Reciprocal Graphs | 反比例函数图像

📚 Reciprocal Graphs | 反比例函数图像

A reciprocal graph is the graph of a function of the form y = k/x, where k is a non-zero constant. It is one of the key graph types you need to recognise and sketch for IGCSE Edexcel Mathematics.

反比例函数图像是形如 y = k/x（k 为非零常数）的函数图像。它是你在 Edexcel IGCSE 数学中需要识别并绘制的关键图像类型之一。

1. What Is a Reciprocal Graph? | 什么是反比例函数图像？

For y = k/x, every y-value is the constant k divided by the x-value. Because x cannot be 0, the graph never touches the y-axis. Similarly, y can never be 0, so the graph never touches the x-axis.

对于 y = k/x，每个 y 值都等于常数 k 除以对应的 x 值。因为 x 不能为 0，图像永远不会接触 y 轴；同理，y 永远不能为 0，所以图像也永远不会接触 x 轴。

The graph has two separate branches. When k is positive, one branch lies in the first quadrant (x > 0, y > 0) and the other in the third quadrant (x < 0, y < 0).

图像分为两条独立的分支。当 k 为正数时，一条分支位于第一象限（x > 0, y > 0），另一条位于第三象限（x < 0, y < 0）。

2. Asymptotes | 渐近线

An asymptote is a straight line that a curve approaches but never touches. For y = k/x, the axes are asymptotes.

渐近线是一条曲线不断靠近但永远不会触及的直线。对于 y = k/x，坐标轴就是渐近线。

The line x = 0 is a vertical asymptote. As x becomes very small and positive, y becomes very large and positive; as x becomes very small and negative, y becomes very large and negative.

直线 x = 0 是垂直渐近线。当 x 从正方向趋近于 0 时，y 会变得很大且为正；当 x 从负方向趋近于 0 时，y 会变得很大且为负。

The line y = 0 is a horizontal asymptote. As x tends to +∞ or -∞, y tends to 0.

直线 y = 0 是水平渐近线。当 x 趋向正无穷或负无穷时，y 趋向 0。

3. Shape of the Curve | 曲线形状

The curve is called a hyperbola. Each branch is smooth and continuous, with no straight-line segments. The curve bends away from both axes without crossing them.

这种曲线称为双曲线。每一条分支都是光滑连续的，没有直线段。曲线向远离两条坐标轴的方向弯曲，但不会与它们相交。

For k > 0, the point on each branch closest to the origin occurs when x = √k and y = √k for the positive branch, and at x = -√k and y = -√k for the negative branch.

当 k > 0 时，每条分支上离原点最近的点分别在 (√k, √k) 和 (-√k, -√k)。

4. The Effect of k | k 的影响

If k is positive, the branches lie in the first and third quadrants. If k is negative, the branches lie in the second and fourth quadrants, because y and x always have opposite signs.

若 k 为正，两条分支位于第一和第三象限；若 k 为负，x 与 y 异号，两条分支位于第二和第四象限。

When |k| increases, the curve moves farther from the origin in each branch: for a given positive x, y has a larger absolute value. The general shape remains the same.

当 |k| 增大时，每条分支离原点更远：在某个正 x 处，y 的绝对值更大。但图像的基本形状不变。

5. Plotting a Reciprocal Graph | 描点绘制反比例函数图像

To plot y = 2/x, choose positive and negative x-values. Do not choose x = 0 because division by zero is undefined.

要绘制 y = 2/x，选择正数和负数的 x 值。不要选 x = 0，因为除数不能为零。

Example table for y = 2/x:

例如 y = 2/x 的取值表：

x	-4	-2	-1	-0.5	0.5	1	2	4
y	-0.5	-1	-2	-4	4	2	1	0.5

Plot these points and connect each quadrant smoothly. Draw the asymptotes as dashed lines if you need to show them.

描出这些点，并用平滑曲线连接每个象限内的点。如需标明渐近线，可用虚线画出 x = 0 和 y = 0。

6. Transformations | 图像变换

You also need to sketch graphs of the form y = a/(x + b) + c. This is a translation of y = a/x. The vertical asymptote shifts to x = -b, and the horizontal asymptote shifts to y = c.

你还需要会画形如 y = a/(x + b) + c 的图像。它是 y = a/x 平移后的结果。垂直渐近线移到 x = -b，水平渐近线移到 y = c。

For example, y = 1/(x – 2) + 3 has vertical asymptote x = 2 and horizontal asymptote y = 3. The graph is the same shape as y = 1/x, shifted right by 2 and up by 3.

例如，y = 1/(x – 2) + 3 的垂直渐近线是 x = 2，水平渐近线是 y = 3。它的形状与 y = 1/x 相同，只是向右平移 2 个单位、向上平移 3 个单位。

When sketching, mark the asymptotes and plot one point on each branch, then draw smooth curves approaching the asymptotes.

画草图时，先标出两条渐近线，再在每条分支上取一个点，然后画出靠近渐近线的平滑曲线。

7. Solving Equations Using Reciprocal Graphs | 利用反比例函数图像解方程

To solve an equation such as 2/x = x – 1, you can sketch y = 2/x and y = x – 1 on the same axes. The x-coordinates of the intersection points are the solutions.

要解方程 2/x = x – 1，可以在同一坐标系中画出 y = 2/x 和 y = x – 1。交点处的 x 坐标就是方程的解。

Algebraic check: multiply both sides by x to get 2 = x² – x, so x² – x – 2 = 0, giving x = 2 or x = -1. These match the x-coordinates of the intersections.

代数验证：两边同乘 x 得 2 = x² – x，即 x² – x – 2 = 0，所以 x = 2 或 x = -1。这与交点的 x 坐标一致。

Graphical solutions are usually approximate. Always check values by substitution where possible.

图解法的结果通常是近似值。尽量通过代入原方程检验所得数值。

8. Worked Example | 典型例题

Sketch y = 6/x. State the equations of the asymptotes. Find the coordinates of the point on the branch in the first quadrant closest to the origin.

画出 y = 6/x 的草图，写出渐近线方程，并求第一象限分支上离原点最近的点的坐标。

The asymptotes are x = 0 and y = 0. For the point closest to the origin on the positive branch, set x = √6 and y = √6. Plot (1, 6), (2, 3), (3, 2), (6, 1) and draw a smooth curve through them, approaching both axes.

渐近线为 x = 0 和 y = 0。对于正分支上离原点最近的点，令 x = √6，y = √6。描出 (1, 6)、(2, 3)、(3, 2)、(6, 1)，用平滑曲线连接，并让曲线逐渐靠近两条坐标轴。

If you cannot use √6, you can simply state that the closest point is approximately (2.45, 2.45), since √6 ≈ 2.45.

如果题目允许，可直接写最接近点约为 (2.45, 2.45)，因为 √6 ≈ 2.45。

9. Comparing with Linear and Quadratic Graphs | 与一次、二次函数图像比较

Linear graphs y = mx + c have constant gradient and no asymptotes. Quadratic graphs y = ax² + bx + c are parabolas with one turning point. Reciprocal graphs have two branches and two asymptotes.

一次函数 y = mx + c 的斜率恒定且没有渐近线；二次函数 y = ax² + bx + c 是只有单个顶点的抛物线；反比例函数有两条分支和两条渐近线。

This distinction is important in multiple-choice questions where you must identify the correct graph from a list.

在做选择题时，正确区分这三类图像非常重要。

10. Common Mistakes | 常见错误

Choosing x = 0 in a table of values for y = k/x. This makes the expression undefined.

在 y = k/x 的取值表中选 x = 0。这会导致表达式无定义。
Drawing one continuous curve that crosses the y-axis. The two branches must remain separate.

把两条分支画成一条穿过 y 轴的连续曲线。两条分支必须保持分离。
Drawing the curve touching or crossing the axes. For y = k/x, the curve only approaches the axes.

画成与坐标轴接触或相交。对于 y = k/x，曲线只能靠近坐标轴，不能接触或相交。
Forgetting that a negative k places the branches in quadrants 2 and 4.

忘记当 k 为负数时，分支位于第二和第四象限。

11. Summary | 小结

Reciprocal graphs have the form y = k/x. They are hyperbolas with two branches, a vertical asymptote x = 0 and a horizontal asymptote y = 0. The sign of k determines which quadrants contain the branches, and |k| affects how far the branches are from the origin.

反比例函数图像形如 y = k/x，是带有两条分支的双曲线，垂直渐近线为 x = 0，水平渐近线为 y = 0。k 的符号决定分支所在的象限，|k| 影响分支离原点的远近。

You should be able to plot, sketch, transform and use reciprocal graphs to solve equations. Practice drawing them by hand so the shape becomes automatic.

你应当能够描点、画草图、进行平移变换，并利用反比例函数图像解方程。多动手画图，才能熟练掌握其形状。

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Reciprocal Graphs | 反比例函数图像