Reciprocal Functions: Graphs and Properties | 反比例函数的图像与性质

📚 Reciprocal Functions: Graphs and Properties | 反比例函数的图像与性质

The reciprocal function, often written as y = k/x, is a fundamental nonlinear function in mathematics. It appears in many areas, from physics to economics, and is a key topic in exam syllabuses. In this article, we will explore its graph, properties, transformations and real-world applications, together with exam tips.

反比例函数，通常写作 y = k/x，是数学中一种基础的非线性函数。它出现在从物理到经济学的许多领域，也是考试大纲中的关键考点。在这篇文章中，我们将探讨它的图像、性质、变换及实际应用，并给出考试技巧。

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1. Definition and Standard Form | 定义与标准形式

A reciprocal function is defined by the equation y = k/x, where k is a non-zero constant. It represents an inverse relationship between x and y: as one variable increases, the other decreases proportionally.

反比例函数由方程 y = k/x 定义，其中 k 是非零常数。它表示 x 与 y 之间的反比关系：当一个变量增大时，另一个变量按比例减小。

The independent variable x appears in the denominator, so x cannot be zero. The function is undefined at x = 0.

自变量 x 出现在分母中，因此 x 不能为零。函数在 x = 0 处没有定义。

An equivalent form is xy = k, which is sometimes useful when solving problems.

等价形式为 xy = k，这在解题时有时很方便。

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

For k > 0, the graph consists of two branches lying in the first and third quadrants. Each branch is a smooth curve called a hyperbola.

当 k > 0 时，图像由位于第一、第三象限的两支曲线组成。每一支都是平滑的曲线，称为双曲线。

For k < 0, the two branches lie in the second and fourth quadrants.

当 k < 0 时，两支曲线位于第二、第四象限。

The curve never touches either axis. As x gets very large, y gets very small; as x gets very close to zero, y gets very large in magnitude.

曲线永远不会与任何坐标轴相交。当 x 变得很大时，y 变得很小；当 x 非常接近零时，y 的绝对值变得很大。

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3. Effect of Constant k | 常数 k 的影响

The absolute value |k| controls the steepness of the curve. Larger |k| moves the branches farther from the origin.

绝对值 |k| 控制曲线的陡峭程度。|k| 越大，分支离原点越远。

The sign of k determines which quadrants the graph occupies. If k is positive, the graph is in the first and third quadrants; if k is negative, it is in the second and fourth quadrants.

k 的符号决定图像所在的象限。若 k 为正，图像位于第一和第三象限；若 k 为负，则位于第二和第四象限。

When k = 1, we have the simplest case y = 1/x, which is the standard reference graph for this family of functions.

当 k = 1 时，我们得到最简单的情形 y = 1/x，这是该函数族的标准参考图像。

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4. Domain and Range | 定义域与值域

Since x appears in the denominator, the domain is all real numbers except 0. In set notation:

由于 x 出现在分母中，定义域为除 0 以外的所有实数。用集合记号表示：

D = { x ∈ ℝ | x ≠ 0 }

Similarly, y can take any real value except 0. No matter what x is, y = k/x can never equal zero because k is non-zero.

类似地，y 可以取除 0 以外的任何实数值。无论 x 取何值，y = k/x 都不可能等于零，因为 k 非零。

R = { y ∈ ℝ | y ≠ 0 }

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5. Asymptotes | 渐近线

The graph of y = k/x has two asymptotes: the vertical line x = 0 and the horizontal line y = 0.

y = k/x 的图像有两条渐近线：铅直线 x = 0 和水平线 y = 0。

As x approaches 0 from the positive side, y tends to +∞ if k > 0; as x approaches 0 from the negative side, y tends to −∞.

当 x 从正方向趋近于 0 时，若 k > 0，则 y 趋向 +∞；当 x 从负方向趋近于 0 时，y 趋向 −∞。

As x → ±∞, y tends to 0. This means the curve gets closer and closer to the x-axis without ever touching it.

当 x → ±∞ 时，y 趋向 0。这意味着曲线越来越接近 x 轴，但永远不会与它相交。

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6. Monotonicity and Symmetry | 单调性与对称性

For k > 0, the function is decreasing on both intervals (−∞, 0) and (0, ∞). For k < 0, it is increasing on both intervals.

当 k > 0 时，函数在 (−∞, 0) 和 (0, ∞) 两个区间上分别递减；当 k < 0 时，函数在这两个区间上分别递增。

The graph is symmetric about the origin. It satisfies f(−x) = −f(x), so it is an odd function.

图像关于原点对称。它满足 f(−x) = −f(x)，因此是奇函数。

In addition, the graph is symmetric about the lines y = x and y = −x.

此外，图像还关于直线 y = x 和 y = −x 对称。

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7. Transformations: y = a/(x − h) + k | 平移变换

The general transformed form of a reciprocal function is y = a/(x − h) + k. The vertical asymptote shifts to x = h, and the horizontal asymptote shifts to y = k.

反比例函数的一般变换形式为 y = a/(x − h) + k。铅直渐近线移动到 x = h，水平渐近线移动到 y = k。

The constant a stretches or compresses the graph vertically. If a is negative, the graph is reflected about the x-axis.

常数 a 会纵向拉伸或压缩图像。若 a 为负，图像关于 x 轴反射。

For example, the function y = 2/(x − 1) + 3 has vertical asymptote x = 1 and horizontal asymptote y = 3.

例如，函数 y = 2/(x − 1) + 3 的铅直渐近线为 x = 1，水平渐近线为 y = 3。

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8. Solving Equations Involving Reciprocal Functions | 反比例函数方程求解

To solve an equation such as k/x = c, multiply both sides by x (remembering x ≠ 0), and then solve for x.

要解形如 k/x = c 的方程，两边乘以 x（记住 x ≠ 0），然后求解 x。

For equations involving variables in the denominator, always check for extraneous solutions after multiplying by the denominator.

对于分母中含有变量的方程，在乘以分母之后一定要检查是否有增根。

Example: Solve 4/x = 2. Multiplying by x gives 4 = 2x, so x = 2. Since x ≠ 0, the solution is valid.

例：解 4/x = 2。两边乘以 x 得 4 = 2x，所以 x = 2。由于 x ≠ 0，该解有效。

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9. Applications in Real Life | 实际应用

Reciprocal relationships appear frequently in science. For example, Boyle’s law states that for a fixed mass of gas at constant temperature, pressure P and volume V satisfy P ∝ 1/V.

反比关系在科学中经常出现。例如，玻意耳定律指出，在恒定温度下，一定质量气体的压强 P 与体积 V 满足 P ∝ 1/V。

In everyday life, the time needed to complete a task is inversely proportional to the number of workers, assuming they work at the same rate.

在日常生活中，假设工人以相同的效率工作，完成一项任务所需的时间与工人数量成反比。

In economics, unit cost and quantity are often inversely related: as production volume increases, the fixed cost per unit decreases.

在经济学中，单位成本与数量常常成反比关系：随着产量增加，每单位分摊的固定成本会降低。

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10. Common Exam Pitfalls | 常见考试误区

Mistake 1: Thinking the graph is continuous across x = 0. It is not – there is a break at x = 0.

误区 1：认为图像在 x = 0 处连续。它并不连续，在 x = 0 处断开。

Mistake 2: Forgetting that x = 0 is excluded from the domain.

误区 2：忘记 x = 0 不在定义域内。

Mistake 3: Confusing the effects of k in the standard form and a in the transformed form.

误区 3：混淆标准形式中 k 与变换形式中 a 的作用。

Mistake 4: Drawing only one branch of the hyperbola instead of two.

误区 4：只绘制双曲线的一支，而遗漏另一支。

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11. Practice Questions | 练习

1. Sketch the graph of y = 3/x. State the domain, range, and equations of the asymptotes.

1. 画出 y = 3/x 的图像，并指出定义域、值域和渐近线方程。

2. Describe the transformations that map y = 1/x to y = −2/(x + 3) − 1.

2. 描述从 y = 1/x 到 y = −2/(x + 3) − 1 的变换过程。

3. Solve the equation 6/x = 3 and verify your answer.

3. 解方程 6/x = 3，并检验你的答案。

Answers: 1. Domain x ≠ 0, range y ≠ 0, asymptotes x = 0 and y = 0. 2. Shift left by 3 units, stretch vertically by factor 2, reflect about the x-axis, then shift down by 1 unit. 3. x = 2.

答案：1. 定义域 x ≠ 0，值域 y ≠ 0，渐近线 x = 0 和 y = 0。2. 向左平移 3 个单位，纵向拉伸 2 倍，关于 x 轴反射，再向下平移 1 个单位。3. x = 2。

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