Simple Rational Functions: Graphs and Properties | 简单有理函数图像与性质

📚 Simple Rational Functions: Graphs and Properties | 简单有理函数图像与性质

Rational functions are quotients of polynomials. In the IB Mathematics curriculum, the ‘simple’ rational functions usually mean the reciprocal function y = k/x and the linear-over-linear form y = (ax+b)/(cx+d). Understanding their graphs and properties is essential for sketching, solving equations and applying transformations.

有理函数是两个多项式之比。在IB数学课程中，“简单”有理函数通常指反比例函数 y = k/x 以及线性比线性的形式 y = (ax+b)/(cx+d)。理解它们的图像和性质对于画图、解方程以及应用变换都至关重要。

1. What Are Rational Functions? | 什么是有理函数？

A rational function is defined as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0. In IB, simple rational functions often have degree 1 in both numerator and denominator, such as f(x) = (2x-1)/(x+3).

有理函数定义为 f(x) = P(x)/Q(x)，其中 P(x) 和 Q(x) 是多项式，且 Q(x) ≠ 0。在IB中，简单有理函数通常分子分母均为一次，例如 f(x) = (2x-1)/(x+3)。

These functions are different from polynomials because they can have breaks in their graphs called asymptotes. They model real-life situations such as average cost, speed, and concentration.

这类函数与多项式不同，因为它们的图像可能出现间断，称为渐近线。它们可以模拟平均成本、速度和浓度等实际问题。

2. The Reciprocal Function y = k/x | 反比例函数 y = k/x

The simplest rational function is y = k/x, where k ≠ 0. Its graph is a hyperbola in two branches. When k > 0, the branches lie in the first and third quadrants; when k < 0, they lie in the second and fourth quadrants.

最简单的有理函数是 y = k/x，其中 k ≠ 0。它的图像是双曲线，分为两支。当 k > 0 时，两支位于第一、第三象限；当 k < 0 时，位于第二、第四象限。

The two axes are asymptotes: the x-axis (y = 0) and the y-axis (x = 0). The function has both rotational symmetry and point symmetry about the origin.

两条坐标轴都是渐近线：x轴 (y = 0) 和 y轴 (x = 0)。该函数具有关于原点的旋转对称性和点对称性。

y = k/x, k ≠ 0

3. The General Linear-Over-Linear Form | 线性比线性的一般形式

The general form studied in IB is f(x) = (ax+b)/(cx+d), where c ≠ 0 and the numerator and denominator have no common factor. If a = 0, the function reduces to a constant over a linear expression, which is still a simple rational function.

IB中研究的一般形式是 f(x) = (ax+b)/(cx+d)，其中 c ≠ 0，且分子分母没有公因式。如果 a = 0，函数就退化为常数除以一次式，这仍然是简单有理函数。

When c = 0, the function becomes linear, not a rational function with a restricted domain. In this article we assume c ≠ 0. The denominator cx+d cannot be zero, so x = -d/c is excluded from the domain.

当 c = 0 时，函数退化为线性函数，不再是定义域受限的有理函数。本文假设 c ≠ 0。分母 cx+d 不能为零，因此 x = -d/c 不在定义域内。

4. Vertical Asymptotes | 垂直渐近线

A vertical asymptote occurs where the denominator equals zero but the numerator does not also equal zero. For f(x) = (ax+b)/(cx+d), the vertical asymptote is the line x = -d/c.

当分母为零而分子不为零时，出现垂直渐近线。对于 f(x) = (ax+b)/(cx+d)，垂直渐近线是直线 x = -d/c。

As x approaches -d/c from either side, the magnitude of f(x) tends to infinity. If the numerator also vanishes at the same point, there is a hole, not an asymptote, because the common factor can be cancelled.

当 x 从两侧趋近 -d/c 时，f(x) 的绝对值趋于无穷大。如果分子也在同一点为零，那么该点是一个空洞，而不是渐近线，因为公因式可以约去。

5. Horizontal Asymptotes | 水平渐近线

For a simple rational function where numerator and denominator have the same degree, the horizontal asymptote is the ratio of the leading coefficients: y = a/c. This is because as |x| → ∞, the terms b and d become negligible compared to ax and cx.

对于分子分母次数相同的简单有理函数，水平渐近线是首项系数之比：y = a/c。这是因为当 |x| → ∞ 时，常数项 b 和 d 相对于 ax 和 cx 变得可忽略。

Thus y = 2 for f(x) = (2x-1)/(x+3). The curve approaches this value but never reaches it for any finite x, except possibly when the equation f(x) = a/c has no solution.

因此 f(x) = (2x-1)/(x+3) 的水平渐近线是 y = 2。曲线会逼近这个值，但不会在有限 x 处取到它，除非方程 f(x) = a/c 恰好有解。

6. Intercepts | 坐标轴交点

To find the y-intercept, set x = 0: y = b/d, provided d ≠ 0. This gives the point (0, b/d) on the y-axis.

求 y 截距时令 x = 0：y = b/d（假设 d ≠ 0）。这给出 y 轴上的点 (0, b/d)。

To find the x-intercept, set y = 0 and solve ax+b = 0. The solution is x = -b/a, provided a ≠ 0. If a = 0, there is no x-intercept unless b = 0, which would make the function identically zero.

求 x 截距时令 y = 0，解 ax+b = 0。解为 x = -b/a，前提是 a ≠ 0。如果 a = 0，除非 b = 0（此时函数恒为零），否则没有 x 截距。

7. Transformations | 图像变换

The graph of f(x) = (ax+b)/(cx+d) can be obtained from y = 1/x by a sequence of vertical and horizontal translations, stretches and a possible reflection. In IB, recognising the transformed hyperbola helps you sketch quickly.

f(x) = (ax+b)/(cx+d) 的图像可以从 y = 1/x 经过一系列垂直、水平平移、伸缩以及可能的反射得到。在IB中，识别变换后的双曲线有助于快速画图。

We can rewrite the function using algebraic division: f(x) = a/c + (bc-ad)/(c(cx+d)). Let k = (bc-ad)/c² and define X = x + d/c. Then f = a/c + k/X, a translation of y = k/X with vertical shift a/c and horizontal shift -d/c.

我们可以通过代数除法改写函数：f(x) = a/c + (bc-ad)/(c(cx+d))。令 k = (bc-ad)/c²，再令 X = x + d/c，则 f = a/c + k/X，这是 y = k/X 经过垂直平移 a/c 和水平平移 -d/c 的结果。

Original Function	Transformation	New Function
y = 1/x	Shift left by d/c	y = 1/(x + d/c)
y = 1/(x + d/c)	Vertical stretch by \|k\| and reflect if k<0	y = k/(x + d/c)
y = k/(x + d/c)	Shift up by a/c	y = a/c + k/(x + d/c)

8. Domain and Range | 定义域与值域

For f(x) = (ax+b)/(cx+d), the domain is all real numbers except x = -d/c. In set notation: {x ∈ ℝ | x ≠ -d/c}.

对于 f(x) = (ax+b)/(cx+d)，定义域为除 x = -d/c 以外的所有实数。用集合记号：{x ∈ ℝ | x ≠ -d/c}。

The range is all real numbers except y = a/c, because the horizontal asymptote is never crossed. This can be shown by solving y = f(x) for x and observing the restriction on y.

值域为除 y = a/c 以外的所有实数，因为水平渐近线永远不会被跨越。这可以通过将 y = f(x) 解出 x 并观察对 y 的限制来证明。

When a = 0, the range is all real numbers except 0, unless b = 0. The domain remains the same restriction.

当 a = 0 时，值域为除 0 以外的所有实数（除非 b = 0）。定义域的限制保持不变。

9. Symmetry and Special Cases | 对称性与特殊情况

The basic function y = k/x is odd: f(-x) = -f(x). Its graph is symmetric about the origin. Once translated to f(x) = a/c + k/(x + d/c), the centre of symmetry moves to the point (-d/c, a/c), which is the intersection of the two asymptotes.

基本函数 y = k/x 是奇函数：f(-x) = -f(x)。它的图像关于原点对称。一旦平移到 f(x) = a/c + k/(x + d/c)，对称中心移动到两条渐近线的交点 (-d/c, a/c)。

If the numerator and denominator have a common factor, the graph is a line with a hole. For example, f(x) = (x+2)(x-1)/(x+2) simplifies to x-1, but x = -2 is not in the domain, so the graph has a hole at (-2, -3).

如果分子分母有公因式，图像是一条直线带一个空洞。例如 f(x) = (x+2)(x-1)/(x+2) 化简为 x-1，但 x = -2 不在定义域内，因此图像在 (-2, -3) 处有一个空洞。

10. Step-by-Step Sketching | 分步画图法

To sketch a simple rational function, first find the vertical asymptote x = -d/c and draw it as a dashed line. Then draw the horizontal asymptote y = a/c as another dashed line.

画简单有理函数的图像时，先找出垂直渐近线 x = -d/c，并用虚线画出；再画出水平渐近线 y = a/c，同样用虚线。

Find intercepts: Mark the y-intercept (0, b/d) and x-intercept (-b/a, 0).
找到交点：标出 y 截距 (0, b/d) 和 x 截距 (-b/a, 0)。
Analyse branches: Use test points in each interval created by the vertical asymptote to determine sign and shape.
分析分支：在垂直渐近线划分的各个区间内取测试点，判断符号和形状。
Draw hyperbola: Connect each branch smoothly, approaching both asymptotes as x moves away from the asymptote.
画出双曲线：将每一支平滑连接，当 x 远离渐近线时使图像逼近两条渐近线。

Finally, label the graph with the equation, asymptotes and important points. Always check whether the curve crosses the horizontal asymptote by solving f(x) = a/c; if a solution exists, the crossing point must be plotted.

最后在图像上标注函数方程、渐近线和重要点。务必通过解 f(x) = a/c 检查曲线是否会与水平渐近线相交；若存在解，需画出交点。

11. Solving Rational Equations and Inequalities | 解有理方程与不等式

Rational equations are solved by multiplying through by the denominator, but this can introduce extraneous roots. Always substitute solutions back into the original equation to verify them.

解有理方程时通常乘以分母以消去分母，但这可能引入增根。务必把解代入原方程验证。

Rational inequalities are solved by identifying critical values: the zeros of the numerator and the zeros of the denominator. Then test intervals on a number line. Remember that the denominator cannot be zero, so open circles are used for excluded values.

解有理不等式时，需要确定临界值：分子的零点和分母的零点。然后在数轴上测试各区间。注意分母不能为零，因此被排除的值用空心圆表示。

12. Common IB Exam Questions and Tips | 常见IB考题与技巧

In IB exams, you may be asked to find the asymptotes and intercepts, sketch the graph, or determine the range from the graph. Sometimes the function is given in the form f(x) = A + B/(x+c).

在IB考试中，你可能会被要求找渐近线和截距、画出图像或从图像求值域。有时函数以 f(x) = A + B/(x+c) 的形式给出。

Remember the formula: Vertical asymptote x = -d/c, horizontal asymptote y = a/c.
记住公式：垂直渐近线 x = -d/c，水平渐近线 y = a/c。
Use GDC wisely: In the calculator paper, you can use the graphing display calculator to check your sketch, but show working for asymptotes.
合理使用计算器：在允许使用计算器的试卷中，可以用图形计算器检查草图，但仍需写出渐近线的求法。
Practice algebraic division: Rewriting as A + B/(x+c) helps translate the graph from y = 1/x.
练习代数除法：改写为 A + B/(x+c) 有助于从 y = 1/x 出发进行图像变换。

Understanding the connection between the algebraic form and the geometric graph is the key to mastering simple rational functions in IB Mathematics.

理解代数形式与几何图像之间的联系，是掌握IB数学中简单有理函数的关键。

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Simple Rational Functions: Graphs and Properties | 简单有理函数图像与性质