Cubic Graphs: Shapes and Key Points | 三次函数图像：形状与关键点

📚 Cubic Graphs: Shapes and Key Points | 三次函数图像：形状与关键点

In A-Level mathematics, understanding the graph of a cubic function is essential for curve sketching, solving equations and applying calculus. A cubic graph has a distinctive S-shaped curve whose features are determined by the coefficients and factors of the function.

在 A-Level 数学中，理解三次函数图像是绘制曲线、解方程以及运用微积分的基础。三次函数图像具有独特的 S 形曲线，其特征由函数的系数和因式决定。

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1. Definition and General Form | 定义与一般形式

A cubic function is a polynomial of degree 3, written in the general form

三次函数是次数为 3 的多项式，其一般形式为

f(x) = ax³ + bx² + cx + d, where a ≠ 0

The degree being 3 means that the highest power of x is x³. The coefficients a, b, c and d are real constants, and a must not be zero.

次数为 3 意味着 x 的最高次幂是 x³。系数 a、b、c、d 都是实数常数，且 a 不能为零。

Coefficient	Role / 作用
a	Leading coefficient; controls end behaviour and vertical stretch / 首项系数，控制端部行为与纵向伸缩
b, c	Affect the positions of stationary points and the point of inflection / 影响驻点和拐点的位置
d	Gives the y-intercept / 给出 y 轴截距

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2. End Behaviour: The Role of the Leading Coefficient | 端部行为：首项系数的作用

The end behaviour of a cubic graph is controlled only by the sign of the leading coefficient a. Other terms become insignificant when x is very large or very small.

三次函数图像的端部行为只由首项系数 a 的符号决定。当 x 非常大或非常小时，其他项的影响变得可以忽略。

If a > 0: as x → −∞, f(x) → −∞; as x → ∞, f(x) → ∞.

The positive cubic starts low on the left and ends high on the right. It rises upwards overall from left to right.

首项系数为正的三次函数，图像左侧很低、右侧很高，整体从左向右上升。

If a < 0: as x → −∞, f(x) → ∞; as x → ∞, f(x) → −∞.

The negative cubic starts high on the left and ends low on the right. It falls downwards overall from left to right.

首项系数为负的三次函数，图像左侧很高、右侧很低，整体从左向右下降。

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3. Roots and the x-axis | 根与 x 轴

A cubic equation can have at most three real roots, and it must have at least one real root. If a real number r satisfies f(r) = 0, then (x − r) is a factor of f(x). This is the factor theorem.

三次方程最多有三个实根，且至少有一个实根。如果实数 r 满足 f(r) = 0，则 (x − r) 是 f(x) 的一个因式，这就是因式定理。

• If a cubic has three distinct linear factors, the graph crosses the x-axis at three distinct points.

  如果三次函数能分解为三个不同的线性因式，则图像在三个不同点穿过 x 轴。

• If one factor is repeated, the graph touches the x-axis at the repeated root and crosses at the other simple root.

  如果某个因式重复出现，则图像在重根处与 x 轴相切，并在另一个单根处穿过 x 轴。

• If the cubic cannot be fully factorised into real linear factors, it may cross the x-axis only once.

  如果三次函数不能完全分解为实线性因式，则它可能只穿过 x 轴一次。

Pattern	Behaviour / 行为
(x − 1)(x − 2)(x − 3)	Crosses at x = 1, 2, 3 / 在 x = 1、2、3 处穿过
(x − 1)²(x + 2)	Touches at x = 1, crosses at x = −2 / 在 x = 1 处相切，x = −2 处穿过
(x − 1)(x² + x + 1)	Crosses once at x = 1 / 只在 x = 1 处穿过

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4. The y-intercept | y 轴截距

The y-intercept is found by setting x = 0. Since f(0) = d, the graph always passes through (0, d).

y 轴截距通过令 x = 0 得到。因为 f(0) = d，所以图像必定经过点 (0, d)。

For example, the graph of f(x) = x³ − 3x² + 2 has a y-intercept at (0, 2). This point is often the easiest point to plot accurately.

例如，函数 f(x) = x³ − 3x² + 2 的图像在 (0, 2) 处有 y 轴截距。这个点通常是最容易准确描出的点。

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5. Repeated Roots: Touching or Crossing | 重根：相切还是穿过

The multiplicity of a root determines how the curve behaves at that root.

根的重数决定了曲线在该根处的行为。

• Double root: the curve touches the x-axis at that point and does not cross. The tangent at the point is horizontal.

  二重根：曲线在该点与 x 轴相切但不过去，切线为水平切线。

• Triple root: the curve passes through the x-axis with a horizontal tangent and a point of inflection at that root.

  三重根：曲线以水平切线和拐点的方式穿过 x 轴。

For the function y = (x − 1)²(x + 2), x = 1 is a double root. The graph touches the x-axis at (1, 0). It crosses the axis at x = −2 because that root has multiplicity 1.

对于函数 y = (x − 1)²(x + 2)，x = 1 是二重根。图像在 (1, 0) 处与 x 轴相切，而在 x = −2 处穿过 x 轴，因为该根的重数为 1。

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6. Stationary Points and Point of Inflection | 驻点与拐点

Stationary points occur where the first derivative is zero. For a cubic function,

驻点出现在一阶导数为零处。对于三次函数，

f'(x) = 3ax² + 2bx + c = 0

This is a quadratic equation in x. Its discriminant is

这是关于 x 的二次方程，其判别式为

Δ = (2b)² − 4(3a)c = 4(b² − 3ac)

• If Δ > 0, the cubic has two distinct stationary points: one local maximum and one local minimum.

  若 Δ > 0，三次函数有两个不同的驻点：一个局部最大值和一个局部最小值。

• If Δ = 0, the cubic has exactly one stationary point, which is also a stationary point of inflection.

  若 Δ = 0，三次函数只有一个驻点，该点同时也是水平拐点。

• If Δ < 0, the cubic has no stationary points and is strictly increasing or decreasing.

  若 Δ < 0，三次函数没有驻点，函数严格递增或严格递减。

Every cubic also has exactly one point of inflection, found by solving f”(x) = 0.

每个三次函数还恰好有一个拐点，可通过解 f”(x) = 0 得到。

f”(x) = 6ax +

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