📚 Graphs 6: Cubic and Reciprocal Graphs | 图形6：三次函数与倒数函数图形
In IGCSE Edexcel Mathematics, Graphs 6 takes you beyond straight lines and parabolas into the world of cubic and reciprocal functions. These non-linear graphs appear frequently in algebra and real-life contexts, and mastering their shapes, key features, and plotting techniques is essential for success in both the core and extended tiers. This article will guide you step by step through cubic curves, reciprocal curves, asymptotes, graphical solutions, and common exam pitfalls.
在IGCSE Edexcel数学中，图形6将带你超越直线和抛物线，进入三次函数和倒数函数的世界。这些非线性图形在代数及实际生活情境中频繁出现，掌握它们的形状、关键特征以及绘制技巧，对于在核心和扩展考试中取得成功至关重要。本文将逐步引导你学习三次曲线、倒数曲线、渐近线、图形解法以及常见的考试陷阱。
1. Introduction to Cubic Graphs | 三次函数图形简介
A cubic function has the general form y = ax³ + bx² + cx + d, where a, b, c and d are constants and a ≠ 0. The simplest cubic graph is y = x³ ³ passes through the origin (0,0) and has rotational symmetry of order 2 about the origin.
三次函数的一般形式为y = ax³ + bx² + cx + d，其中a、b、c和d为常数，且a ≠ 0。最简单的三次图形是y = x³。与对称的U形二次函数不同，三次图形是一条光滑的S形曲线，在y轴的正负方向上都延伸至无穷。y = x³ 的图形经过原点(0,0)，并且关于原点具有二阶旋转对称性。
2. Plotting Cubic Graphs | 绘制三次函数图形
To plot a cubic graph accurately, you need to create a table of values for a range of x-values, typically from –3 to 3. For example, consider y = x³ – 3x:
要精确绘制三次图形，你需要为一系列x值（通常从–3到3）创建一个数值表。例如，考虑y = x³ – 3x：
| x | –3 | –2 | –1 | 0 | 1 | 2 | 3 |
| y | –18 | –2 | 2 | 0 | –2 | 2 | 18 |
Plot these points on a grid and join them with a smooth curve. Notice the symmetry: the y-values for x and –x are opposites, which reflects the fact that this is an odd function.
在网格上标出这些点，并用一条光滑的曲线将它们连接起来。注意其中的对称性：x和–x对应的y值互为相反数，这反映了该函数是一个奇函数。
3. Key Characteristics of Cubic Curves | 三次曲线的主要特征
ight. The y-intercept is always at (0, d), and the curve may cross the x-axis at up to three points, which are the roots of the equation ax³ + bx² + cx + d = 0.
三次图形最多可以有两个转折点（一个局部极大值和一个局部极小值），也可以一个都没有，具体取决于各项系数。首项系数a的符号决定了整体形状：若a > 0，图形左端下降、右端上升；若a < 0，则左端上升、右端下降。y轴截距始终位于(0, d)，曲线最多可在三个点与x轴相交，这些交点也就是方程 ax³ + bx² + cx + d = 0 的根。
4. Reciprocal Graphs: y = k/x | 倒数函数图形: y = k/x
The reciprocal function y = k/x (where k is a non-zero constant) produces a hyperbola made up of two separate branches. When k > 0, the branches lie in the first and third quadrants; when k < 0, they lie in the second and fourth quadrants. The graph never touches the x-axis or the y-axis; instead, it approaches them infinitely closely. These boundary lines are called asymptotes.
倒数函数y = k/x（k为非零常数）生成一条由两个独立分支组成的双曲线。当k > 0时，两个分支分别位于第一和第三象限；当k < 0时，它们位于第二和第四象限。该图形永远不与x轴或y轴相交，而是无限趋近于它们。这些边界线被称为渐近线。
5. Reciprocal Graphs: y = k/x² | 倒数函数图形: y = k/x²
Another important type is y = k/x² different from the y = k/x hyperbola because the curve approaches the y-axis more steeply.
另一种重要的类型是y = k/x²。当k > 0时，图形有两个分支，都位于x轴上方，分处第一和第二象限，并且关于y轴对称。当k < 0时，两个分支都位于x轴下方，处于第三和第四象限。其渐近线仍然为x = 0和y = 0，但形状与y = k/x的双曲线不同，因为曲线更陡峭地逼近y轴。
6. Understanding Asymptotes | 理解渐近线
zero is undefined and y can never equal zero. When sketching, always draw these as dashed lines and label them. As x gets very large positively or negatively, the value of y approaches 0; as x approaches 0 from either side, the value of y becomes very large positively or negatively depending on the sign of k and the direction of approach.
渐近线是图形无限接近但永不触及的直线。对于倒数图形，由于除以零无意义，且y永远不可能等于零，因此既有水平渐近线（y = 0）也有垂直渐近线（x = 0）。在绘制草图时，务必用虚线画出这些渐近线，并加以标注。当x在正方向或负方向上变得非常大时，y的值趋近于0；当x从任意一侧趋近于0时，y的值根据k的符号和趋近方向变得非常大（正或负）。
7. Sketching Cubic and Reciprocal Graphs | 草绘三次与倒数图形
For a cubic graph, identify the y-intercept, the sign of a, and use a table of values if necessary to find the general shape and any turning points. Mark the x-intercepts if you can solve the cubic equation. For reciprocal graphs, note the value of k, the quadrants of the branches, and draw the asymptotes first. Then sketch the two branches getting closer to the asymptotes without touching them. Practice helps to recognise the standard shapes quickly.
对于三次图形，先确定y轴截距和a的符号，必要时使用数值表来找到整体形状及转折点。如果能解出三次方程，则标出x轴截距。对于倒数图形，注意k的值以及分支所在的象限，并首先画出渐近线，然后勾画两条逐渐接近但永不触及渐近线的分支。通过练习可以快速识别这些标准形状。
8. Solving Equations Graphically | 图解法解方程
Graphs can be used to solve equations that may be difficult to tackle algebraically. For instance, to solve x³ – 3x = 1, you can draw the graph of y = x³ – uation so that one side is a cubic you have already plotted, and the other is a simpler function such as a straight line.
可以利用图形来求解那些可能难以用代数方法处理的方程。例如，要求解x³ – 3x = 1，你可以在同一坐标系中画出y = x³ – 3x以及水平线y = 1的图形，其交点的x坐标就是方程的解。另一种做法是：对方程进行移项，使一边是你已经画好的三次函数，另一边则是一个更简单的函数，比如一条直线。
9. Intersection Points and Simultaneous Equations | 交点与联立方程
ctions between a reciprocal graph and another curve can be used to solve more complex equations. When using graphs, always read values to the accuracy allowed by the grid and clearly indicate the points of intersection.
三次曲线与直线的交点坐标给出了方程组 y = f(x) 和 y = mx + c 的联立解。同样，倒数图形与另一条曲线的交点可用于求解更复杂的方程。在使用图形时，应始终按网格所允许的精度读取数值，并清晰标明交点。
10. Transformations of Graphs | 图形变换
f(x) + a shifts the graph vertically by a units, y = f(x + a) shifts it horizontally, y = –f(x) reflects it in the x-axis, and y = f(– y. Recognising these can help you sketch related graphs quickly without a table of values.
三次和倒数图形可以利用标准函数记号进行变换。例如，y = f(x) + a 将图形垂直平移a个单位，y = f(x + a) 将其水平平移，y = –f(x) 将其关于x轴反射，y = f(–x) 则关于y轴反射。伸缩和其他变换也同样适用。辨认这些变换能够帮助你无需数值表即可快速绘制相关图形。
11. Common Pitfalls and Tips | 常见错误与技巧
bel the intersection points clearly and ensure your curve is smooth, not pointed. Also, do not confuse the reciprocal graphs y = 1/x and y = 1/x² – they look different and the latter is always positive when k > 0.
许多学生会忘记倒数图形由两个独立的分支组成，从而错误地将它们跨过渐近线连接起来。另一些人在三次图形存在两个转折点时，仍将其画成简单的S形曲线。当不确定时，务必检查a的符号并使用数值表。在用图解法求解时，应清晰标出交点，并确保曲线光滑、没有尖角。此外，切勿混淆 y = 1/x 和 y = 1/x² 的图形——它们的外观不同，后者在k > 0时始终为正。
12. Summary and Key Points | 总结与要点
ts, understanding asymptotes, and using transformations are essential skills. Practice reading off solutions from intersection points and always label axes, asymptotes, and key features in an exam.
三次函数图形 y = ax³ + bx² + cx + d 是至多有两个转折点的S形曲线，其末端走向取决于a的符号。倒数图形 y = k/x 和 y = k/x² 均具有两个分支，并以 x = 0 和 y = 0 为渐近线。绘制点、理解渐近线以及使用变换都是必备技能。要练习通过交点读取方程的解，并在考试中始终标注坐标轴、渐近线和关键特征。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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