Reciprocal Function Graphs: Asymptotes and Properties | 倒数函数图像：渐近线与性质

📚 Reciprocal Function Graphs: Asymptotes and Properties | 倒数函数图像：渐近线与性质

Reciprocal functions are a core topic in Edexcel A-Level Mathematics. They appear in coordinate geometry, rational functions, and even in calculus, for example when differentiating x⁻¹ or integrating 1/x. A solid grasp of the asymptotes and shape of y = 1/x is essential for sketching more complex rational expressions.

倒数函数是 Edexcel A-Level 数学的核心内容。它们出现在坐标几何、有理函数，甚至微积分中，例如对 x⁻¹ 求导或积分 1/x。牢固理解 y = 1/x 的渐近线和形状，对于绘制更复杂的有理表达式图像至关重要。

1. What is a Reciprocal Function? | 什么是倒数函数？

The standard reciprocal function is defined by f(x) = 1/x for all real x except x = 0. It is also commonly written as y = x⁻¹. The graph is a hyperbola made up of two separate branches: one in the first quadrant and one in the third quadrant.

标准倒数函数定义为 f(x) = 1/x，其中 x 为实数且 x ≠ 0。它通常也写作 y = x⁻¹。图像是一条由两条独立分支组成的双曲线：一条在第一象限，另一条在第三象限。

Because the numerator is a constant, the product x × f(x) is always 1. This means that as one variable becomes very large, the other becomes very small, and vice versa.

由于分子是常数，x × f(x) 始终等于 1。这意味着当一个变量变得很大时，另一个变量会变得很小，反之亦然。

2. The Graph of y = 1/x | y = 1/x 的图像

For x > 0, as x increases, f(x) decreases steadily toward 0. For x < 0, as x decreases, f(x) also moves toward 0 from below. The curve never touches either axis, because neither x nor y can ever be zero.

当 x > 0 时，随着 x 增大，f(x) 逐步减小并趋近于 0。当 x < 0 时，随着 x 减小，f(x) 同样从下方趋近于 0。曲线永远不会接触坐标轴，因为 x 和 y 都不可能为零。

The following table shows some values for the first quadrant branch.

下表显示了第一象限分支的一些取值。

This clearly shows the decreasing trend. In the third quadrant, the values are symmetric in magnitude but negative.

这清楚展示了递减趋势。在第三象限中，数值在大小上对称，但符号为负。

3. Vertical Asymptote | 垂直渐近线

A vertical asymptote is a vertical line that the graph approaches but never crosses or touches. For y = 1/x, the vertical asymptote is x = 0.

垂直渐近线是一条图像趋近但永不跨越或接触的竖直直线。对于 y = 1/x，垂直渐近线是 x = 0。

As x approaches 0 from the right, y

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）