A-Level Maths: Reciprocal Function Graphs and Asymptotes | A-Level 数学：反比例函数图像与渐近线

2026-08-30 22:47 · 本文部分内容由AI辅助创作

📚 A-Level Maths: Reciprocal Function Graphs and Asymptotes | A-Level 数学：反比例函数图像与渐近线

Among the many functions studied at A-Level, the reciprocal function y = k/x stands out for its simple equation yet rich graphical behaviour. Its graph is a hyperbola with two clearly defined asymptotes. Mastering how to sketch this graph, locate its asymptotes, and apply transformations is essential for Pure Mathematics and many applied topics.

在 A-Level 学习的众多函数中，反比例函数 y = k/x 以其简洁的表达式和丰富的图形特征而格外突出。它的图像是带有两条明确渐近线的双曲线。掌握该图像的画法、渐近线的求法以及平移变换，是纯数学与许多应用主题的基础。

1. What is an Inverse Proportional Function? | 什么是反比例函数？

A variable y is said to be inversely proportional to x when y = k/x, where k is a non-zero constant. This is also written as y ∝ 1/x. The constant k determines both the steepness of the curve and the quadrant in which each branch lies. If k > 0, the product xy = k is positive, so x and y must have the same sign: the branches lie in the first and third quadrants. If k < 0, the branches lie in the second and fourth quadrants.

当 y = k/x（k 为非零常数）时，称变量 y 与 x 成反比，也可写作 y ∝ 1/x。常数 k 同时决定了曲线的陡峭程度以及各分支所在的象限。当 k > 0 时，乘积 xy = k 为正，x 与 y 必须同号，因此两个分支位于第一、三象限；当 k < 0 时，分支位于第二、四象限。

y = k/x ⇔ xy = k, k ≠ 0

The equivalent form xy = k is often the most useful one in solving inverse proportion problems. At every point on the graph, the product of the coordinates equals k. Notice that the graph never passes through the origin, because xy = 0 would require k = 0, which is forbidden.

等价的 xy = k 形式在解决反比例问题中最为常用。图像上的每一点，其横纵坐标之积都等于 k。注意图像永远不会经过原点，因为 xy = 0 将要求 k = 0，而这是不允许的。

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