Reciprocal Graphs | 倒数函数图像

📚 Reciprocal Graphs | 倒数函数图像

Reciprocal functions appear throughout Edexcel A Level Mathematics, especially in Pure Mathematics topics on curve sketching, transformations, asymptotes and inequalities. A solid understanding of y = k/x and y = k/x² is essential for drawing accurate graphs and solving related equations.

倒数函数在 Edexcel A Level 数学中经常出现，尤其是在纯数学的曲线绘制、图像变换、渐近线与不等式等内容中。熟练掌握 y = k/x 和 y = k/x² 的图像，对于准确作图和求解相关方程非常重要。

1. Definition of a Reciprocal Function | 倒数函数的定义

A reciprocal function is any function that can be written in the form y = k / x, where k is a non-zero constant. It is often expressed using a negative power as y = kx⁻¹.

倒数函数是指可以写成 y = k / x 形式的函数，其中 k 是非零常数。它也常用负指数形式表示为 y = kx⁻¹。

More generally, after applying transformations, we study functions of the form y = a / (x + b) + c. These are translated reciprocal hyperbolas and still have two separate branches.

更一般地，经过变换后我们会研究 y = a / (x + b) + c 这样的函数。它们是平移后的倒数双曲线，仍然具有两个独立的分支。

y = k / x, x ≠ 0

2. Basic Graph of y = k/x | y = k/x 的基本图像

For k > 0, the graph of y = k / x is a rectangular hyperbola with two branches. One branch lies in the first quadrant and the other lies in the third quadrant.

当 k > 0 时，y = k / x 的图像是一条等轴双曲线，由两个分支组成。一个分支位于第一象限，另一个分支位于第三象限。

The graph never touches the x-axis or the y-axis. These axes are asymptotes: the curve approaches them infinitely closely but never crosses them.

图像永远不会接触 x 轴或 y 轴。这两条轴是渐近线：曲线无限接近它们，但永远不会穿过它们。

y = k / x, k > 0

Key features of y = k / x:

y = k / x 的主要特征如下：

Domain: x ≠ 0	定义域：x ≠ 0
Range: y ≠ 0	值域：y ≠ 0
Vertical asymptote: x = 0	竖直渐近线：x = 0
Horizontal asymptote: y = 0	水平渐近线：y = 0
No x-intercept or y-intercept	没有 x 轴截距和 y 轴截距

3. Effect of the Sign of k | k 的符号影响

When k > 0, the two branches of y = k / x lie in the first and third quadrants. The graph is decreasing on each separate interval of its domain.

当 k > 0 时，y = k / x 的两个分支位于第一和第三象限。图像在其定义域的每个独立区间上都是递减的。

When k < 0, the branches are reflected in the axes and lie in the second and fourth quadrants. The graph is then increasing on each separate interval.

当 k < 0 时，分支关于坐标轴反射，位于第二和第四象限。此时图像在其定义域的每个独立区间上是递增的。

Changing the sign of k reflects the graph in both the x-axis and the y-axis. This is equivalent to a rotation of 180° about the origin because y = k / x is an odd function.

改变 k 的符号会使图像同时关于 x 轴和 y 轴反射。这相当于绕原点旋转 180°，因为 y = k / x 是奇函数。

y = −k / x is the reflection of y = k / x in both axes

4. Asymptotes and Behaviour at Infinity | 渐近线与无穷远处的行为

For y = k / x with k > 0, as x approaches 0 from the positive side, y tends to positive infinity. As x approaches 0 from the negative side, y tends to negative infinity.

对于 k > 0 的 y = k / x，当 x 从正侧趋近 0 时，y 趋向正无穷；当 x 从负侧趋近 0 时，y 趋向负无穷。

x → 0⁺ ⇒ y → +∞ and x → 0⁻ ⇒ y → −∞

As x tends to positive infinity, y tends to 0 from above. As x tends to negative infinity, y tends to 0 from below. This confirms that y = 0 is a horizontal asymptote.

当 x 趋向正无穷时，y 从上方趋近 0；当 x 趋向负无穷时，y 从下方趋近 0。这进一步说明 y = 0 是一条水平渐近线。

x → +∞ ⇒ y → 0⁺ and x → −∞ ⇒ y → 0⁻

5. Transformations of y = k/x | y = k/x 的图像变换

The general transformed reciprocal function can be written as y = a / (x − p) + q. Here p and q translate the graph, while a stretches or reflects it vertically.

一般的倒数函数变换形式可以写成 y = a / (x − p) + q。其中 p 和 q 使图像发生平移，a 则使图像发生纵向伸缩或反射。

The vertical asymptote is now x = p, and the horizontal asymptote is now y = q. The graph has the same basic hyperbolic shape but is shifted.

此时竖直渐近线为 x = p，水平渐近线为 y = q。图像保持相同的基本双曲线形状，但发生了平移。

y = a / (x − p) + q has asymptotes x = p and y = q

For example, y = 2 / (x − 3) + 1 has a vertical asymptote at x = 3 and a horizontal asymptote at y = 1. The graph is a translation of y = 2 / x by vector (3, 1).

例如，y = 2 / (x − 3) + 1 的竖直渐近线为 x = 3，水平渐近线为 y = 1。该图像是 y = 2 / x 按向量 (3, 1) 平移后的结果。

6. Graph of y = k/x² | y = k/x² 的图像

The function y = k / x² is another important reciprocal graph. Since x² is always positive for x ≠ 0, the sign of y depends only on the sign of k.

函数 y = k / x² 是另一种重要的倒数函数图像。由于当 x ≠ 0 时 x² 始终为正，所以 y 的符号只取决于 k 的符号。

When k > 0, both branches lie above the x-axis in the first and second quadrants. When k < 0, both branches lie below the x-axis in the third and fourth quadrants.

当 k > 0 时，两个分支都位于 x 轴上方，分别在第一和第二象限。当 k < 0 时，两个分支都位于 x 轴下方，分别在第三和第四象限。

y = k / x², x ≠ 0

The graph has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. Since y = k / x² is an even function, its graph is symmetrical about the y-axis.

该图像有一条竖直渐近线 x = 0 和一条水平渐近线 y = 0。由于 y = k / x² 是偶函数，其图像关于 y 轴对称。

7. Sketching Reciprocal Graphs Step by Step | 绘制倒数函数图像的步骤

To sketch a transformed reciprocal graph such as y = a / (x − p) + q, follow a clear method. Begin by identifying the asymptotes and intercepts before considering the sign of each branch.

绘制 y = a / (x − p) + q 这类变换后的倒数函数图像时，需要遵循清晰的方法。先确定渐近线和截距，再考虑每个分支的符号。

Step 1: Write down the vertical asymptote by setting the denominator equal to zero.

步骤 1：令分母等于零，写出竖直渐近线。

Step 2: Write down the horizontal asymptote from the constant term q.

步骤 2：根据常数项 q 写出水平渐近线。

Step 3: Find the y-intercept by substituting x = 0, if x = 0 is not the vertical asymptote.

步骤 3：如果 x = 0 不是竖直渐近线，代入 x = 0 求 y 轴截距。

Step 4: Find the x-intercept by setting y = 0 and solving the resulting equation.

步骤 4：令 y = 0，解方程求 x 轴截距。

Step 5: Choose a test point on each side of the vertical asymptote to determine whether the branch lies above or below the horizontal asymptote.

步骤 5：在竖直渐近线两侧各取一个测试点，判断分支位于水平渐近线的上方还是下方。

Step 6: Draw the two smooth hyperbolic branches, making sure they approach but never cross the asymptotes.

步骤 6：画出两条光滑的双曲线分支，确保它们接近但永不穿过渐近线。

8. Intersections of Reciprocal Graphs with Lines | 倒数函数图像与直线的交点

To find the intersection points of a reciprocal graph and a straight line, set their equations equal to each other. Then multiply through by x or by the denominator to form a polynomial equation.

要求倒数函数图像与直线的交点，需要将两个方程联立。然后两边同乘 x 或分母，转化为多项式方程。

For example, solve y = 2 / x and y = x + 1 simultaneously. Multiplying by x gives x² + x − 2 = 0, which factorises as (x + 2)(x − 1) = 0.

例如，联立求解 y = 2 / x 与 y = x + 1。两边同乘 x 得到 x² + x − 2 = 0，分解因式得 (x + 2)(x − 1) = 0。

2 / x = x + 1 ⇒ x² + x − 2 = 0 ⇒ x = −2 or x = 1

Always check that the solutions do not make any denominator zero. Here x = −2 and x = 1 are both valid, giving intersection points (−2, −1) and (1, 2).

始终要检查解是否会使任何分母为零。此处 x = −2 和 x = 1 均有效，对应的交点分别为 (−2, −1) 和 (1, 2)。

9. Reciprocal Inequalities | 倒数不等式

When solving an inequality such as 2 / x < 1, do not multiply both sides by x directly unless you know the sign of x. Instead, rearrange with a common denominator and use a sign table.

解 2 / x < 1 这类不等式时，不要直接两边同乘 x，除非已知 x 的符号。正确做法是通分整理，并使用符号表。

Rearrange to (2 − x) / x < 0. The critical values are x = 2 and x = 0, where the numerator or denominator changes sign.

将不等式整理为 (2 − x) / x < 0。临界值为 x = 2 和 x = 0，因为分子或分母在这些点变号。

(2 − x) / x < 0 ⇒ x < 0 or x > 2

The solution is x < 0 or x > 2. Notice that x = 0 must be excluded automatically because it makes the original expression undefined.

解集为 x < 0 或 x > 2。注意 x = 0 必须自动排除，因为它会使原式无意义。

10. Common Mistakes and Exam Tips | 常见错误与考试提示

One common mistake is drawing y = k / x² as two branches in the first and third quadrants. For k > 0 the branches of y = k / x² must both be above the x-axis.

一个常见错误是把 y = k / x² 画成第一和第三象限的两个分支。对于 k > 0，y = k / x² 的两个分支必须都在 x 轴上方。

Another mistake is forgetting to state the domain x ≠ 0 or the range y ≠ 0. These restrictions follow directly from the asymptotes and are often examined.

另一个常见错误是忘记写出定义域 x ≠ 0 或值域 y ≠ 0。这些限制由渐近线直接得出，也经常是考点。

When solving equations or inequalities, always check for division by zero. A solution that makes a denominator zero must be rejected.

在解方程或不等式时，始终检查是否存在除以零的情况。任何使分母为零的解都必须舍去。

Draw asymptotes as dashed lines when sketching, and label them clearly. This shows the examiner that you understand the behaviour of the graph.

作图时请将渐近线画成虚线，并清晰标注。这样能向阅卷人展示你理解图像的变化趋势。

11. Worked Example | 例题精讲

Sketch the curve y = 3 / (x − 2) + 1, and state its asymptotes, intercepts, domain and range.

绘制曲线 y = 3 / (x − 2) + 1，并写出其渐近线、截距、定义域和值域。

The vertical asymptote occurs when the denominator is zero, so x = 2. The horizontal asymptote is y = 1 because the constant term is 1.

当分母为零时得到竖直渐近线，因此 x = 2。由于常数项为 1，水平渐近线为 y = 1。

For the y-intercept, substitute x = 0:

求 y 轴截距时，代入 x = 0：

y = 3 / (0 − 2) + 1 = −3/2 + 1 = −1/2

For the x-intercept, set y = 0:

求 x 轴截距时，令 y = 0：

3 / (x − 2) + 1 = 0 ⇒ 3 / (x − 2) = −1 ⇒ x − 2 = −3 ⇒ x = −1

The domain is all real numbers except x = 2. The range is all real numbers except y = 1.

定义域为除 x = 2 以外的所有实数。值域为除 y = 1 以外的所有实数。

To determine the branches, test points on either side of x = 2. When x = 3, y = 3 + 1 = 4, so the right branch lies above y = 1. When x = 1, y = −3 + 1 = −2, so the left branch lies below y = 1.

为了确定分支位置，在 x = 2 两侧取测试点。当 x = 3 时，y = 3 + 1 = 4，因此右分支位于 y = 1 上方。当 x = 1 时，y = −3 + 1 = −2，因此左分支位于 y = 1 下方。

12. Summary | 小结

Reciprocal graphs are built around vertical and horizontal asymptotes, with two separate hyperbolic branches. The most basic forms are y = k / x and y = k / x².

倒数函数图像的核心特征是竖直和水平渐近线，以及两个独立的双曲线分支。最基本的形式是 y = k / x 和 y = k / x²。

Transformations change the position of the asymptotes but not the essential shape. Solving intersections and inequalities requires careful handling of denominators and signs.

图像变换会改变渐近线的位置，但不会改变其基本形状。求解交点和不等式时需要谨慎处理分母与符号。

Always sketch asymptotes first, label intercepts, and check for

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

更多咨询请联系16621398022（同微信）

Reciprocal Graphs | 倒数函数图像