Cubic Graphs | 三次函数图像

📚 Cubic Graphs | 三次函数图像

A cubic graph is the graph of a polynomial of degree three. For Edexcel A-Level Mathematics, you need to be able to sketch these graphs, interpret repeated roots, use differentiation to find stationary points, and understand transformations.

三次函数图像是指三次多项式的图像。在 Edexcel A-Level 数学中，你需要能够绘制这些图像、解释重根、利用微分求驻点，并理解图像变换。

1. What Is a Cubic Graph? | 什么是三次函数图像？

A cubic function has the general form f(x) = ax³ + bx² + cx + d, where a ≠ 0. Because a is non-zero, the highest power of x is 3.

三次函数的一般形式为 f(x) = ax³ + bx² + cx + d，其中 a ≠ 0。由于 a 不为零，x 的最高次数是 3。

The graph is a smooth, continuous curve with no gaps, jumps or sharp corners. Its domain and range are both the set of all real numbers.

图像是一条光滑、连续的曲线，没有间断、跳跃或尖角。其定义域和值域都是全体实数。

Every cubic equation has at least one real root, so every cubic graph crosses or touches the x-axis at least once.

每个三次方程至少有一个实数根，因此每条三次函数图像至少与 x 轴相交或相切一次。

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2. The Leading Coefficient and End Behaviour | 首项系数与图像远端走向

The sign of the leading coefficient a determines the end behaviour of a cubic graph.

首项系数 a 的符号决定了三次函数图像在左右两端的走向。

If a > 0, then as x → ∞, y → ∞, and as x → −∞, y → −∞. The graph runs from bottom-left to top-right.

若 a > 0，则当 x → ∞ 时，y → ∞；当 x → −∞ 时，y → −∞。图像从左下方向右上方延伸。

If a < 0, then as x → ∞, y → −∞, and as x → −∞, y → ∞. The graph runs from top-left to bottom-right.

若 a < 0，则当 x → ∞ 时，y → −∞；当 x → −∞ 时，y → ∞。图像从左上方向右下方延伸。

Always check the sign of a before drawing the two ends of a cubic.

在绘制三次函数图像的两端之前，一定要先检查 a 的符号。

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3. Roots and x-Intercepts | 根与 x 轴截距

The roots of the cubic equation ax³ + bx² + cx + d = 0 are the x-coordinates where the graph cuts or touches the x-axis.

三次方程 ax³ + bx² + cx + d = 0 的根就是图像与 x 轴相交或相切时的横坐标。

A cubic can have one, two or three distinct real roots, but it cannot have more than three.

三次函数可以有一个、两个或三个不同的实数根，但不可能超过三个。

If the cubic is factorised as y = a(x − p)(x − q)(x − r), then the roots are p, q and r.

如果三次函数因式分解为 y = a(x − p)(x − q)(x − r)，那么根就是 p、q 和 r。

A single root, such as x = p, makes the graph cross the x-axis at that point.

单根（例如 x = p）会使图像在该点穿过 x 轴。

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4. Repeated Roots and Their Graphical Effect | 重根及其图像表现

If a factor is repeated, the behaviour at that root changes. A squared factor (x − p)² gives a repeated root where the graph touches the x-axis and turns back, without crossing.

如果某个因式重复出现，该根处的图像行为会改变。平方因式 (x − p)² 产生一个重根，图像在该点接触 x 轴并折返，但不穿过。

A cubed factor (x − p)³ gives a triple root. The graph crosses the x-axis at p, but it has a stationary point of inflection there and flattens as it passes through.

立方因式 (x − p)³ 产生一个三重根。图像在 p 处穿过 x 轴，但在那里有一个水平拐点，通过时图像变得平缓。

In general, an even-multiplicity root causes a touch, while an odd-multiplicity root causes a crossing.

一般来说，偶数重根导致图像与 x 轴相切，奇数重根导致图像穿过 x 轴。

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5. The y-Intercept | y 轴截距

The y-intercept is found by substituting x = 0 into the cubic equation.

将 x = 0 代入三次方程即可求得 y 轴截距。

For f(x) = ax³ + bx² + cx + d, the y-intercept is simply (0, d).

对于 f(x) = ax³ + bx² + cx + d，y 轴截距就是 (0, d)。

Always label this point clearly on a sketch, and check that it is consistent with the constant term after expanding.

在草图上务必清晰标出这一点，并检查展开后它是否与常数项一致。

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6. Factorised Form and Sketching | 因式分解形式与草图绘制

To sketch a cubic, first factorise the expression fully. This gives the x-intercepts and reveals whether each root is single or repeated.

要绘制三次函数草图，首先将表达式完全因式分解。这能给出 x 轴截距，并显示每个根是单根还是重根。

Next, determine the sign of a to draw the two end branches correctly.

接下来，确定 a 的符号，以正确绘制图像的两端走向。

Plot the roots and the y-intercept, then join these points with a smooth curve that follows the end behaviour and the touch-or-cross rule at each root.

标出根和 y 轴截距，然后用光滑曲线连接这些点，并使其满足两端走向以及每个根处的相切或穿过规则。

For example, y = (x + 1)(x − 2)² has a single root at x = −1 and a repeated root at x = 2. It crosses at −1 and touches at 2.

例如，y = (x + 1)(x − 2)² 在 x = −1 处有一个单根，在 x = 2 处有一个重根。它在 −1 处穿过 x 轴，在 2 处接触 x 轴。

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7. Stationary Points and Turning Points | 驻点与转折点

Stationary points occur where the derivative is zero, so solve f'(x) = 0.

驻点出现在导数为零的位置，因此需要解方程 f'(x) = 0。

For f(x) = ax³ + bx² + cx + d, the derivative is f'(x) = 3ax² + 2bx + c, which is a quadratic.

对于 f(x) = ax³ + bx² + cx + d，导数为 f'(x) = 3ax² + 2bx + c，这是一个二次函数。

A cubic can have two stationary points, one stationary point, or no stationary points, depending on the discriminant of the derivative quadratic.

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