βγ + γα = c/a，以及 αβγ = −d/a。这就是三次方程的韦达定理，在已知根之间关系的问题中非常有用。
For example, if the roots are consecutive integers, we can set β = α + 1, γ = α + 2 and use the sum formula to solve for α. This technique is widely used in polynomial-related questions in A-Level papers.
例如，若三个根为连续整数，可设 β = α + 1，γ = α + 2，再结合求和公式解出 α。这类技巧在 A-Level 的多项式考题中十分常见。
7. Factor Theorem and Synthetic Division | 因式定理与综合除法
The Factor Theorem states that (x − k) is a factor of f(x) if and only if f(k) = 0. For a cubic equation, once one root is found by trial or inspection, the remaining quadratic factor can be obtained through algebraic division or synthetic division, after which the other roots can be solved using the quadratic formula.
因式定理指出：若 f(k) = 0，则 x − k 是 f(x) 的因式，反之亦然。求解三次方程时，先通过试根或观察找到一个根，然后利用长除法或综合除法得到剩下的二次因式，再用二次公式求出其余根。
When a cubic has rational roots, possible candidates are of the form p/q, where p divides d and q divides a (the Rational Root Theorem). In A-Level questions, the first root is almost always an integer, often from the set {±1, ±2, ±3}.
若三次函数有有理根，则满足 p/q 的形式，其中 p 整除常数项 d，q 整除首项系数 a（有理根定理）。在 A-Level 的考题中，第一个根往往是整数，常见于 {±1, ±2, ±3} 等数值。
8. Transformations of Cubic Graphs | 三次函数图像的变换
Similar to quadratic functions, cubic graphs can be transformed through translations, reflections and stretches. The general vertex form for a cubic with a repeated root is y = a(x − h)³ + k, where (h, k) is the point of inflection. Changing h shifts the graph left or right, and changing k shifts it up or down.
与二次函数类似，三次函数图像也可以进行平移、反射和伸缩变换。三次函数的顶点形式可以表示为 y = a(x − h)³ + k，其中 (h, k) 为拐点坐标。调整 h 可使图像左右平移，调整 k 则使图像上下移动。
A negative value of a reflects the graph in the x-axis. Replacing x by x − h gives a horizontal shift of h units, while replacing y by y − k gives a vertical shift of k units. A horizontal stretch by a factor of 1/q is obtained by replacing x with qx.
当 a 为负数时，图像关于 x 轴翻转。将 x 替换为 x − h 可实现水平平移 h 个单位；将 y 替换为 y − k 可实现垂直平移 k 个单位。将 x 替换为 qx 则可实现水平压缩或拉伸（压缩系数为 1/q）。
9. Solving Cubic Inequalities | 三次不等式求解
Once the roots are known, solving a cubic inequality such as f(x) > 0 or f(x) < 0 follows the same sign-analysis method used for quadratics. Sketching the graph is usually the quickest route: mark the x-intercepts and note the sign of the leading coefficient to determine where the function is positive or negative.
在求出实根之后，解三次不等式（如 f(x) > 0 或 f(x) < 0）与二次不等式的符号分析法一致。先画草图通常是最快的方法：标注 x 轴交点，再由首项系数的符号确定各区间内函数的正负。
For a cubic with three distinct roots α < β < γ, if a > 0, the function is negative on (−∞, α), positive on (α, β), negative on (β, γ), and positive on (γ, +∞). This alternating pattern directly yields the solution interval.
对于有三个根 α < β < γ 且 a > 0 的三次函数，其函数值在 (−∞, α) 为负，在 (α, β) 为正，在 (β, γ) 为负，在 (γ, +∞) 为正。这种正负交替的模式可以直接给出不等式的解集。
10. Applications in Real-World Problems | 实际应用问题
Cubic functions appear in physical and geometric contexts. In volume problems, an open-top box constructed by cutting squares of side x from a rectangular sheet has a volume expressed as a cubic function of x. Optimising the volume requires finding the stationary point of the cubic, a classic application of differentiation.
三次函数广泛出现在物理与几何背景中。例如，在无盖盒子的体积问题里，从长方形纸板的四角剪去边长为 x 的小正方形后，盒子的体积是 x 的三次函数。最大体积的求解需要找到三次函数的驻点，这是微分法的经典应用。
Cubic models also appear in economics for cost and revenue functions, in kinematics for displacement-time relationships, and in biology for population growth under limited resources. In A-Level exam questions, the cubic model is usually given with a meaningful domain restriction.
三次模型还出现在经济学的成本与收益函数、运动学中位移与时间的关系，以及生物学中资源受限的人口增长等问题中。A-Level 考题中的三次模型通常会附带一个有实际意义的定义域限制。
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