Cubic Function Graphs: Trends and Key Points | 三次函数图像：变化趋势与关键点

📚 Cubic Function Graphs: Trends and Key Points | 三次函数图像：变化趋势与关键点

Cubic functions are among the most rewarding curves to study in IGCSE Mathematics. Their S-shaped graphs contain turning points, points of inflection, and up to three x-intercepts, which makes them far richer than quadratics. In this revision article, we will unpack the shape, the key features, and a reliable sketching procedure, so you can handle any cubic graph question with confidence.

三次函数是 IGCSE 数学中最值得研究的曲线之一。它们呈 S 形的图像包含驻点、拐点以及最多三个 x 轴截距，因此远比二次函数丰富。在这篇复习文章中，我们将系统讲解其形状、关键特征以及一套可靠的作图步骤，帮助你自信应对任何三次函数图像问题。

1. Getting to Know Cubic Functions | 认识三次函数

A cubic function is any polynomial of degree 3. In its most general form, it can be written as

三次函数是任何次数为 3 的多项式。其最一般的形式可以写成

y = ax³ + bx² + cx + d, where a ≠ 0

The coefficients b, c and d may be zero, but the leading coefficient a must never be zero; otherwise the function would drop to a quadratic or lower. The highest power is 3, which tells us the graph can change direction up to twice, producing two turning points at most.

系数 b、c 和 d 可以为零，但首项系数 a 绝不能为零；否则函数就会降为二次或更低次。最高次数为 3，这告诉我们图像最多可以改变两次方向，从而产生最多两个转折点。

Unlike a straight line or a parabola, a cubic graph is not symmetric about any vertical line. Its signature shape is a gentle wave: it falls, flattens, rises, then continues on one side of the coordinate plane in a smooth curve. The exact appearance depends mainly on the sign of a.

与直线或抛物线不同，三次图像关于任何竖直线都不对称。它的标志性形状是一个平缓的波浪：曲线先下降、趋平、再上升，然后向坐标平面的一侧平滑延伸。具体外观主要取决于 a 的正负。

2. End Behaviour and the Leading Coefficient | 端值趋势与首项系数

End behaviour describes what happens to y as x becomes very large positive or very large negative. For a cubic, the term ax³ dominates when |x| is huge, so we only need to look at a.

端值趋势描述的是当 x 变为很大正数或很大负数时 y 的变化。对三次函数而言，当 |x| 很大时，ax³ 这一项占主导地位，因此我们只需看 a。

Leading coefficient \| 首项系数	As x → +∞ \| 当 x → +∞	As x → −∞ \| 当 x → −∞	Overall shape \| 整体形状
a > 0	y → +∞	y → −∞	Rises from lower left to upper right (upward S)
a < 0	y → −∞	y → +∞	Falls from upper left to lower right (downward S)

A common mnemononic: “positive a starts low and ends high; negative a starts high and ends low.” This instantly tells you which arm of the graph goes upwards and which goes downwards, even before you locate any intercepts.

一个常用口诀是：“a 为正时左低右高；a 为负时左高右低。”即使在定位任何截距之前，这个规律也能立刻告诉你图像的哪一端向上、哪一端向下。

3. Roots: Where the Curve Meets the x-axis | 零点：曲线与 x 轴的交点

The roots of a cubic equation f(x) = 0 are the x-coordinates of the points where the graph crosses or touches the x-axis. In factorised form,

三次方程 f(x) = 0 的根，就是图像穿过或接触 x 轴时对应点的 x 坐标。在因式分解形式中，

y = a(x − r₁)(x − r₂)(x − r₃)

the roots are r₁, r₂ and r₃. A cubic can have one, two or three real roots. If the discriminant analysis is too advanced, rely on the graph: an S-shaped curve may cross the x-axis once, twice (with one repeated root), or three times.

其中根为 r₁、r₂ 和 r₃。三次函数可以有一个、两个或三个实根。如果判别式分析太难，完全可以借助图像：S 形曲线可能穿过 x 轴一次、两次（含一个重根）或三次。

Multiplicity matters:

重根非常重要：

Simple root (multiplicity 1): the curve crosses the x-axis at that point. The sign of y changes on either side.

单根（重数 1）：曲线在该点穿过 x 轴，y 在两侧变号。
Repeated root (multiplicity 2): the curve touches the x-axis and turns back. This creates a stationary point on the axis.

重根（重数 2）：曲线接触 x 轴后折返，这会在轴上形成一个驻点。
Triple root (multiplicity 3): the curve crosses the x-axis but flattens out momentarily, appearing almost horizontal at that point.

三重根（重数 3）：曲线穿过 x 轴但短暂变平，在该点看起来几乎是水平的。

To find roots, try the factor theorem first: test small integer values such as ±1, ±2, ±3. If f(p) = 0, then (x − p) is a factor. Then divide or factorise the remaining quadratic.

求根时，先尝试因式定理：代入小整数如 ±1、±2、±3。如果 f(p) = 0，则 (x − p) 是一个因式，然后对剩余二次式进行除法或因式分解。

4. The y-intercept | y 轴截距

The y-intercept is the point where the graph crosses the y-axis. It is found simply by setting x = 0:

y 轴截距是图像与 y 轴相交的点，只需令 x = 0 即可求出：

y = a(0)³ + b(0)² + c(0) + d = d

So the y-intercept is always (0, d). In factorised form, it equals −a(r₁r₂r₃). This point is often the easiest to plot, and it serves as a useful anchor when you sketch the curve during an exam.

因此 y 轴截距总是 (0, d)。在因式分解形式中，它等于 −a(r₁r₂r₃)。这个点通常最容易画出，也是考试作图时非常有用的锚点。

Do not forget that the y-intercept can be positive, negative, or zero. If d = 0, then x = 0 is also a root, so the graph passes through the origin.

不要忘记 y 轴截距可以是正、负或零。如果 d = 0，则 x = 0 也是一个根，因此图像穿过原点。

5. Stationary Points: Maxima and Minima | 驻点：极大值与极小值

Stationary points occur where the gradient of the curve is zero. To find them, differentiate the cubic and solve

驻点出现在曲线斜率为零的地方。求法是对三次函数求导并解方程

dy/dx = 3ax² + 2bx + c = 0

This is a quadratic equation, so a cubic has at most two stationary points. The nature of each turning point is decided by the second derivative:

这是一个二次方程，因此三次函数最多有两个驻点。每个转折点的性质由二阶导数决定：

If d²y/dx² < 0 at the point, the gradient is decreasing, so it is a local maximum.

如果该点处 d²y/dx² < 0，说明斜率在减小，因此是局部极大值。
If d²y/dx² > 0 at the point, the gradient is increasing, so it is a local minimum.

如果该点处 d²y/dx² > 0，说明斜率在增大，因此是局部极小值。
If d²y/dx² = 0, use the sign table for dy/dx to decide whether a stationery point exists at all.

如果 d²y/dx² = 0，则需要用 dy/dx 的符号表判断该处是否真的是驻点。

Do not confuse the point of inflection with a stationary point. A stationary point has zero gradient; a point of inflection does not necessarily have zero gradient.

不要混淆拐点与驻点。驻点的斜率为零；而拐点的斜率不一定为零。

6. The Point of Inflection | 拐点

Every cubic curve has exactly one point of inflection, where the curvature changes from concave down to concave up, or vice versa. This occurs when

每一条三次曲线都恰好有一个拐点，即曲线的凹凸性发生改变的位置，由凹向下变为凹向上，或反过来。它出现在

d²y/dx² = 6ax + 2b = 0, giving x = −b/(3a)

At this x-value, the curve changes from bending one way to bending the other way. For the standard cubic y = x³, the inflection point is at (0, 0), exactly halfway between the two turning points when they exist.

在该 x 值处，曲线的弯曲方向发生转换。对于标准三次函数 y = x³，拐点在 (0, 0)，恰好位于两个转折点（若存在）的正中间。

The point of inflection is also the centre of rotational symmetry for many cubics: if you rotate the graph by 180° about this point, it maps onto itself. This symmetry is exact when the cubic is in the form y = a(x − h)³ + k; otherwise the symmetry is approximate but still noticeable.

拐点也是许多三次图像的旋转对称中心：将图像绕该点旋转 180° 后会与自身重合。当三次函数写成 y = a(x − h)³ + k 时，这种对称是精确的；否则对称性只是近似但依然可辨。

7. Increasing and Decreasing Intervals | 递增与递减区间

Analysing the sign of dy/dx tells you where the function is increasing or decreasing. For a cubic with a > 0 and two distinct stationary points at x = p and x = q (p < q), the behaviour follows this pattern:

分析 dy/dx 的符号可以判断函数在哪些区间递增、哪些区间递减。对 a > 0 且有两个不同驻点 x = p 和 x = q（p < q）的三次函数，其变化模式如下：

Interval \| 区间	x < p	p < x < q	x > q
Sign of dy/dx	Positive	Negative	Positive
Behaviour	Increasing	Decreasing	Increasing

If a < 0, the pattern reverses: decreasing, then increasing, then decreasing again. If the derivative has no real roots, then the cubic is monotonic: it is either always increasing or always decreasing across its whole domain.

如果 a < 0，模式则相反：先递减、再递增、最后再递减。如果导数没有实根，则三次函数是单调的：在定义域内要么始终递增，要么始终递减。

When writing your answers, use inequality notation, for example “y is increasing for x < p and x > q.” A single continuous interval must not be rewritten as “p > x > q”, because that notation is ambiguous.

书写答案时，请使用不等式记号，例如“y 在 x < p 和 x > q 时递增”。不要写成“p > x > q”，因为这种连写记号有歧义。

8. A Step-by-Step Sketching Framework | 逐步绘图框架

Follow these six steps to sketch any cubic in an exam. They work universally:

在考试中绘制任意三次函数图像时，请遵循以下六个步骤，它们具有普遍适用性：

Determine the sign of a to decide the end behaviour.

判断 a 的符号以决定端值趋势。
Find all x-intercepts by solving f(x) = 0, and note their multiplicities.

通过解 f(x) = 0 找出所有 x 轴截距，并注意它们的重数。
Write down the y-intercept (0, d).

写出 y 轴截距 (0, d)。
Solve dy/dx = 0 to locate stationary points; use d²y/dx² to classify them.

解 dy/dx = 0 以定位驻点；用 d²y/dx² 对它们进行分类。
Find the point of inflection if the question asks for it, or if it helps you balance the curve.

如果题目要求或有助于你平衡曲线，则求出拐点。
Plot all key points on axes, then join them with a smooth, continuous S-shaped curve, respecting end behaviour.

在坐标轴上标出所有关键点，然后用平滑连续的 S 形曲线连接它们，并满足端值趋势。

Always label turning points with their exact coordinates. If a coordinate is a surd, leave it in surd form unless the question requests a decimal approximation.

务必用精确坐标标注转折点。如果坐标是无理数，除非题目要求小数近似值，否则保留根式形式。

9. Worked Example: Sketch y = x³ − 3x² + 2 | 例题：绘制 y = x³ − 3x² + 2

Let us apply the framework to a typical IGCSE question.

让我们将上述框架应用于一道典型的 IGCSE 题目。

Step 1 – End behaviour: Here a = 1 > 0, so the curve rises from the lower left and ends in the upper right.

第一步 —— 端值趋势：这里 a = 1 > 0，所以曲线从左下方向上升起，终点朝向右上方。

Step 2 – Roots: Test x = 1: f(1) = 1 − 3 + 2 = 0, so (x − 1) is a factor. Dividing gives

第二步 —— 零点：代入 x = 1：f(1) = 1 − 3 + 2 = 0，因此 (x − 1) 是一个因式。做除法得

x³ − 3x² + 2 = (x − 1)(x² − 2x − 2)

The quadratic gives x = 1 ± √3. Hence the x-intercepts are x = 1, x = 1 − √3 ≈ −0.73, and x = 1 + √3 ≈ 2.73.

二次式给出 x = 1 ± √3。因此 x 轴截距为 x = 1、x = 1 − √3 ≈ −0.73 和 x = 1 + √3 ≈ 2.73。

Step 3 – y-intercept: f(0) = 2, so the graph passes through (0, 2).

第三步 —— y 轴截距：f(0) = 2，所以图像经过 (0, 2)。

Step 4 – Stationary points: Differentiate: dy/dx = 3x² − 6x = 3x(x − 2). Thus x = 0 and x = 2. Their y-values are f(0) = 2 and f(2) = 8 − 12 + 2 = −2. The second derivative is d²y/dx² = 6x − 6, giving −6 at x = 0 (local maximum) and +6 at x = 2 (local minimum). So the turning points are (0, 2) and (2, −2).

第四步 —— 驻点：求导：dy/dx = 3x² − 6x = 3x(x − 2)，因此 x = 0 和 x = 2。对应的 y 值为 f(0) = 2 和 f(2) = 8 − 12 + 2 = −2。二阶导数为 d²y/dx² = 6x − 6，在 x = 0 处为 −6（局部极大值），在 x = 2 处为 +6（局部极小值）。因此转折点为 (0, 2) 和 (2, −2)。

Step 5 – Inflection point: Set 6x − 6 = 0, giving x = 1. Then f(1) = 0. The point of inflection is (1, 0).

第五步 —— 拐点：令 6x − 6 = 0，得 x = 1，则 f(1) = 0。拐点为 (1, 0)。

Step 6 – Sketch: Plot (−0.73, 0), (0, 2), (1, 0), (2, −2) and (2.73, 0). Join these with a smooth curve that comes from the lower left, reaches a maximum at (0, 2), passes through the inflection point (1, 0), reaches a minimum at (2, −2), and finally rises to the upper right.

第六步 —— 作图：标出 (−0.73, 0)、(0, 2)、(1, 0)、(2, −2) 和 (2.73, 0)。用平滑曲线连接这些点：曲线从左下方而来，在 (0, 2) 达到极大值，经过拐点 (1, 0)，在 (2, −2) 达到极小值，最后向右上方上升。

10. Common Exam Pitfalls and Tips | 常见考试误区和技巧

Even top students lose marks on cubic graphs through small but avoidable mistakes. Here are the most common traps:

即使是优秀学生，也常常因一些微小但可避免的错误而在三次函数图像上失分。以下是最常见的陷阱：

Forgetting the sign of a: Drawing the S-curve upside down loses the end-behaviour mark instantly. Always check whether the leading coefficient is positive or negative.

忘记 a 的符号：把 S 形曲线画反会立刻丢掉端值趋势的分数。务必先检查首项系数的正负。
Misreading multiplicity: A repeated root means touching, not crossing. If you draw a crossing at a double root, examiners will penalise the shape.

误读重根：重根意味着“接触”而非“穿过”。如果在二重根处画成穿过，阅卷人会因形状错误而扣分。
Writing the y-value of a turning point incorrectly: Substitute the x-value back into the original cubic, not into the derivative. The derivative gives slope, not the y-coordinate.

把驻点 y 值写错：应将 x 值代回原三次函数，而不是代入导数。导数给出的是斜率，而不是 y 坐标。
Forgetting that dy/dx = 0 may have no real solutions: Not every cubic has turning points. If 3ax² + 2bx + c has a negative discriminant, then the function is monotonically increasing or decreasing.

忘记 dy/dx = 0 可能没有实解：并非每个三次函数都有转折点。如果 3ax² + 2bx + c 的判别式为负，则函数单调递增或递减。
Joining points with straight lines: A cubic graph is smooth and continuous. Connecting key points with ruler-straight segments is a visual disaster in exams.

用直线连接各点：三次图像是平滑连续的。用直尺把关键点连成直线段，在考试中会造成视觉灾难。

Finally, always show your working: factorisations, derivative equations, and sign tests. Even if your final sketch is slightly off, the examiner can award method marks from your written steps.

最后，务必展示你的过程：因式分解、导数方程和符号测试。即使最终图像略有偏差，阅卷人也能根据你的书面步骤给出方法分。

Mastering cubic graphs is about combining algebra with a clear mental image of the S-curve. Practise sketching at least one cubic every revision session, and you will soon recognise the shape, predict the turning points, and hit full marks on this topic.

掌握三次函数图像的关键，在于将代数与清晰的 S 形图像结合起来。每次复习时至少练习绘制一个三次函数，你很快就能识别形状、预判转折点，并在这一考点上获得满分。

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Cubic Function Graphs: Trends and Key Points | 三次函数图像：变化趋势与关键点