📚 Reciprocal Graphs | 倒数函数图像
Reciprocal graphs and the behaviour of functions such as y = 1/x sit at the very centre of the Edexcel A-Level Mathematics specification. They appear in curve-sketching questions, transformation problems, inequality work and graphical equation solving, and they carry a fixed vocabulary that examiners reward in written answers. This article builds from the basic rectangular hyperbola towards transformed rational functions, and it supplies the exact notation, reference points and asymptote reasoning you need.
倒数函数图像以及 y = 1/x 这类函数的性质，是 Edexcel A-Level 数学大纲的核心内容，频繁出现在曲线作图、函数变换、不等式以及利用图像解方程等题型中，并伴随着一套固定的数学语言，考试评分时十分看重。本文从基本等轴双曲线出发，逐步拓展到变换后的有理函数，并给出答题所需的精确记号、参考点与渐近线推理方法。
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 is not one connected curve but a pair of separate branches: one branch lies in the first quadrant, where both x and y are positive, and the other lies in the third quadrant, where both are negative.
y = 1/x 的图像称为等轴双曲线（rectangular hyperbola）。它不是一条连通的曲线，而是由两条彼此分离的分支组成：一条位于第一象限，此时 x 与 y 都为正；另一条位于第三象限，此时 x 与 y 都为负。
The domain is every real number except x = 0, and the range is every real number except y = 0. Division by zero is undefined, and no finite value of x can ever make 1/x equal to zero, so the curve never meets either coordinate axis.
定义域是所有实数中除去 x = 0 的部分，值域是所有实数中除去 y = 0 的部分。除以零没有定义，而且任何有限的 x 值都不可能使 1/x 等于零，因此曲线永远不与坐标轴相交。
The curve passes through (1, 1) and (−1, −1). Other reliable reference points are (2, 0.5), (0.5, 2), (−2, −0.5) and (−0.5, −2). These points are extremely useful when sketching transformed versions of the graph.
曲线经过 (1, 1) 和 (−1, −1)。其他可靠的参考点包括 (2, 0.5)、(0.5, 2)、(−2, −0.5) 和 (−0.5, −2)。在绘制变换后的图像时，这些点非常有用。
Because f(−x) = −1/x = −f(x), the function is odd. Its graph therefore has 180° rotational symmetry about the origin.
由于 f(−x) = −1/x = −f(x)，该函数为奇函数，因此其图像关于原点具有 180° 旋转对称性。
y = 1/x, x ≠ 0, y ≠ 0
2. Asymptotes and Behaviour at Infinity | 渐近线与无穷远处行为
An asymptote is a straight line that a curve approaches arbitrarily closely but never reaches. Every reciprocal graph of the form y = 1/(x − a) + b has exactly one vertical and one horizontal asymptote.
渐近线是指曲线可以任意接近、但永远无法到达的直线。任何形如 y = 1/(x − a) + b 的倒数函数图像都恰有一条垂直渐近线和一条水平渐近线。
The vertical line x = 0 is a vertical asymptote of y = 1/x, because y → +∞ as x → 0⁺ and y → −∞ as x → 0⁻. The two branches therefore shoot off in opposite vertical directions beside the y-axis.
竖直线 x = 0 是 y = 1/x 的垂直渐近线，因为当 x → 0⁺ 时 y → +∞，而当 x → 0⁻ 时 y → −∞。因此两条分支在 y 轴两侧沿相反方向竖直延伸。
The horizontal line y = 0 is a horizontal asymptote, because y → 0⁺ as x → +∞ and y → 0⁻ as x → −∞. The curve flattens towards the x-axis at both ends without ever touching it.
水平线 y = 0 是水平渐近线，因为当 x → +∞ 时 y → 0⁺，当 x → −∞ 时 y → 0⁻。曲线在两端都趋于扁平、无限贴近 x 轴，但始终不相交。
as x → 0⁺, y → +∞  as x → 0⁻, y → −∞  as x → ±∞, y → 0
3. Key Features, Monotonicity and Symmetry | 关键特征、单调性与对称性
Differentiating gives dy/dx = −1/x², which is negative for every permitted value of x. There are no stationary points, no maximum and no minimum, so the curve is decreasing on each branch separately.
求导得 dy/dx = −1/x²，对任何允许的 x 值它都是负的。函数没有驻点、没有极大值也没有极小值，因此曲线在每一条分支上分别递减。
Be careful: the function is not decreasing over its whole domain. Taking x₁ = −0.5 and x₂ = 0.5 gives x₁ < x₂ but f(x₁) = −2 < 2 = f(x₂), so a single decreasing statement covering both branches is wrong. Always say "decreasing on (−∞, 0) and on (0, +∞)".
要小心：该函数并非在整个定义域上递减。取 x₁ = −0.5、x₂ = 0.5，有 x₁ < x₂，但 f(x₁) = −2 < 2 = f(x₂)，所以用一个"递减"覆盖两条分支是错误的。应当写作"在 (−∞, 0) 上和 (0, +∞) 上分别递减"。
The graph is also symmetric about the lines y = x and y = −x. Swapping x and y in y = 1/x returns x = 1/y, which rearranges to the same equation, confirming self-inverse behaviour.
图像还关于直线 y = x 和 y = −x 对称。把 y = 1/x 中的 x 与 y 互换得到 x = 1/y，整理后仍是同一个方程，这证明该函数是自反函数。
- Intercepts: none, because x ≠ 0 and y ≠ 0. | 截距：没有，因为 x ≠ 0 且 y ≠ 0。
- Self-intersections: none. | 自交点：没有。
- Odd function with 180° rotational symmetry about O. | 奇函数，关于原点 O 有 180° 旋转对称性。
- Both axes act as asymptotes. | 两条坐标轴同时充当渐近线。
4. Translations: y = 1/(x − a) + b | 平移变换：y = 1/(x − a) + b
Replacing x by x − a translates the graph a units to the right, and adding b translates it b units upwards. The combined transformation is a translation by the vector (a, b).
把 x 换成 x − a 使图像向右平移 a 个单位，再加上 b 使图像向上平移 b 个单位。两者合成的变换就是按向量 (a, b) 的平移。
The asymptotes move with the curve. The vertical asymptote becomes x = a and the horizontal asymptote becomes y = b. Their point of intersection (a, b) is the centre of the hyperbola and the centre of its 180° rotational symmetry.
渐近线随曲线一起移动。垂直渐近线变为 x = a，水平渐近线变为 y = b。它们的交点 (a, b) 就是双曲线的中心，也是 180° 旋转对称的中心。
The domain is x ≠ a and the range is y ≠ b. The reference points (1, 1) and (−1, −1) shift to (a + 1, b + 1) and (a − 1, b − 1).
定义域为 x ≠ a，值域为 y ≠ b。参考点 (1, 1) 和 (−1, −1) 分别平移到 (a + 1, b + 1) 和 (a − 1, b − 1)。
y = 1/x shifted by (a, b) gives y = 1/(x − a) + b
5. Vertical Scaling and the Family y = k/x | 纵向缩放与 y = k/x 族
Multiplying the whole function by a constant k produces y = k/x, a one-way stretch of scale factor |k| parallel to the y-axis. The asymptotes are unchanged: still x = 0 and y = 0.
把整个函数乘以常数 k 得到 y = k/x，这相当于沿 y 轴方向、缩放因子为 |k| 的单向拉伸。渐近线保持不变，仍然是 x = 0 和 y = 0。
When k > 0, both branches lie in the first and third quadrants, and the curve passes through (1, k), (k, 1) and (2, k/2). The larger k is, the further the branches sit from the origin.
当 k > 0 时，两条分支位于第一、第三象限，曲线经过 (1, k)、(1, k) 的水平对应点 (k, 1) 以及 (2, k/2)。k 越大，分支离原点越远。
When k < 0, the branches move into the second and fourth quadrants. This is exactly the reflection of the k > 0 case in the x-axis, and it is also a 180° rotation about the origin.
当 k < 0 时，分支移到第二、第四象限。这正好是 k > 0 情形关于 x 轴的反射，同时也等价于关于原点作 180° 旋转。
- k > 0: branches in quadrants 1 and 3. | k > 0：分支在第一、第三象限。
- k < 0: branches in quadrants 2 and 4. | k < 0：分支在第二、第四象限。
- Asymptotes always x = 0 and y = 0. | 渐近线始终是 x = 0 与 y = 0。
6. Reciprocals of Linear Functions: y = 1/(ax + b) | 一次函数的倒数：y = 1/(ax + b)
For y = 1/(ax + b), factorise the denominator as a(x + b/a) so that y = (1/a) × 1/(x + b/a). This is the standard hyperbola translated b/a units horizontally and stretched vertically by
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