Reciprocal Graphs: Asymptotes and Symmetry | 倒数函数图像：渐近线与对称性

📚 Reciprocal Graphs: Asymptotes and Symmetry | 倒数函数图像：渐近线与对称性

The reciprocal function is one of the most important non-linear graphs in the IGCSE syllabus. Its distinctive two-branch shape, central symmetry, and asymptotic behaviour appear frequently in exam questions. In this revision guide, you will learn how to sketch reciprocal graphs, how to locate their asymptotes, and how to identify their symmetry properties with confidence.

倒数函数是 IGCSE 考纲中最重要的非直线图形之一。它独特的双支形状、中心对称性和渐近行为在考试题目中经常出现。在本复习指南中，你将学习如何绘制倒数函数图像、如何确定渐近线，以及如何准确识别其对称性质。

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

The simplest reciprocal function is defined by y = 1/x, where x cannot be zero. As x takes any non-zero real value, y is its multiplicative inverse. The graph of this function is a hyperbola with two separate branches: one in the first quadrant and one in the third quadrant.

最简单的倒数函数由 y = 1/x 定义，其中 x 不能为零。当 x 取任何非零实数时，y 是它的乘法倒数。这个函数的图像是一条双曲线，由两条分离的分支组成：一条在第一象限，另一条在第三象限。

y = 1/x (x ≠ 0)

Because the product x × y is always equal to 1, both x and y must have the same sign. This explains why the graph never appears in the second or fourth quadrant for the basic function.

由于 x × y 始终等于 1，所以 x 与 y 必须同号。这就解释了为什么基本函数的图像永远不会出现在第二或第四象限。

2. Key Features of y = 1/x | y = 1/x 的关键特征

For the graph of y = 1/x, the domain is all real numbers except x = 0, and the range is all real numbers except y = 0. The curve approaches but never touches the coordinate axes.

对于 y = 1/x 的图像，定义域是所有不等于 0 的实数，值域是所有不等于 0 的实数。曲线无限逼近但永远不会触碰坐标轴。

Domain: x ∈ ℝ, x ≠ 0
Range: y ∈ ℝ, y ≠ 0
Asymptotes: x = 0 and y = 0

定义域：x ∈ ℝ，x ≠ 0
值域：y ∈ ℝ，y ≠ 0
渐近线：x = 0 与 y = 0

The graph has no x-intercept and no y-intercept because the numerator is constant and non-zero. The only way to find intersections with the axes is through transformations or solving equations.

该图像没有 x 截距也没有 y 截距，因为分子是常数且不为零。通过变换或解方程才能找到与坐标轴的交点。

3. Vertical Asymptote: x = 0 | 垂直渐近线：x = 0

A vertical asymptote is a vertical line that the graph approaches as the input x gets closer to a value where the function is undefined. For y = 1/x, the denominator becomes zero at x = 0, so the line x = 0 is a vertical asymptote.

垂直渐近线是当自变量 x 趋近于某个使函数无定义的值时，图像所逼近的垂直直线。对于 y = 1/x，当 x = 0 时分母为零，因此直线 x = 0 是一条垂直渐近线。

As x → 0⁺ (from the positive side), 1/x becomes very large positive: y → +∞. As x → 0⁻ (from the negative side), 1/x becomes very large negative: y → −∞.

当 x → 0⁺（从正方向趋近）时，1/x 变成非常大的正数：y → +∞。当 x → 0⁻（从负方向趋近）时，1/x 变成非常大的负数：y → −∞。

x → 0⁺ ⇒ y → +∞，x → 0⁻ ⇒ y → −∞

This behaviour shows that the graph has two branches that shoot upward and downward respectively as they approach the y-axis.

这种性质表明图像的两条分支在接近 y 轴时分别向上和向下无限延伸。

4. Horizontal Asymptote: y = 0 | 水平渐近线：y = 0

A horizontal asymptote describes the value that y approaches when x becomes extremely large in magnitude. For y = 1/x, as x → +∞ or x → −∞, the value of y gets closer and closer to 0.

水平渐近线描述的是当 x 的绝对值变得极大时 y 所逼近的值。对于 y = 1/x，当 x → +∞ 或 x → −∞ 时，y 的值越来越接近 0。

x → ±∞ ⇒ y → 0

The line y = 0 is therefore a horizontal asymptote. The graph never crosses this line because no finite value of x can make 1/x equal to 0.

因此直线 y = 0 是一条水平渐近线。图像永远不会穿过这条线，因为没有任何有限的 x 值能使 1/x 等于 0。

In the first quadrant, the curve approaches the x-axis from above; in the third quadrant, it approaches from below.

在第一象限，曲线从上方接近 x 轴；在第三象限，曲线从下方接近 x 轴。

5. Graph Transformations | 图像变换

You need to understand how parameters a, h and k affect the reciprocal graph y = a/(x − h) + k. Each parameter has a clear geometric meaning.

你需要理解参数 a、h、k 如何影响倒数函数图像 y = a/(x − h) + k。每个参数都有明确的几何意义。

Parameter 参数	Effect 效果
a	Vertical stretch or compression; if a < 0 the graph is reflected in the x-axis.
a	垂直拉伸或压缩；若 a < 0，图像关于 x 轴反射。
h	Horizontal translation: the vertical asymptote moves to x = h.
h	水平平移：垂直渐近线移动到 x = h。
k	Vertical translation: the horizontal asymptote moves to y = k.
k	垂直平移：水平渐近线移动到 y = k。

Note that the opposite sign of h in the denominator means that the vertical asymptote is x = h, not x = −h. For example, y = 1/(x − 3) has vertical asymptote x = 3.

注意分母中 h 的符号相反，因此垂直渐近线是 x = h，而不是 x = −h。例如，y = 1/(x − 3) 的垂直渐近线是 x = 3。

6. Finding Asymptotes from the Equation | 从方程求渐近线

For any function written in the form y = a/(x − h) + k, the asymptotes can be read directly:

对于任何写成 y = a/(x − h) + k 形式的函数，我们可以直接读出渐近线：

Vertical asymptote: set the denominator to zero: x − h = 0, so x = h.
Horizontal asymptote: as x → ±∞, the fraction tends to 0, so y → k. Therefore y = k.

垂直渐近线：令分母为零：x − h = 0，所以 x = h。
水平渐近线：当 x → ±∞ 时，分式趋近于 0，所以 y → k。因此 y = k。

y = a/(x − h) + k ⇒ x = h，y = k

For example, y = 3/(x + 2) − 5 has vertical asymptote x = −2 and horizontal asymptote y = −5.

例如，y = 3/(x + 2) − 5 的垂直渐近线是 x = −2，水平渐近线是 y = −5。

7. Symmetry of y = 1/x | y = 1/x 的对称性

The graph of y = 1/x has three important symmetry properties. First, it is an odd function because f(−x) = −f(x). This means the graph has rotational symmetry of order 2 about the origin: rotating it by 180° produces the same graph.

y = 1/x 的图像具有三个重要的对称性质。首先，它是一个奇函数，因为 f(−x) = −f(x)。这意味着图像关于原点具有二阶旋转对称性：将其旋转 180° 后得到相同的图像。

Second, the graph is symmetrical about the line y = x. If you reflect the curve over the line y = x, every point (a, b) maps to (b, a), and since b = 1/a, the reflected point satisfies y = 1/x because a = 1/b.

其次，图像关于直线 y = x 对称。如果把曲线关于直线 y = x 反射，每个点 (a, b) 映射到 (b, a)，由于 b = 1/a，反射后的点满足 y = 1/x，因为 a = 1/b。

Third, the graph is also symmetrical about the line y = −x. Reflecting (a, b) over y = −x gives (−b, −a), and the point satisfies (−b) × (−a) = ab = 1.

第三，图像还关于直线 y = −x 对称。将 (a, b) 关于 y = −x 反射得到 (−b, −a)，而该点满足 (−b) × (−a) = ab = 1。

对称轴：y = x，y = −x；对称中心：原点 (0, 0)

These symmetry properties help you sketch the graph quickly if you have one branch, because the other branch is obtained by reflection or rotation.

这些对称性质可以帮助你快速绘制图像：如果已经画出一条分支，另一条分支可以通过反射或旋转得到。

8. Symmetry of Transformed Reciprocal Graphs | 变换后图像的对称性

For y = a/(x − h) + k, the centre of symmetry is the point where the two asymptotes intersect: (h, k). The graph is rotationally symmetric about this point by 180°.

对于 y = a/(x − h) + k，对称中心是两条渐近线的交点：(h, k)。图像关于该点具有 180° 旋转对称性。

The two lines of reflectional symmetry are the diagonals that pass through the centre and have slopes +1 and −1. Their equations are:

两条反射对称轴是通过中心且斜率分别为 +1 和 −1 的对角线。它们的方程为：

y − k = x − h 与 y − k = −(x − h)

Equivalently, the axes of symmetry are y = x + (k − h) and y = −x + (k + h). The value of a does not affect these axes; it only stretches the branches and determines which quadrants they occupy.

等价地，对称轴为 y = x + (k − h) 与 y = −x + (k + h)。a 的值不会影响对称轴；它只负责拉伸分支，并决定分支位于哪些象限。

9. How to Sketch a Reciprocal Graph | 如何绘制倒数函数图像

Follow a reliable step-by-step method to sketch y = a/(x − h) + k accurately.

按照以下可靠的分步方法，可以准确绘制 y = a/(x − h) + k 的图像。

Identify the vertical asymptote x = h and draw it as a dashed vertical line.

确定垂直渐近线 x = h，并用虚线画出垂直直线。
Identify the horizontal asymptote y = k and draw it as a dashed horizontal line.

确定水平渐近线 y = k，并用虚线画出水平直线。
Find the intercepts by substitution: set x = 0 to find the y-intercept (if possible), and set y = 0 to find the x-intercept (if possible).

通过代入法求交点：令 x = 0 求 y 截距（如可能），令 y = 0 求 x 截距（如可能）。
Plot a few points on each side of the vertical asymptote, especially x = h ± 1 and x = h ± 2.

在垂直渐近线两侧各取几个点，特别是 x = h ± 1 和 x = h ± 2。
Draw two smooth branches that approach both asymptotes, making sure they pass through the plotted points.

画出两条光滑分支，使它们逼近两条渐近线，并确保经过所取的点。
If a > 0, the branch on the right of the vertical asymptote is above the horizontal asymptote when the signs of (x − h) and a are positive; use test points to decide the correct quadrants.

若 a > 0，垂直渐近线右侧的分支在水平渐近线上方；可用测试点判断正确的象限。

Always label the asymptotes with their equations on your graph in an exam.

考试中一定要在图像上标出渐近线的方程。

10. Worked Example | 例题精讲

Question: Sketch y = 2/(x − 3) + 1. State the equations of the asymptotes and describe the symmetry.

题目：画出 y = 2/(x − 3) + 1 的图像，写出渐近线方程并描述其对称性。

Step 1: Compare with y = a/(x − h) + k. Here a = 2, h = 3, k = 1.

第一步：与 y = a/(x − h) + k 比较。这里 a = 2，h = 3，k = 1。

Step 2: Vertical asymptote is x = 3. Horizontal asymptote is y = 1.

第二步：垂直渐近线是 x = 3，水平渐近线是 y = 1。

Step 3: Find intercepts. When x = 0, y = 2/(0 − 3) + 1 = −2/3 + 1 = 1/3. So the y-intercept is (0, 1/3). When y = 0, 0 = 2/(x − 3) + 1 ⇒ 2/(x − 3) = −1 ⇒ x − 3 = −2 ⇒ x = 1. So the x-intercept is (1, 0).

第三步：求交点。当 x = 0 时，y = 2/(0 − 3) + 1 = −2/3 + 1 = 1/3。所以 y 截距为 (0, 1/3)。当 y = 0 时，0 = 2/(x − 3) + 1 ⇒ 2/(x − 3) = −1 ⇒ x − 3 = −2 ⇒ x = 1。所以 x 截距为 (1, 0)。

Step 4: The centre of symmetry is (3, 1). The lines of symmetry are y − 1 = x − 3 and y − 1 = −(x − 3), which simplify to y = x − 2 and y = −x + 4.

第四步：对称中心为 (3, 1)。对称轴为 y − 1 = x − 3 与 y − 1 = −(x − 3)，化简为 y = x − 2 与 y = −x + 4。

Step 5: Because a = 2 > 0, the branch to the right of x = 3 is above y = 1, and the branch to the left is below y = 1.

第五步：因为 a = 2 > 0，x = 3 右侧的分支在 y = 1 上方，左侧的分支在 y = 1 下方。

11. Common Mistakes and Tips | 常见错误与提示

Many students confuse the sign of h. Remember that in y = a/(x − h) + k, the vertical asymptote is x = h even though the denominator is written as x − h. If you see y = a/(x + p), then the asymptote is x = −p.

许多学生会混淆 h 的符号。请记住，在 y = a/(x − h) + k 中，虽然分母写作 x − h，但垂直渐近线是 x = h。如果看到 y = a/(x + p)，那么渐近线是 x = −p。

Another common error is drawing a graph that crosses an asymptote. By definition, the graph approaches asymptotes but never touches or crosses them (for reciprocal functions). Always leave a small gap or extend the branch toward infinity.

另一个常见错误是画出穿过渐近线的图像。根据定义，图像无限接近渐近线但永远不会接触或穿过它们（对于倒数函数）。务必留有间隙或向无穷远处延伸。

When working with a negative value of a, the branches swap quadrants. For example, y = −1/x has one branch in the second quadrant and one in the fourth quadrant.

当 a 为负数时，两条分支会交换象限。例如，y = −1/x 的一条分支在第二象限，另一条在第四象限。

Always use test points to confirm the location of each branch. A single point on each side of the vertical asymptote is usually enough.

务必使用测试点确认每条分支的位置。通常在垂直渐近线两侧各取一个点就足够了。

12. Conclusion | 总结

Reciprocal graphs are determined by their vertical asymptote x = h, horizontal asymptote y = k, and the centre of symmetry (h, k). The sign and magnitude of a control the orientation and steepness of the branches. Understanding the symmetry of y = 1/x and its transformed versions allows you to sketch accurately and answer exam questions with speed.

倒数函数图像由垂直渐近线 x = h、水平渐近线 y = k 以及对称中心 (h, k) 决定。a 的符号和大小控制分支的方向与陡峭程度。理解 y = 1/x 及其变换形式的对称性，可以帮助你准确作图并在考试中快速作答。

Remember the golden rules: set the denominator to zero for the vertical asymptote, let x tend to infinity for the horizontal asymptote, and use the centre (h, k) to analyse symmetry.

请记住黄金法则：令分母为零求垂直渐近线，令 x 趋向无穷求水平渐近线，利用中心 (h, k) 分析对称性。

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Reciprocal Graphs: Asymptotes and Symmetry | 倒数函数图像：渐近线与对称性