Cubic Graphs | 三次函数图像

📚 Cubic Graphs | 三次函数图像

A cubic graph is the graphical representation of a cubic function, one of the most important non-linear functions in the IGCSE Edexcel Mathematics syllabus. Understanding its shape, roots, and turning points is essential for sketching and interpreting these graphs confidently in exams.

三次函数图像是三次函数的图形表示，也是 IGCSE Edexcel 数学考纲中最重要的一类非线性函数之一。理解其形状、根和转折点，对于在考试中自信地绘制和分析这类图像至关重要。

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1. What Is a Cubic Function? | 什么是三次函数？

A cubic function is any function of the form y = ax³ + bx² + cx + d, where a ≠ 0, and a, b, c, d are constants. The highest power of x is 3, which is what gives the function its name. The coefficient a is called the leading coefficient, and it determines the overall orientation of the graph.

三次函数是形如 y = ax³ + bx² + cx + d 的函数，其中 a ≠ 0，a、b、c、d 均为常数。变量 x 的最高次数为 3，这也是该函数名称的由来。系数 a 称为首项系数，它决定了图像的整体方向。

For example, y = x³, y = 2x³ − 3x² + x − 5, and y = −x³ + 4x are all cubic functions. Note that if a = 0, the function becomes quadratic, not cubic.

例如，y = x³、y = 2x³ − 3x² + x − 5 和 y = −x³ + 4x 都是三次函数。注意：如果 a = 0，该函数就变成二次函数，而不是三次函数。

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2. The Shape of Cubic Graphs | 三次函数图像的形状

The most distinctive feature of a cubic graph is its “S” shape or snake-like curve. Unlike a quadratic graph, which has a single turning point (a U-shape or n-shape), a cubic graph can have up to two turning points: one local maximum and one local minimum.

三次函数图像最显著的特征是其“S”形或蛇形曲线。与只有一个转折点（U 形或 n 形）的二次函数图像不同，三次函数图像最多可以有两个转折点：一个局部最大值和一个局部最小值。

When the leading coefficient a is positive (a > 0), the graph rises from the bottom-left and exits toward the top-right. When a is negative (a < 0), the graph descends from the top-left and exits toward the bottom-right.

当首项系数 a 为正数（a > 0）时，图像从左下方向上升起，向右上方延伸。当 a 为负数（a < 0）时，图像从左上方向右下方下降延伸。

a > 0: falls to the left, rises to the right
a < 0: rises to the left, falls to the right

It is worth memorising these two end behaviours, as examiners frequently test whether students know which way the curve starts and ends.

记忆这两种端部行为非常重要，因为考官经常考查学生是否知道曲线的起始和结束方向。

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3. Roots, x-Intercepts and the y-Intercept | 根、x 截距和 y 截距

The roots of a cubic equation are the x-values where the graph crosses (or touches) the x-axis, i.e., where y = 0. A cubic equation can have 1, 2, or 3 real roots. This depends on the discriminant and the positions of the turning points.

三次方程的根是图像与 x 轴相交（或相切）处的 x 值，即 y = 0 的位置。三次方程可以有 1 个、2 个或 3 个实数根，这取决于判别式和转折点的位置。

To find the x-intercepts algebraically, set y = 0 and solve ax³ + bx² + cx + d = 0. If the expression factorises, this is straightforward. For example:

要求 x 截距，令 y = 0 并解 ax³ + bx² + cx + d = 0。如果表达式可以因式分解，这一步就很简单。例如：

y = x³ − 4x = x(x² − 4) = x(x − 2)(x + 2)

Setting each factor to zero gives x = 0, x = 2, and x = −2. These are the three x-intercepts. The y-intercept is found by substituting x = 0, giving y = d. In the example above, the y-intercept is (0, 0).

令每个因式为零，得 x = 0、x = 2 和 x = −2，这三个值就是 x 截距。y 截距通过代入 x = 0 求得，即 y = d。在上例中，y 截距为 (0, 0)。

If a cubic cannot be factorised easily, you may need to use the factor theorem: if f(p) = 0, then (x − p) is a factor. This is a key technique in the Edexcel IGCSE syllabus.

若三次式不易因式分解，可能需要使用因式定理：若 f(p) = 0，则 (x − p) 是其中一个因式。这是 Edexcel IGCSE 考纲中的关键技巧。

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4. Turning Points and Stationary Points | 转折点与驻点

A cubic graph can have up to two turning points. These occur where the gradient of the curve changes sign. To find them, we differentiate: dy/dx = 3ax² + 2bx + c, then set dy/dx = 0 and solve the resulting quadratic equation.

三次函数图像最多可以有两个转折点，它们出现在曲线斜率改变符号的位置。求法是对函数求导：dy/dx = 3ax² + 2bx + c，然后令 dy/dx = 0 并解所得的一元二次方程。

Consider y = x³ − 3x. Differentiating gives dy/dx = 3x² − 3 = 3(x² − 1). Setting this to zero gives x = 1 and x = −1. Substituting back into the original function:

以 y = x³ − 3x 为例，求导得 dy/dx = 3x² − 3 = 3(x² − 1)，令其为零得 x = 1 和 x = −1。代回原函数：

At x = 1, y = 1 − 3 = −2, so the local minimum is at (1, −2). At x = −1, y = −1 + 3 = 2, so the local maximum is at (−1, 2). The x-coordinate of the midpoint of the two turning points also gives the x-coordinate of the point of inflection, if one exists.

当 x = 1 时，y = 1 − 3 = −2，因此局部最小值在 (1, −2)。当 x = −1 时，y = −1 + 3 = 2，因此局部最大值在 (−1, 2)。两个转折点中点的 x 坐标也给出拐点的 x 坐标（如果存在的话）。

To determine whether a stationary point is a maximum or a minimum, you can use the second derivative: if d²y/dx² > 0, it is a minimum; if d²y/dx² < 0, it is a maximum.

要判断驻点是最大值还是最小值，可以使用二阶导数：若 d²y/dx² > 0，则为最小值；若 d²y/dx² < 0，则为最大值。

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5. Sketching Cubic Graphs | 绘制三次函数图像

When sketching a cubic graph in the exam, follow this systematic procedure. First, identify the sign of the leading coefficient to determine the end behaviour. Second, find the y-intercept by setting x = 0. Third, find the x-intercepts by setting y = 0 and solving. Fourth, locate the turning points using differentiation. Finally, plot these key points and draw a smooth curve through them.

在考试中绘制三次函数图像时，按以下系统步骤进行。第一步，判断首项系数的符号以确定端部行为。第二步，令 x = 0 求 y 截距。第三步，令 y = 0 并求解方程，得到 x 截距。第四步，通过求导找到转折点。最后，标出这些关键点并用平滑曲线将它们连接起来。

Here is a summary table for sketching:

以下是绘图的步骤汇总表：

Step | 步骤	Action | 操作
1	Check sign of a for end behaviour | 检查 a 的符号确定端部行为
2	Find y-intercept (0, d) | 求 y 截距 (0, d)
3	Factorise and find x-intercepts | 因式分解求 x 截距
4	Differentiate to find turning points | 求导找转折点
5	Draw a smooth curve | 绘制平滑曲线

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

Cubic graphs follow the same transformation rules as other functions. The general form y = a(x − h)³ + k represents a cubic graph that has been translated h units horizontally and k units vertically. The point (h, k) is the point of inflection of the transformed graph.

三次函数图像遵循与其他函数相同的变换规则。一般形式 y = a(x − h)³ + k 表示一个水平平移 h 个单位、垂直平移 k 个单位的三次函数图像，点 (h, k) 是变换后图像的拐点。

Specifically, y = f(x) + c shifts the graph vertically upward by c units. y = f(x + c) shifts the graph horizontally to the left by c units. y = −f(x) reflects the graph in the x-axis, and y = f(−x) reflects it in the y-axis.

具体来说，y = f(x) + c 将图像垂直向上平移 c 个单位；y = f(x + c) 将图像水平向左平移 c 个单位；y = −f(x) 将图像关于 x 轴作反射；y = f(−x) 将图像关于 y 轴作反射。

For example, starting from y = x³, the graph y = (x − 2)³ − 1 is shifted 2 units right and 1 unit down, so its point of inflection moves from (0, 0) to (2, −1).

例如，从 y = x³ 出发，图像 y = (x − 2)³ − 1 向右平移 2 个单位、向下平移 1 个单位，因此其拐点从 (0, 0) 移动到 (2, −1)。

Vertical stretches are represented by multiplying the whole function by a constant: y = kf(x). A horizontal stretch is represented by y = f(kx). Practice identifying these transformations from the equation alone.

垂直拉伸通过将整个函数乘以常数来表示：y = kf(x)。水平拉伸表示为 y = f(kx)。请练习仅从方程出发识别这些变换。

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7. Solving Cubic Equations Graphically | 用图像法解三次方程

Graphical methods can be used to approximate the roots of a cubic equation. For example, to solve x³ − 3x + 1 = 0, you can plot the graph of y = x³ − 3x + 1 and read off the x-coordinates where the curve crosses the x-axis.

图像法可以用来近似求三次方程的根。例如，要解 x³ − 3x + 1 = 0，可以绘制 y = x³ − 3x + 1 的图像，然后读出曲线与 x 轴相交处的 x 坐标。

Alternatively, you can rearrange the equation into two separate functions and find their intersection points. For instance, x³ = 3x − 1 can be solved by plotting y = x³ and y = 3x − 1 on the same axes; the x-coordinates of their intersections are the solutions.

另一种方法是把方程重新整理为两个函数，并求它们的交点。例如，x³ = 3x − 1 可以通过在同一坐标系中绘制 y = x³ 和 y = 3x − 1 来求解；交点处的 x 坐标就是方程的解。

In exam questions, you may be given a completed graph and asked to read off the roots to one decimal place. Always check that your approximated answers satisfy the original equation when substituted back.

在考试题中，你可能会得到一张已绘制好的图像，并被要求精确到一位小数读出根。务必检查近似答案代回原方程后是否成立。

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8. Worked Example | 例题解析

Let us work through a full example together. Sketch the graph of y = x³ − 3x² + 2.

让我们完整地演练一道例题：绘制 y = x³ − 3x² + 2 的图像。

Step 1: The leading coefficient is positive (a = 1), so the graph falls to the left and rises to the right. Step 2: The y-intercept is at (0, 2). Step 3: To find x-intercepts, set y = 0:

第一步：首项系数为正（a = 1），所以图像在左侧下降、右侧上升。第二步：y 截距为 (0, 2)。第三步：求 x 截距，令 y = 0：

x³ − 3x² + 2 = 0

By the factor theorem, test x = 1: 1 − 3 + 2 = 0, so (x − 1) is a factor. Dividing gives x³ − 3x² + 2 = (x − 1)(x² − 2x − 2). Solving x² − 2x − 2 = 0 using the quadratic formula gives x = 1 ± √3. Therefore, the x-intercepts are x = 1, x ≈ 2.73, and x ≈ −0.73.

根据因式定理，试 x = 1：1 − 3 + 2 = 0，所以 (x − 1) 是一个因式。相除得 x³ − 3x² + 2 = (x − 1)(x² − 2x − 2)。用求根公式解 x² − 2x − 2 = 0，得 x = 1 ± √3。因此 x 截距为 x = 1、x ≈ 2.73 和 x ≈ −0.73。

Step 4: Differentiate: dy/dx = 3x² − 6x = 3x(x − 2). Setting dy/dx = 0 gives x = 0 or x = 2. Substituting back: at x = 0, y = 2 (local maximum); at x = 2, y = 8 − 12 + 2 = −2 (local minimum). Now we plot these points and draw the smooth S-shaped curve.

第四步：求导得 dy/dx = 3x² − 6x = 3x(x − 2)。令 dy/dx = 0 得 x = 0 或 x = 2。代回原函数：当 x = 0 时，y = 2（局部最大值）；当 x = 2 时，y = 8 − 12 + 2 = −2（局部最小值）。现在标出这些点并绘制平滑的 S 形曲线。

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9. Common Exam Pitfalls | 常见考试误区

One common mistake is drawing a cubic graph that looks like a quadratic. Remember that a cubic graph must have the characteristic S-shape, and it must extend beyond the turning points in opposite directions. Another frequent error is misidentifying the end behaviour when the leading coefficient is negative.

常见错误之一是画出看起来像二次函数的三次图像。请记住，三次图像必须具有特征性的 S 形，并且必须从转折点向相反方向延伸。另一个常见错误是在首项系数为负时错误判断端部行为。

Students also often forget to find all three x-intercepts when they exist. A cubic equation can have a repeated root, in which case the graph touches the x-axis rather than crossing it. For example, y = (x − 1)²(x + 2) touches the x-axis at x = 1 and crosses at x = −2.

学生们还常常忘记求所有存在的 x 截距。三次方程可能有重根，此时图像与 x 轴相切而不是穿过。例如，y = (x − 1)²(x + 2) 在 x = 1 处与 x 轴相切，在 x = −2 处穿过。

Finally, be careful when sketching: the curve must pass through all identified intercepts and turning points smoothly, without sharp corners or sudden kinks. Use a pencil and plot additional points if necessary for accuracy.

最后，绘图时务必小心：曲线必须平滑地通过所有已标出的截距和转折点，不能有尖锐的拐角或突然的弯折。建议使用铅笔，如需提高准确性可多取几个点。

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10. Summary | 总结

To master cubic graphs for the IGCSE Edexcel exam, remember these key points. A cubic function has the form y = ax³ + bx² + cx + d with a ≠ 0. Its graph is S-shaped with up to two turning points. The leading coefficient determines end behaviour: positive a means the graph falls left and rises right; negative a means the reverse.

要在 IGCSE Edexcel 考试中掌握三次函数图像，请记住以下要点。三次函数的形式为 y = ax³ + bx² + cx + d，其中 a ≠ 0。其图像呈 S 形，最多有两个转折点。首项系数决定端部行为：a 为正时图像左降右升；a 为负时相反。

To sketch a cubic graph, find the y-intercept, factorise to find x-intercepts, differentiate to find turning points, and then draw a smooth curve. Practise factorisation using the factor theorem, and be comfortable with graph transformations. With consistent practice, cubic graphs will become one of your strongest topics.

绘制三次函数图像时，先求 y 截距，再因式分解求 x 截距，接着求导找转折点，最后绘制平滑曲线。熟练运用因式定理进行因式分解，并熟悉图像变换。通过持续练习，三次函数图像将成为你最有把握的考点之一。

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