Properties and Transformations of Reciprocal Function Graphs | 倒数函数图像的性质与变换

📚 Properties and Transformations of Reciprocal Function Graphs | 倒数函数图像的性质与变换

The reciprocal function is one of the most fundamental rational functions in A-Level mathematics. Its graph, a rectangular hyperbola, appears in many physical contexts such as inverse proportion, electrical resistance in parallel circuits, and gravitational force. Understanding its properties and transformations is essential for sketching curves, solving equations graphically, and analysing asymptotic behaviour.

倒数函数是 A-Level 数学中最基础的有理函数之一。它的图像是一条等轴双曲线，出现在许多物理情境中，例如反比例关系、并联电路中的电阻以及万有引力。理解其性质与变换，对于画曲线草图、用图像法解方程和分析渐近线行为至关重要。

1. Definition and Basic Form | 定义与基本形式

The simplest reciprocal function is written as:

最简单的倒数函数写作：

y = 1/x , x ≠ 0

Here the variable x appears in the denominator. Because division by zero is undefined, the value x = 0 is excluded from the domain. In a more general form, the reciprocal function can be written as y = a/(x − h) + k, where a, h and k are constants. This general form allows us to describe any rectangular hyperbola whose asymptotes are horizontal and vertical.

这里变量 x 出现在分母中。由于除以零没有意义，因此 x = 0 被排除在定义域之外。在更一般的形式中，倒数函数可以写作 y = a/(x − h) + k，其中 a、h 和 k 为常数。这种一般形式可以描述任何渐近线为水平线和垂直线的等轴双曲线。

2. Key Properties of y = 1/x | y = 1/x 的关键性质

The graph of y = 1/x has several important features that must be memorised:

y = 1/x 的图像有几个必须牢记的重要特征：

Domain: all real numbers except x = 0, i.e. x ∈ ℝ, x ≠ 0.

定义域：除 x = 0 以外的所有实数，即 x ∈ ℝ，x ≠ 0。
Range: all real numbers except y = 0, i.e. y ∈ ℝ, y ≠ 0.

值域：除 y = 0 以外的所有实数，即 y ∈ ℝ，y ≠ 0。
Asymptotes: the vertical line x = 0 (the y-axis) and the horizontal line y = 0 (the x-axis).

渐近线：垂直渐近线 x = 0（y 轴）和水平渐近线 y = 0（x 轴）。
Symmetry: the graph is odd, so it has rotational symmetry of order 2 about the origin. That is, f(−x) = −f(x).

对称性：图像关于原点具有二阶旋转对称性，即 f(−x) = −f(x)。
Sign behaviour: when x > 0, y > 0 (first quadrant); when x < 0, y < 0 (third quadrant).

符号行为：当 x > 0 时，y > 0（第一象限）；当 x < 0 时，y < 0（第三象限）。

3. The Rectangular Hyperbola Shape | 等轴双曲线的形状

The reciprocal graph consists of two separate branches. As x approaches 0 from the right, y increases without bound toward +∞. As x approaches 0 from the left, y decreases without bound toward −∞. Simultaneously, as x → +∞, y approaches 0 from above; as x → −∞, y approaches 0 from below.

倒数函数的图像由两条分离的曲线分支构成。当 x 从右侧趋近 0 时，y 无限增大趋向 +∞；当 x 从左侧趋近 0 时，y 无限减小趋向 −∞。同时，当 x → +∞ 时，y 从上方趋近 0；当 x → −∞ 时，y 从下方趋近 0。

The maximum or minimum value does not exist for this function. However, there are two notable points: (1, 1) and (−1, −1), which correspond to y = 1/x when the numerator equals the denominator in absolute value. These points help position the curve during sketching.

该函数不存在最大值或最小值。但有两个值得注意的点：(1, 1) 和 (−1, −1)，它们对应于绝对值相等时 y = 1/x 的情形。这些点有助于在作图时确定曲线的位置。

f(1) = 1, f(−1) = −1

4. Vertical Translations: y = 1/x + k | 垂直平移：y = 1/x + k

Adding a constant k outside the fraction shifts the graph vertically. The horizontal asymptote changes from y = 0 to y = k, while the vertical asymptote remains x = 0. The domain is unchanged, but the range becomes y ≠ k.

在分式外加上常数 k 会使图像垂直平移。水平渐近线从 y = 0 变为 y = k，而垂直渐近线仍为 x = 0。定义域不变，但值域变为 y ≠ k。

For example, consider y = 1/x + 2:

例如，考虑 y = 1/x + 2：

The horizontal asymptote is y = 2, so the curve approaches this line as x → ±∞.

水平渐近线为 y = 2，因此当 x → ±∞ 时曲线趋近于这条线。
The two branches now lie in the regions above and below the line y = 2. The whole graph is moved upward by 2 units.

两条分支现在位于直线 y = 2 的上方和下方区域。整个图像向上平移 2 个单位。

5. Horizontal Translations: y = 1/(x − h) | 水平平移：y = 1/(x − h)

Replacing x by (x − h) shifts the graph horizontally. The vertical asymptote changes from x = 0 to x = h, while the horizontal asymptote remains y = 0. The domain becomes x ≠ h, and the range remains y ≠ 0.

将 x 替换为 (x − h) 会使图像水平平移。垂直渐近线从 x = 0 变为 x = h，而水平渐近线仍为 y = 0。定义域变为 x ≠ h，值域仍为 y ≠ 0。

For example, y = 1/(x − 3) is the graph of y = 1/x shifted 3 units to the right. The vertical asymptote is now x = 3. The branch in the first quadrant moves to the region x > 3, y > 0, and the branch in the third quadrant moves to x < 3, y < 0.

例如，y = 1/(x − 3) 是 y = 1/x 向右平移 3 个单位的图像。垂直渐近线现在为 x = 3。第一象限的分支移动到 x > 3，y > 0 的区域，第三象限的分支移动到 x < 3，y < 0 的区域。

6. Vertical Stretch and Reflection: y = a/x | 垂直伸缩与反射：y = a/x

The constant a affects the steepness of the curve and determines whether reflection occurs. If a > 0, the branches retain the same orientation as y = 1/x. If a < 0, the graph is reflected across the x-axis, so one branch lies in the second quadrant and the other in the fourth quadrant.

常数 a 影响曲线的陡峭程度，并决定是否发生反射。若 a > 0，两条分支与 y = 1/x 保持相同方向。若 a < 0，图像关于 x 轴反射，因此一条分支位于第二象限，另一条位于第四象限。

For a > 0, the point (1, a) lies on the graph because f(1) = a. Similarly, (−1, −a) also lies on the graph. For a < 0, those two points are reflected to (−1, a) and (1, −a) in the usual coordinate sense.

当 a > 0 时，点 (1, a) 在图像上，因为 f(1) = a。类似地，(−1, −a) 也在图像上。当 a < 0 时，这两个点相应地变为 (−1, a) 和 (1, −a)。

The larger the absolute value of a, the faster the graph moves away from the axes. For instance, y = 5/x rises more steeply near the y-axis than y = 1/x, and it falls more slowly toward the x-axis at large x.

|a| 越大，图像离开坐标轴的速度越快。例如，y = 5/x 在 y 轴附近上升得比 y = 1/x 更陡，而在 x 趋向无穷大时，它向 x 轴下降得更缓慢。

7. Combined Transformations: y = a/(x − h) + k | 组合变换：y = a/(x − h) + k

The most general reciprocal function combines a horizontal shift, a vertical stretch/reflection, and a vertical shift. The asymptotes are x = h and y = k. The domain is x ≠ h, and the range is y ≠ k. The constant a controls the shape and orientation.

最一般的倒数函数结合了水平平移、垂直伸缩/反射和垂直平移。渐近线为 x = h 和 y = k。定义域为 x ≠ h，值域为 y ≠ k。常数 a 控制形状和方向。

y = a/(x − h) + k, a ≠ 0

A useful way to rewrite this is as a single fraction:

一个有用的方法是将它写成一个分式：

y = [a + k(x − h)] / (x − h)

This form clearly shows that the function is a rational function of the form (linear)/(linear), and that the graph is a hyperbola. The horizontal asymptote is the ratio of the leading coefficients in the numerator and denominator. The vertical asymptote occurs where the denominator is zero.

这种形式清楚地表明该函数是一个分子分母均为一次式的有理函数，其图像是双曲线。水平渐近线是分子和分母中最高次项系数之比。垂直渐近线出现在分母为零处。

8. Finding Intercepts and Asymptotes | 求截距与渐近线

To sketch any reciprocal function y = a/(x − h) + k systematically, follow these steps:

要系统地绘制任意倒数函数 y = a/(x − h) + k 的图像，请按以下步骤操作：

Vertical asymptote: set the denominator equal to zero. For y = a/(x − h) + k, the vertical asymptote is x = h.

垂直渐近线：令分母等于零。对于 y = a/(x − h) + k，垂直渐近线为 x = h。
Horizontal asymptote: as x → ±∞, the term a/(x − h) tends to 0, so y → k. Thus the horizontal asymptote is y = k.

水平渐近线：当 x → ±∞ 时，a/(x − h) 趋向 0，因此 y → k。所以水平渐近线为 y = k。
x-intercept: set y = 0 and solve for x. This requires a/(x − h) + k = 0, giving x = h − a/k provided k ≠ 0.

x 截距：令 y = 0，解方程。由 a/(x − h) + k = 0 得 x = h − a/k（前提是 k ≠ 0）。
y-intercept: set x = 0 and evaluate y = a/(−h) + k = k − a/h, provided h ≠ 0.

y 截距：令 x = 0，代入得 y = a/(−h) + k = k − a/h（前提是 h ≠ 0）。

For example, sketch y = 4/(x − 2) + 1. The vertical asymptote is x = 2 and the horizontal asymptote is y = 1. The x-intercept is found from 0 = 4/(x − 2) + 1, which gives 4/(x − 2) = −1, so x − 2 = −4, hence x = −2. The y-intercept is y = 4/(−2) + 1 = −2 + 1 = −1. These two intercepts and the two asymptotes are enough to draw a reliable sketch.

例如，绘制 y = 4/(x − 2) + 1 的草图。垂直渐近线为 x = 2，水平渐近线为 y = 1。x 截距由 0 = 4/(x − 2) + 1 求得：4/(x − 2) = −1，所以 x − 2 = −4，即 x = −2。y 截距为 y = 4/(−2) + 1 = −2 + 1 = −1。这两个截距和两条渐近线足以绘制出可靠的草图。

9. Behaviour Near Asymptotes | 渐近线附近的行为

When sketching, it is important to know which side of an asymptote each branch lies on. Consider the general function y = a/(x − h) + k. If a > 0, then near x = h:

在绘制草图时，了解每条分支位于渐近线的哪一侧非常重要。考虑一般函数 y = a/(x − h) + k。若 a > 0，则在 x = h 附近：

When x is slightly greater than h, the denominator is positive, so a/(x − h) is positive and large; hence y → +∞.

当 x 略大于 h 时，分母为正，a/(x − h) 为正且很大，因此 y → +∞。
When x is slightly less than h, the denominator is negative, so a/(x − h) is negative and large in magnitude; hence y → −∞.

当 x 略小于 h 时，分母为负，a/(x − h) 为负且绝对值很大，因此 y → −∞。

If a < 0, the directions are reversed: the branch on the right of the vertical asymptote goes to −∞, and the branch on the left goes to +∞. This is a quick check to avoid common sign errors.

若 a < 0，方向相反：垂直渐近线右侧的分支趋向 −∞，左侧的分支趋向 +∞。这是一个快速检验方法，可避免常见的符号错误。

10. Relationship Between y = 1/x and Its Transformations | y = 1/x 与其变换的关系

Every transformed reciprocal function can be obtained from y = 1/x by applying the following sequence of transformations:

每个经过变换的倒数函数都可以通过对 y = 1/x 按以下顺序施加变换而得到：

Transformation / 变换	Equation / 方程	Effect / 效果
Vertical stretch by factor \|a\| and reflection if a < 0 / 纵向伸缩 \|a\| 倍，若 a < 0 则反射	y = a/x	Changes steepness and orientation / 改变陡峭程度和方向
Horizontal shift by h / 水平平移 h	y = a/(x − h)	Moves vertical asymptote to x = h / 将垂直渐近线移至 x = h
Vertical shift by k / 垂直平移 k	y = a/(x − h) + k	Moves horizontal asymptote to y = k / 将水平渐近线移至 y = k

This sequence is also the order you should use when transforming the graph by hand. First stretch, then shift horizontally, then shift vertically.

这个顺序也适用于手动变换图形：先伸缩，再水平平移，最后垂直平移。

11. Common Exam Pitfalls | 常见考试陷阱

Students often lose marks on reciprocal graph questions for avoidable reasons. Here are the most frequent errors:

学生在倒数函数图像题目中常因可避免的原因失分。以下是最常见的错误：

Forgetting the excluded value: the domain of y = 1/x does not include 0. When a graph is drawn, the vertical asymptote must be clearly labelled.

忘记排除值：y = 1/x 的定义域不包含 0。画图时必须清楚标注垂直渐近线。
Mixing up horizontal and vertical asymptotes: for y = a/(x − h) + k, the vertical asymptote is x = h and the horizontal asymptote is y = k. Some students write y = h incorrectly.

混淆水平与垂直渐近线：对于 y = a/(x − h) + k，垂直渐近线是 x = h，水平渐近线是 y = k。有些学生会错误地写成 y = h。
Incorrect sign in x-intercept: when solving a/(x − h) + k = 0, the result is x = h − a/k, not x = h + a/k. Always check with a simple example such as y = 1/x + 1, whose x-intercept is −1.

x 截距的符号错误：解 a/(x − h) + k = 0 时，结果为 x = h − a/k，而不是 x = h + a/k。应始终用简单例子检验，例如 y = 1/x + 1 的 x 截距为 −1。
Drawing only one branch: a reciprocal graph always has two branches unless the domain is restricted. Remember to draw both branches around the vertical asymptote.

只画一条分支：除非定义域受到限制，否则倒数函数图像始终有两条分支。记得在垂直渐近线的两侧各画一条分支。

12. Summary and Quick Revision Table | 总结与快速复习表

The table below summarises the essential facts for the reciprocal function and its general transformation. Use it as a quick revision aid before the exam.

下表总结了倒数函数及其一般变换的基本事实。可将其用作考试前的快速复习工具。

Item / 项目	y = 1/x	y = a/(x − h) + k
Vertical asymptote / 垂直渐近线	x = 0	x = h
Horizontal asymptote / 水平渐近线	y = 0	y = k
Domain / 定义域	x ∈ ℝ, x ≠ 0	x ∈ ℝ, x ≠ h
Range / 值域	y ∈ ℝ, y ≠ 0	y ∈ ℝ, y ≠ k
Key points / 关键点	(1, 1), (−1, −1)	(1 + h, a + k), (h − 1, k − a)
Symmetry / 对称性	Odd: f(−x) = −f(x)	Point symmetry about (h, k)

For the general form, the point symmetry is about the intersection of the two asymptotes, i.e. the point (h, k). This is a useful check when sketching the graph: if you rotate the graph by 180° about (h, k), it should map onto itself.

对于一般形式，点对称中心是两条渐近线的交点，即点 (h, k)。这是绘制图形时一个有用的检验方法：如果将图像绕 (h, k) 旋转 180°，它应与自身重合。

By mastering the basic shape of y = 1/x and understanding how each parameter in y = a/(x − h) + k transforms the graph, you can rapidly sketch any reciprocal function and correctly identify its asymptotes, intercepts, domain and range. Practice with multiple values of a, h and k until the procedure becomes automatic.

通过掌握 y = 1/x 的基本形状，并理解 y = a/(x − h) + k 中每个参数如何变换图像，你就能快速绘制任意倒数函数的草图，并正确识别其渐近线、截距、定义域和值域。多练习不同的 a、h 和 k 值，直到这个过程变得熟练自然。

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Properties and Transformations of Reciprocal Function Graphs | 倒数函数图像的性质与变换