Cubic Graphs | 三次函数图像

📚 Cubic Graphs | 三次函数图像

A cubic graph is the graph of a cubic function, a polynomial of degree 3. In A-Level Mathematics, especially within the Edexcel specification, you are required to recognise, sketch, and interpret cubic graphs, using factorisation to find x-intercepts and calculus to locate stationary points. Understanding the key features of cubic graphs will also help you to solve equations and inequalities involving cubic polynomials.

三次函数图像是指三次多项式函数的图形。在 A-Level 数学（尤其是 Edexcel 考试大纲）中，你需要识别、绘制并解释三次函数图像，利用因式分解求 x 轴截距，并使用微积分求驻点。理解三次函数图像的主要特征也将帮助你解决涉及三次多项式的方程与不等式。

1. Introduction to Cubic Functions | 三次函数简介

A cubic function is any function of the form f(x) = ax³ + bx² + cx + d, where a, b, c and d are constants and a ≠ 0. The term with x³ determines the fundamental shape and end behaviour of the graph. Cubic functions are continuous and smooth, with no breaks or sharp corners.

三次函数是指形如 f(x) = ax³ + bx² + cx + d 的函数，其中 a、b、c、d 为常数且 a ≠ 0。含 x³ 的项决定了图形的基本形状和远端走向。三次函数是连续且光滑的，没有间断点或尖角。

2. General Shape and End Behaviour | 一般形状与末端行为

The leading coefficient a controls the global direction. If a > 0, the graph falls to the left and rises to the right (like y = x³). If a < 0, the graph rises to the left and falls to the right (like y = −x³). In both cases the curve can have up to two turning points, but it may also have none (e.g. y = x³ has a stationary point of inflection).

首项系数 a 控制整体走向。若 a > 0，图像左端下降、右端上升（类似于 y = x³）。若 a < 0，图像左端上升、右端下降（类似于 y = −x³）。两种情况下曲线最多可有两个转折点，也可能没有（例如 y = x³ 只有一个拐点型的驻点）。

The extreme ends (as x → ±∞) are dominated by the x³ term. Therefore the long-run behaviour is simply: for large |x|, the graph behaves like ax³.

当 x → ±∞ 时，图形主要由 x³ 项主导。因此远端行为很简单：当 |x| 很大时，图形近似于 ax³。

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3. Roots and Factors | 根与因式

A cubic equation ax³ + bx² + cx + d = 0 can have up to three real roots (x-intercepts). It must have at least one real root because a cubic graph crosses the x-axis at least once. If the polynomial can be factorised, the roots are easy to read: each linear factor gives an x-intercept.

三次方程 ax³ + bx² + cx + d = 0 最多可有三个实根（即 x 轴截距）。由于三次函数图像至少穿过 x 轴一次，因此至少有一个实根。若多项式可因式分解，根就容易读出：每个一次因式对应一个 x 轴截距。

Factor theorem is often used to find the first root by testing small integer values. Once one factor (x − p) is known, polynomial division reduces the cubic to a quadratic, which can then be solved or factorised further.

通常使用因式定理，通过试小整数来找到第一个根。一旦确定了因式 (x − p)，利用多项式除法可将三次式降为二次式，再进一步求解或因式分解。

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4. Sketching Cubic Graphs from Factorised Form | 从因式分解形式绘制三次函数图像

When a cubic is fully factorised as y = a(x − p)(x − q)(x − r), the x-intercepts are p, q and r. The leading coefficient a tells you the overall shape. If a > 0, the graph starts below the x-axis on the left and finishes above on the right, crossing the axis at each distinct root.

当三次函数完全因式分解为 y = a(x − p)(x − q)(x − r) 时，x 轴截距为 p、q 和 r。首项系数 a 给出整体形状。若 a > 0，图像在左端位于 x 轴下方，右端位于 x 轴上方，并在每个不同的根处穿过 x 轴。

To sketch: (1) Identify the roots and mark them on the x-axis. (2) Compute the y-intercept by setting x = 0. (3) Check the sign of a to determine the start and end quadrants. (4) Join the points smoothly, making sure the curve crosses at simple roots and touches at repeated roots (discussed later).

绘图步骤：(1) 找出根并在 x 轴上标出；(2) 计算 y 轴截距 (令 x = 0)；(3) 检查 a 的正负以确定起始和结束象限；(4) 用平滑曲线连接各点，注意在单根处穿过 x 轴，在重根处接触 x 轴（稍后讨论）。

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5. Finding the y-intercept | 求y轴截距

The y-intercept of any graph is found by substituting x = 0 into the function. For a cubic y = ax³ + bx² + cx + d, this simply gives y = d. This point is (0, d) and is always easy to plot. In factorised form multiply the constant terms: y = a(0 − p)(0 − q)(0 − r) = −a p q r.

任何函数图像的 y 轴截距都可以通过代入 x = 0 求得。对于三次函数 y = ax³ + bx² + cx + d，直接得到 y = d。该点坐标为 (0, d)，总是很容易标出。若为因式分解形式，则常数为 y = a(0 − p)(0 − q)(0 − r) = −a p q r。

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6. Repeated Roots and Their Effect on the Graph | 重复根及其对图像的影响

If a linear factor appears twice, e.g. y = (x − p)²(x − q), the root x = p is a repeated root. Here the graph touches the x-axis at p and turns, rather than crossing. This creates a turning point exactly on the x-axis. If the factor appears three times, y = (x − p)³, the curve has a stationary point of inflection at x = p, where it crosses the axis and the gradient is zero.

若某个一次因式出现两次，例如 y = (x − p)²(x − q)，则根 x = p 为重复根。此时图像在 p 点接触 x 轴并折返而不穿过，在此形成一个恰好在 x 轴上的转折点。若因式出现三次，即 y = (x − p)³，则曲线在 x = p 处有一个拐点型驻点，既穿过 x 轴，切线又呈水平。

Repeated roots are very common in exam questions. Always check whether the graph just touches or crosses at a given intercept. A squared factor means touching; a single factor means crossing; a cubed factor means crossing with zero gradient.

重复根在考题中非常常见。务必检查图形在给定截距处是接触还是穿过。平方因式意味着接触；一次因式意味着穿过；立方因式意味着以零梯度穿过。

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7. Turning Points and Points of Inflection | 转折点与拐点

A cubic graph can have either two turning points (one local maximum and one local minimum) or just one stationary point of inflection, depending on the discriminant of the derivative. For example, y = x³ has a stationary point of inflection at (0, 0), where the curve changes concavity but keeps increasing.

三次函数图像可以有两个转折点（一个极大值和一个极小值），也可以只有一个拐点型驻点，这取决于导数的判别式。例如 y = x³ 在原点 (0,0) 有一个拐点型驻点，此处凹凸性改变，但函数整体保持递增。

If the derivative 3ax² + 2bx + c has two distinct real roots, the cubic has a local maximum and a local minimum. If the derivative has a repeated root (or no real roots), the cubic has a non‑turning stationary point of inflection or is strictly monotonic, though in the latter case it still has an inflection point where concavity changes.

若导数 3ax² + 2bx + c 有两个相异实根，则三次函数有一个局部极大值和一个局部极小值。若导数有重根（或没有实根），则三次函数有一个非极值的拐点型驻点，或者曲线严格单调，但即便如此仍存在凹凸性改变的拐点。

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8. Using Calculus to Find Stationary Points | 用微积分求驻点

To locate stationary points on a cubic y = f(x), find f'(x) = 3ax² + 2bx + c and set it to zero. Solve 3ax² + 2bx + c = 0. The solutions give the x‑coordinates of stationary points. Substitute these back into f(x) to get the corresponding y‑coordinates.

要求三次函数 y = f(x) 的驻点，先求导 f'(x) = 3ax² + 2bx + c 并令其为零。解方程 3ax² + 2bx + c = 0，所得解即为驻点的 x 坐标。再代回 f(x) 求得对应的 y 坐标。

Classify each stationary point using the second derivative f”(x) = 6ax + 2b. If f”(x) > 0, the point is a local minimum; if f”(x) < 0, it is a local maximum. If f”(x) = 0, the test is inconclusive and you may need to check the gradient on either side.

利用二阶导数 f”(x) = 6ax + 2b 判断驻点类型。若 f”(x) > 0，则为局部极小值；若 f”(x) < 0，则为局部极大值。若 f”(x) = 0，该判别法失效，此时需检查点两侧的梯度符号。

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9. Transformations of Cubic Graphs | 三次函数图像的变换

Cubic graphs can be transformed using standard rules. Replacing x by (x − h) translates the graph h units to the right. Adding k to the function shifts it up by k units. For instance, y = (x − 2)³ + 3 is the graph of y = x³ shifted 2 units right and 3 units up. Its point of inflection moves from (0, 0) to (2, 3).

三次函数图像可以按标准规则进行变换。用 (x − h) 替换 x 会将图像向右平移 h 个单位；函数式加 k 则上移 k 个单位。例如 y = (x − 2)³ + 3 是 y = x³ 向右平移 2 个单位、向上平移 3 个单位后的图像。其拐点也由 (0, 0) 移至 (2, 3)。

Stretches also apply. y = a f(x) stretches vertically by factor a, and y = f(bx) stretches horizontally by factor 1/b. These transformations can alter the steepness and the positions of roots, so always re‑calculate key points when sketching transformed cubics.

伸缩也同样适用。y = a f(x) 以 a 为倍数纵向拉伸；y = f(bx) 以 1/b 为倍数横向伸缩。这些变换会改变曲线的陡度以及根的位置，因此在绘制变换后的三次图像时，务必重新计算关键点。

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10. Solving Cubic Inequalities Graphically | 用图像法解三次不等式

Once a sketch of a cubic graph is drawn, solving an inequality such as (x − 2)(x + 1)(x − 4) > 0 becomes straightforward. Identify the intervals where the graph is above the x‑axis. If the cubic is positive with three distinct roots, the pattern of signs alternates: +, −, +, − from right to left (or vice versa depending on leading coefficient).

在绘出三次函数草图后，求解诸如 (x − 2)(x + 1)(x − 4) > 0 的不等式就变得简单了。只需找出图像位于 x 轴上方的区间。若三次函数首项系数为正且有三个不同单根，则区间符号自右向左交替为 +, −, +, −（或反之，取决于首项系数符号）。

Simply write the solution as a union of intervals. For the above example with a > 0, the solution is x < −1 or 2 < x < 4. Be careful with repeated roots: the sign may not change at a repeated root unless the factor’s power is odd.

将解写成区间的并集即可。对于上例且 a > 0，解为 x < −1 或 2 < x < 4。注意重复根：除非因式的指数为奇数，否则符号在重复根处不会改变。

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11. Worked Example: Sketching a Cubic Graph | 实例：绘制三次函数图像

Sketch the graph of y = x³ − 4x² + x + 6. First, find a root by trial. Trying x = −1 gives (−1)³ − 4(−1)² + (−1) + 6 = −1 − 4 − 1 + 6 = 0, so (x + 1) is a factor. Dividing the cubic by (x + 1) gives x² − 5x + 6, which factorises as (x − 2)(x − 3). Hence y = (x + 1)(x − 2)(x − 3).

绘制 y = x³ − 4x² + x + 6 的图像。先通过试根求一个根。尝试 x = −1： (−1)³ − 4(−1)² + (−1) + 6 = 0，因此 (x + 1) 为因式。将三次式除以 (x + 1) 得 x² − 5x + 6，进一步分解为 (x − 2)(x − 3)。因此 y = (x + 1)(x − 2)(x − 3)。

The x‑intercepts are −1, 2 and 3. The y‑intercept is y = 6. The leading coefficient is positive (1), so the graph falls to the left and rises to the right. Mark the intercepts, draw a smooth curve crossing at each root. To find turning points, differentiate: y’ = 3x² − 8x + 1. Set to zero: 3x² − 8x + 1 = 0. The solutions are x = (8 ± √(64 − 12)) / 6 = (8 ± √52)/6 = (8 ± 2√13)/6 = (4 ± √13)/3. Approximate values are 2.54 and 0.13. Substitute back to find y‑coordinates and classify using second derivative y” = 6x − 8. At x ≈ 2.54, y” > 0 → local minimum; at x ≈ 0.13, y” < 0 → local maximum. Include these on the sketch.

x 轴截距为 −1、2 和 3。y 轴截距为 6。首项系数为正（1），因此图像左降右升。标出截距，画出在每根处穿过的平滑曲线。为求转折点，求导： y’ = 3x² − 8x + 1。令其为零：3x² − 8x + 1 = 0。解得 x = (8 ± √(64 − 12))/6 = (8 ± √52)/6 = (8 ± 2√13)/6 = (4 ± √13)/3，近似值约为 2.54 和 0.13。代回求 y 坐标，并用二阶导数 y” = 6x − 8 判别：在 x ≈ 2.54 处 y” > 0 → 局部极小；在 x ≈ 0.13 处 y” < 0 → 局部极大。在草图上标出这些点。

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12. Summary and Key Points | 总结与要点

A cubic graph y = ax³ + bx² + cx + d must be approached systematically: check the sign of a to determine end behaviour, find intercepts by factorising or substituting x = 0, use calculus to locate and classify stationary points, and interpret repeated roots correctly. Practising with a variety of factorised and expanded forms will build confidence for the Edexcel A-Level exam.

处理三次函数图像 y = ax³ + bx² + cx + d 需按部就班：检查 a 的正负以确定末端行为；通过因式分解或代 x = 0 求截距；使用微积分定位并判别驻点；正确解读重复根。针对因式形式和展开形式进行多样化练习，将有助于自信应对 Edexcel A-Level 考试。

Key facts: at most 3 real roots; at least 1; multiplicity 2 touches, multiplicity 3 crosses with zero slope; maximum 2 turning points; leading coefficient controls direction; transformations follow standard rules. Solving inequalities reduces to reading intervals from a sketch.

关键要点：最多三个实根；至少一个实根；二重根触轴，三重根以零斜率穿轴；最多两个转折点；首项系数决定图形方向；图像变换遵循标准规则；解不等式可归结为从草图上读取区间。

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