1. What is a Cubic Function? 什么是三次函数？

Edexcel IGCSE数学 Cubic Graphs考点精讲

Introduction 引言

Cubic graphs are a fundamental topic in the Edexcel IGCSE Mathematics syllabus. Understanding their shape, key features, and how to sketch them is essential for success in both Foundation and Higher tier exams. This article provides a complete, bilingual guide covering everything from basic definitions to advanced problem-solving techniques for cubic functions. Whether you are preparing for your first exposure to this topic or revising before the final exam, this guide will help you build confidence and accuracy.

三次函数图像（Cubic Graphs）是Edexcel IGCSE数学考纲中的核心知识点。从基础定义到高阶解题技巧，本文提供中英双语完整指南。无论你是初次接触该知识点还是考前复习，这篇指南都将帮助你建立信心、提升准确性。三次函数图像贯穿代数与函数两大模块，出现在历年考卷的简答题和绘图题中，是不可失分的必考内容。

1. What is a Cubic Function? 什么是三次函数？

A cubic function is a polynomial function of degree 3, meaning the highest power of the variable x is 3. Its general form is written as:

三次函数是一个三次多项式函数，即变量x的最高次幂为3。其一般形式写作：

y = f(x) = ax³ + bx² + cx + d

where a, b, c, and d are real constants, and the leading coefficient a must not equal zero (a ≠ 0). If a = 0, the x³ term disappears and the function reduces to a quadratic, losing its cubic character. This is the single most important condition — always check that a ≠ 0 when identifying a cubic function in an exam question.

其中 a、b、c 和 d 为实常数，且首项系数a必须不等于零（a ≠ 0）。如果a = 0，x³项消失，函数退化为二次函数，失去三次特征。这是最重要的条件——在考试中识别三次函数时，一定要先检查a ≠ 0。

The degree of the polynomial tells us three crucial things about the graph: first, it can have at most 3 real roots (also called x-intercepts or zeros); second, it can have at most 2 turning points (where the gradient equals zero); and third, it always has exactly one point of inflection where the curvature changes direction. These structural properties define the "skeleton" of every cubic graph.

多项式的次数告诉我们关于图像的三个关键信息：第一，最多有3个实根（也称x轴截距或零点）；第二，最多有2个拐点（梯度为零的位置）；第三，始终恰好有1个拐点（反曲点），曲率在此改变方向。这些结构属性定义了每张三次函数图像的"骨架"。

2. End Behavior: The Role of Leading Coefficient a 图像端点行为：首项系数a的作用

The sign of the leading coefficient a is the single most important factor determining the overall shape of a cubic graph. You should determine this first whenever you sketch a cubic — it tells you which way the graph "opens" at its ends.

首项系数a的符号是决定三次图像整体形状的最重要因素。每次绘制三次图像时，都应该首先确定a的符号——它告诉你图像两端"开口"的方向。

Case 1: a > 0 (Positive Leading Coefficient 正首项系数)

When a is positive, as x approaches negative infinity, y also approaches negative infinity. As x approaches positive infinity, y approaches positive infinity. The graph emerges from the bottom-left quadrant and exits through the top-right quadrant. Visually, this forms an upward-opening S-curve. This is sometimes described as the "standard" cubic shape because the simplest cubic y = x³ follows this pattern.

当a为正时，x趋近负无穷时y也趋近负无穷；x趋近正无穷时y趋近正无穷。图像从左下象限出发，从右上象限离开。视觉上形成向上开口的S形曲线。这有时被称为"标准"三次形状，因为最简单的三次函数y = x³就遵循此规律。

Key behavior summary for a > 0: Left end goes DOWN, right end goes UP. Think: "Positive a pulls the right side up like a smile."

a > 0 关键行为总结：左端向下 ↓，右端向上 ↑。记法："正a像微笑一样把右端拉起来。"

Case 2: a < 0 (Negative Leading Coefficient 负首项系数)

When a is negative, the behavior is reversed. As x approaches negative infinity, y approaches positive infinity. As x approaches positive infinity, y approaches negative infinity. The graph starts at the top-left and finishes at the bottom-right — a downward-opening S-curve. This is the mirror image of the positive case, reflected across the x-axis.

当a为负时，行为完全相反。x趋近负无穷时y趋近正无穷；x趋近正无穷时y趋近负无穷。图像从左上方开始，到右下方结束——向下开口的S形曲线。这是正系数情况关于x轴的镜像。

Key behavior summary for a < 0: Left end goes UP, right end goes DOWN. Think: "Negative a drags the right side down like a frown."

a < 0 关键行为总结：左端向上 ↑，右端向下 ↓。记法："负a像皱眉一样把右端拉下来。"

Memory Aid 记忆口诀： "a正右上翘，a负右下掉。" Just look at the sign of a and you instantly know which way the right-hand end of the graph points. The left-hand end always goes the opposite way.

"a正右上翘，a负右下掉。"只需看一眼a的符号，你就能立即知道图像右端指向的方向。左端总是走向相反方向。

3. Finding the y-Intercept 求y轴截距

The y-intercept is perhaps the simplest feature to find on any cubic graph. It is the point where the curve crosses the y-axis, which occurs exactly when x = 0. Substitute x = 0 into the general equation:

y轴截距可能是三次图像上最容易找到的特征。它是曲线与y轴的交点，恰好发生在x = 0时。将x = 0代入一般方程：

y = a(0)³ + b(0)² + c(0) + d = d

So the y-intercept is simply the constant term: (0, d). This means that every cubic function of the form y = ax³ + bx² + cx + d crosses the y-axis at height d. This is always your starting point when plotting — mark (0, d) first, and you already have one anchor on your coordinate plane.

因此y轴截距就是常数项：(0, d)。这意味着每个形如y = ax³ + bx² + cx + d的三次函数都在高度d处穿过y轴。绘图时这始终是你的出发点——先标出(0, d)，坐标平面上就已经有了一个锚点。

Note: If d = 0, the graph passes through the origin (0, 0), which is both the y-intercept and a root of the equation. This is a useful special case to recognize quickly in exam problems.

注意：如果d = 0，图像经过原点(0, 0)，这既是y轴截距也是方程的一个根。在考试中快速识别这种特殊情况非常有用。

4. Finding Roots: The Factor Theorem and Polynomial Division 求根：因式定理与多项式除法

Finding the x-intercepts — also called roots, zeros, or solutions to f(x) = 0 — is the most algebraically demanding part of analyzing a cubic graph. The standard approach for IGCSE Edexcel is the Factor Theorem combined with polynomial division.

求x轴截距——也称为根、零点或f(x) = 0的解——是分析三次图像中代数要求最高的部分。IGCSE Edexcel的标准方法是因式定理结合多项式除法。

The Factor Theorem 因式定理

The Factor Theorem states: For a polynomial f(x), if f(p) = 0, then (x − p) is a factor of f(x). This gives us a systematic way to break down a cubic into simpler pieces.

因式定理指出：对于多项式f(x)，若f(p) = 0，则(x − p)是f(x)的一个因式。这为我们提供了一种系统地将三次函数分解为更简单部分的方法。

Step-by-Step Factorization Process 因式分解步骤：

Step 1: Try small integer values for x. The most likely candidates are factors of the constant term d divided by factors of the leading coefficient a. For simple IGCSE problems where a = 1, just try x = ±1, ±2, ±3, and the factors of d.

步骤1：尝试小整数值。最可能的候选值是常数项d的因子除以首项系数a的因子。对于a = 1的简单IGCSE题目，只需尝试x = ±1、±2、±3以及d的因子。

Step 2: Once you find a value p where f(p) = 0, you know (x − p) is a factor. Now perform polynomial long division: divide the cubic ax³ + bx² + cx + d by (x − p). The result will be a quadratic expression.

步骤2：一旦找到f(p) = 0的值p，就知道(x − p)是一个因式。现在进行多项式长除法：将三次式ax³ + bx² + cx + d除以(x − p)，结果是一个二次表达式。

Step 3: Factor the quadratic (or use the quadratic formula if it does not factor nicely). Each linear factor gives one root.

步骤3：因式分解二次式（如果不能轻松分解则使用求根公式）。每个一次因式给出一个根。

Worked Example 例题：

Find all roots of f(x) = x³ − 6x² + 11x − 6 and sketch its key features.

求f(x) = x³ − 6x² + 11x − 6的所有根并勾画关键特征。

Step 1 — Try values: Constant term d = −6. Factors: ±1, ±2, ±3, ±6.

Try x = 1: f(1) = 1 − 6 + 11 − 6 = 0. Success! So (x − 1) is a factor.

验证 x = 1：f(1) = 1 − 6 + 11 − 6 = 0。成功！因此(x − 1)是一个因式。

Step 2 — Polynomial division: (x³ − 6x² + 11x − 6) ÷ (x − 1)

Using long division: x³ ÷ x = x², multiply back: x²(x − 1) = x³ − x², subtract: −5x² + 11x − 6. Next: −5x² ÷ x = −5x, multiply: −5x(x − 1) = −5x² + 5x, subtract: 6x − 6. Finally: 6x ÷ x = 6, multiply: 6(x − 1) = 6x − 6, subtract: 0. Quotient: x² − 5x + 6.

Step 3 — Factor the quadratic: x² − 5x + 6 = (x − 2)(x − 3).

Roots: x = 1, x = 2, x = 3. The graph crosses the x-axis at (1, 0), (2, 0), and (3, 0).

根：x = 1, x = 2, x = 3。图像在(1, 0)、(2, 0)和(3, 0)处穿过x轴。

Key Insight 关键见解： When a cubic has three distinct real roots, the graph must cross the x-axis three times. Since cubics are continuous, the curve alternates between being above and below the x-axis between consecutive roots. This creates the characteristic wavy shape that distinguishes cubics from quadratics.

当三次函数有三个不同实根时，图像必须三次穿过x轴。由于三次函数是连续的，曲线在相邻根之间交替处于x轴上方和下方。这形成了区分三次函数与二次函数的特征波浪形状。

5. Turning Points: Using the First Derivative 拐点：使用一阶导数

Turning points are where the gradient of the curve is momentarily zero — the graph stops rising or falling and reverses direction. For a cubic, there can be 0, 1, or 2 turning points, and finding them requires calculus.

拐点处曲线的梯度暂时为零——图像停止上升或下降并反转方向。三次函数可以有0、1或2个拐点，求解需要微积分。

Start by differentiating the cubic function:

首先对三次函数求导：

y = ax³ + bx² + cx + d

dy/dx = 3ax² + 2bx + c

Set dy/dx = 0 to find the x-coordinates of turning points:

令dy/dx = 0来求拐点的x坐标：

3ax² + 2bx + c = 0

This is a quadratic equation. The discriminant Δ = (2b)² − 4(3a)(c) = 4b² − 12ac determines how many turning points exist:

这是一个二次方程。判别式Δ = (2b)² − 4(3a)(c) = 4b² − 12ac决定了存在多少个拐点：

Δ > 0: Two distinct real solutions → 2 turning points (one maximum, one minimum). This is the most common case in IGCSE problems.

Δ > 0：两个不同实解 → 2个拐点（一个极大值，一个极小值）。这是IGCSE题目中最常见的情况。

Δ = 0: One repeated real solution → 1 turning point which is also a stationary point of inflection (the graph flattens but does not change direction).

Δ = 0：一个重根实解 → 1个拐点，同时也是驻点拐点（图像变平但不改变方向）。

Δ < 0: No real solutions → 0 turning points. The graph is strictly monotonic, either always increasing or always decreasing. Example: y = x³ + x has no turning points.

Δ < 0：无实解 → 0个拐点。图像严格单调，始终递增或始终递减。例如：y = x³ + x无拐点。

Classifying Turning Points 拐点类型判定：

Use the second derivative to determine whether each turning point is a maximum or minimum:

使用二阶导数判定每个拐点是极大值还是极小值：

d²y/dx² = 6ax + 2b

At a turning point x = p: if d²y/dx² > 0, the point is a local minimum; if d²y/dx² < 0, the point is a local maximum; if d²y/dx² = 0, further investigation is needed (it could be a point of inflection).

在拐点x = p处：若d²y/dx² > 0，该点为局部极小值；若d²y/dx² < 0，该点为局部极大值；若d²y/dx² = 0，需要进一步分析（可能是拐点/反曲点）。

6. The Point of Inflection 拐点（反曲点）

Every cubic graph has exactly one point of inflection — a point where the curvature changes from concave upward (holding water like a cup) to concave downward (shedding water like an umbrella), or vice versa. Mathematically, this occurs where the second derivative equals zero:

每个三次函数图像恰好有一个拐点（反曲点）——曲率从凹向上（像杯子盛水）变为凹向下（像伞面排水）或反之的位置。数学上，这发生在二阶导数为零处：

d²y/dx² = 6ax + 2b = 0

x = −b/(3a)

This is a remarkably simple formula. The x-coordinate of the inflection point depends only on the coefficients a and b. Importantly, the inflection point lies exactly halfway between the two turning points (when they exist), making it a center of rotational symmetry for the cubic graph.

这是一个非常简洁的公式。拐点的x坐标仅取决于系数a和b。重要的是，拐点恰好位于两个拐点（如果存在）的正中间，使其成为三次图像的旋转对称中心。

Substitute x = −b/(3a) back into the original cubic equation to get the y-coordinate. Together, (x, y) gives the coordinates of the inflection point.

将x = −b/(3a)代回原三次方程得到y坐标。合在一起(x, y)即为拐点坐标。

7. Complete Sketching Strategy 完整绘图策略

When an IGCSE Edexcel question asks you to sketch a cubic graph, follow this systematic seven-step approach. Method marks are awarded for each step, so even if your final drawing is not perfect, showing your working earns substantial credit.

当IGCSE Edexcel题目要求你绘制三次函数图像时，请按以下七步系统方法进行。每个步骤都计算方法分，所以即使最终图形不完美，展示解题过程也能获得大量分数。

Step 1 — End Behavior（端点行为）： Check the sign of a. If a > 0, the graph rises on the right; if a < 0, the graph falls on the right. Draw light arrows on your axes to indicate the direction at both ends.

检查a的符号。若a > 0，右端上升；若a < 0，右端下降。在坐标轴上画浅色箭头指示两端方向。

Step 2 — y-Intercept（y轴截距）： Find (0, d) and mark it clearly on the y-axis. This is always your first plotted point.

求(0, d)并在y轴上清晰标出。这始终是你绘制的第一个点。

Step 3 — Roots（根）： Solve f(x) = 0 using the Factor Theorem and polynomial division. Mark each root on the x-axis.

使用因式定理和多项式除法解f(x) = 0。在x轴上标出每个根。

Step 4 — Turning Points（拐点）： Find dy/dx, set it to zero, solve the quadratic, and classify each turning point using the second derivative. Mark the (x, y) coordinates.

求dy/dx，令其为零，解二次方程，使用二阶导数判定每个拐点类型。标注(x, y)坐标。

Step 5 — Inflection Point（反曲点）： Use x = −b/(3a) and substitute back for y. This gives you the symmetry center.

使用x = −b/(3a)并代回求y。这给出了对称中心。

Step 6 — Draw the Curve（绘制曲线）： Connect all points with a smooth continuous curve. The curve must pass through every marked point and follow the end behavior arrows. Do NOT use straight line segments between points — cubic graphs are smooth curves.

用平滑连续曲线连接所有点。曲线必须经过每个标注点并遵循端点行为箭头。不要用直线段连接点——三次图像是平滑曲线。

Step 7 — Label Everything（标注一切）： Label the axes (x, y), the coordinates of intercepts and turning points, and the equation of the curve. Neat labelling is part of the presentation marks.

标注坐标轴（x, y）、截距和拐点坐标以及曲线方程。整洁标注是卷面分的一部分。

8. Special Cases and Variations 特殊情况与变化形式

Repeated Roots 重根： When a cubic has a repeated root, the behavior at that root changes. For a double root like y = (x − p)²(x − q), the graph touches the x-axis at x = p without crossing (it "bounces" or is tangent to the axis), and crosses at x = q normally. This is a common exam trap — students often draw the graph crossing at the repeated root when it should only touch.

当三次函数有重根时，该根处的行为会改变。对于形如y = (x − p)²(x − q)的二重根，图像在x = p处接触x轴但不穿过（"弹回"或与轴相切），在x = q处正常穿过。这是常见的考试陷阱——学生常常在重根处画出穿过的图像，而实际上应该只是接触。

For a triple root y = (x − p)³, the graph passes through the x-axis at x = p but with a flattened inflection at that point, not a sharp crossing.

对于三重根y = (x − p)³，图像在x = p处穿过x轴，但在该点处有一个平坦的拐点，而非尖锐的穿越。

Cubics with One Real Root 只有一个实根的三次函数： Some cubics like y = x³ + x² + x + 1 have only one real root. In these cases, the graph crosses the x-axis exactly once. The other two roots exist but are complex (non-real), so they do not appear on the graph. You can recognize these cases when the cubic does not factor nicely and the discriminant of the derivative is negative or zero.

某些三次函数如y = x³ + x² + x + 1只有一个实根。此时图像恰好一次穿过x轴。另外两个根存在但是复数（非实数），因此不会出现在图像上。当三次函数不能轻松因式分解且导数判别式为负或零时，可以识别出这种情况。

Cubics in Factorized Form 因式分解形式的三次函数： If a cubic is given as y = (x − r₁)(x − r₂)(x − r₃), you can read the roots directly. Multiply out (expand) to find the coefficients a, b, c, d if needed for further analysis. The roots are simply r₁, r₂, and r₃.

如果三次函数以y = (x − r₁)(x − r₂)(x − r₃)的形式给出，你可以直接读出根。如需进一步分析，将其乘开（展开）求系数a, b, c, d。根就是r₁、r₂和r₃。

9. Common Mistakes to Avoid 常见错误避免

Mistake 1: Forgetting that a ≠ 0. Always verify that the leading coefficient is non-zero before applying cubic techniques. A quadratic disguised as a cubic is a classic exam trick.

错误1：忘记a ≠ 0。在应用三次函数技巧之前，务必验证首项系数非零。伪装成三次的二次函数是经典考试陷阱。

Mistake 2: Drawing the wrong end behavior. Check the sign of a first, every time. This is the fastest way to lose all the shape marks.

错误2：画出错误的端点行为。每次都先检查a的符号。这是最快丢光形状分的方式。

Mistake 3: Using straight lines between key points. Cubic graphs are smooth curves — your sketch should have gentle bends, not sharp corners.

错误3：在关键点之间使用直线。三次图像是平滑曲线——你的草图应有柔和的弯曲，而非尖角。

Mistake 4: Not labelling coordinates. Edexcel awards marks for correctly stated coordinates of intercepts and turning points, not just for marking dots on the graph.

错误4：不标注坐标。Edexcel对正确写出的截距和拐点坐标给分，而不仅仅是在图上点个点。

Mistake 5: Confusing the discriminant of the derivative with the discriminant of the original cubic. They serve different purposes: the cubic's own factorization tells you about roots; the derivative's discriminant tells you about turning points.

错误5：混淆导数判别式和原三次函数判别式。它们服务于不同目的：三次函数本身的因式分解告诉你根的信息；导数的判别式告诉你拐点的信息。

10. Practice Questions with Guidance 练习题与指导

For each of the following cubic functions, find: (i) the y-intercept, (ii) all roots, (iii) the turning points and their nature, (iv) the point of inflection, and (v) sketch the graph. Solutions are available at aleveler.com.

对以下每个三次函数，求：(i) y轴截距，(ii) 所有根，(iii) 拐点及其类型，(iv) 反曲点，(v) 绘制图像。答案见aleveler.com。

Q1: f(x) = x³ − 6x² + 11x − 6 (Hint: try x = 1, then factor the quadratic)

Q2: f(x) = −x³ + 3x² + 9x + 5 (Hint: a is negative, try x = −1)

Q3: f(x) = x³ − 3x (Hint: factor out x first, then use difference of squares)

Q4: f(x) = (x + 1)(x − 2)² (Hint: already factorized — identify the repeated root)

Q5: f(x) = x³ + 3x² + 3x + 1 (Hint: recognize this as a perfect cube expansion)

Summary and Final Tips 总结与最后建议

Cubic graphs are highly predictable once you understand their underlying structure. The four pillars of cubic analysis are: end behavior (from the sign of a), y-intercept (from the constant d), roots (from the Factor Theorem and polynomial division), and turning points (from the first and second derivatives). Master these four elements, apply the systematic sketching strategy, and you can confidently tackle any cubic graph question on the Edexcel IGCSE Mathematics exam.

三次函数图像一旦理解了其底层结构，就变得高度可预测。三次分析的四大支柱是：端点行为（来自a的符号）、y轴截距（来自常数d）、根（来自因式定理和多项式除法）以及拐点（来自一阶和二阶导数）。掌握这四个要素，应用系统绘图策略，你就能自信地应对Edexcel IGCSE数学考试中的任何三次图像题目。

Remember: practice is the key to mastery. Sketch five different cubics from the practice questions above, check your work against the solutions on aleveler.com, and you will find that cubic graphs become second nature. Best of luck in your examinations!

记住：练习是掌握的关键。从上述练习题中选五个不同的三次函数进行绘图，在aleveler.com上对照答案检查你的结果，你会发现三次图像变得轻而易举。祝考试顺利！

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