Reciprocal Graphs | 倒数函数图像

📚 Reciprocal Graphs | 倒数函数图像

Reciprocal graphs are a core part of the Edexcel A-Level Mathematics specification, appearing in curve sketching, transformations, inequalities and equation solving. This article builds up from y = 1/x to transformed rational functions and gives the exact vocabulary needed for exam answers.

倒数函数图像是 Edexcel A-Level 数学大纲的核心内容，出现在曲线绘图、函数变换、不等式以及解方程等题型中。本文从 y = 1/x 逐步拓展到变换后的有理函数，并给出答题所需的精确数学语言。

1. The Basic Reciprocal Function y = 1/x | 基本倒数函数 y = 1/x

The graph of y = 1/x is called a rectangular hyperbola. It consists of two separate branches, one in the first quadrant and one in the third quadrant.

y = 1/x 的图像称为等轴双曲线。它由两个独立分支组成，分别位于第一象限和第三象限。

The domain is all real x except x = 0, and the range is all real y except y = 0. This is because division by zero is undefined and no value of x makes 1/x equal to zero.

定义域是所有实数 x 除了 x = 0，值域是所有实数 y 除了 y = 0。这是因为除以零没有定义，而且没有任何 x 值能使 1/x 等于零。

The graph passes through (1, 1) and (−1, −1). These are useful reference points when sketching transformations.

图像经过点 (1, 1) 和 (−1, −1)。这些是绘制变换图像时很有用的参考点。

Since f(−x) = −1/x = −f(x), the function is odd, so its graph has 180° rotational symmetry about the origin.

由于 f(−x) = −1/x = −f(x)，该函数为奇函数，因此图像关于原点有 180° 旋转对称性。

y = 1/x, x ≠ 0

2. Asymptotes and Behaviour at Infinity | 渐近线与无穷远处行为

The vertical line x = 0 is a vertical asymptote because y → +∞ as x → 0⁺ and y → −∞ as x → 0⁻.

竖直线 x = 0 是一条垂直渐近线，因为当 x → 0⁺ 时 y → +∞，当 x → 0⁻ 时 y → −∞。

The horizontal line y = 0 is a horizontal asymptote because y → 0⁺ as x → +∞ and

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

更多咨询请联系16621398022（同微信）