📚 Rational Functions | 有理函数
A rational function is any function that can be written as the ratio of two polynomials. In IB Mathematics, understanding the behaviour of these functions — their asymptotes, intercepts, and transformations — is essential for mastering the calculus and algebra components of the syllabus. This article provides a thorough revision of rational functions, from definitions to advanced techniques such as partial fractions and inequality solving.
有理函数是可以写成两个多项式之比的函数。在 IB 数学中，理解这些函数的行为——它们的渐近线、截距和变换——对于掌握微积分和代数部分至关重要。本文全面复习有理函数，从定义到高级技巧如部分分式和解不等式，一应俱全。
1. Definition and Basic Form | 定义与基本形式
A rational function has the form f(x) = P(x)/Q(x) ax + b)/(cx + d). In IB, you will encounter both proper fractions (degree of P < degree of Q) and improper fractions where the degree of the numerator is greater than or equal to the denominator.
有理函数的形式为 f(x) = P(x)/Q(x)，其中 P(x) 和 Q(x) 是多项式且 Q(x) 不是零多项式。最简单的有理函数包括倒数函数如 f(x) = 1/x，以及线性除以线性的形式如 f(x) = (ax + b)/(cx + d)。在 IB 中，你既会遇到真分式（分子的次数小于分母的次数），也会遇到分子的次数大于或等于分母的假分式。
2. Domain and Vertical Asymptotes | 定义域与垂直渐近线
s ±∞. If a factor cancels, there is a hole (removable discontinuity) at that point instead of a vertical asymptote.
有理函数的定义域包括所有实数，但使分母为零的值除外。要找到这些限制，解方程 Q(x) = 0。分母的每一个不与分子因子抵消的实根都会产生一条垂直渐近线，即直线 x = a，函数在附近趋向 ±∞。如果因子抵消，则该点存在一个“洞”（可去间断点），而不是垂直渐近线。
3. Horizontal and Oblique Asymptotes | 水平与斜渐近线
a/c, where a and c are the leading coefficients. If the degree of P is exactly one greater than the degree of Q, the function has an oblique (slant) asymptote, found by polynomial long division. If the degree difference is greater than 1, there is no linear asymptote.
远端行为由水平或斜渐近线描述。比较 P(x) 和 Q(x) 的次数：如果 P 的次数 < Q 的次数，则水平渐近线为 y = 0。如果次数相等，水平渐近线为 y = a/c，其中 a 和 c 是首项系数。如果 P 的次数恰好比 Q 的次数大 1，则该函数有一条斜渐近线，可通过多项式长除法求得。如果次数差大于 1，则没有线性渐近线。
4. Finding Intercepts | 求截距
nding denominator value is not zero. Always check whether the zero of the numerator is also a zero of the denominator to distinguish between an intercept and a hole. Write intercepts as coordinates: (0, b) for y-intercept and (a, 0) for x-intercept.
y 截距通过计算 f(0) 求得，前提是 0 在定义域内。x 截距发生在分子 P(x) = 0 时，但只有当对应的分母值不为零时才是真正的截距。务必检查分子的零点是否也是分母的零点，以区分截距和洞。将截距写为坐标形式：y 截距 (0, b)，x 截距 (a, 0)。
5. Sketching Graphs | 绘制图像
To sketch a rational function, follow a systematic approach: (1) Simplify the expression if possible and note any holes. (2) Find vertical asymptotes. (3) Determine horizontal or oblique asymptotes. (4) Locate x- and y-intercepts. (5) Compute sign test values in intervals between vertical asymptotes and intercepts to know where the graph lies above or below the x-axis. (6) Plot a few additional points if needed and sketch the curve approaching asymptotes.
绘制有理函数图像要遵循系统方法：(1) 如果可能，先化简表达式并注意所有洞。(2) 找出垂直渐近线。(3) 确定水平或斜渐近线。(4) 定位 x 截距和 y 截距。(5) 在垂直渐近线和截距之间的区间内计算符号测试值，了解图像在 x 轴上方还是下方。(6) 若需要，多描几个点并绘制接近渐近线的曲线。
6. Transformations of Rational Functions | 有理函数的变换
x-axis if a is negative. Recognising these patterns helps quickly sketch functions such as f(x) = 2/(x + 3) - 1 without heavy computation.
有理函数可以像其他函数一样通过平移、反射和伸缩进行变换。对于倒数函数 f(x) = 1/x，g(x) = a/(x - h) + k 的图像可通过向右平移 h，向上平移 k，并以因子 a 垂直伸缩得到，如果 a 为负则关于 x 轴反射。识别这些模式有助于快速绘制如 f(x) = 2/(x + 3) - 1 之类的函数，无需大量计算。
7. Solving Rational Equations | 解有理方程
To solve an equation involving rational functions, first find the least common denominator (LCD) of all fractions and multiply every term by it, being careful to exclude values that make any denominator zero. Simplify to a polynomial equation and solve. Always check solutions against the original domain restrictions to discard extraneous roots that make denominators zero. IB examiners expect you to state the restriction and verify solutions.
解含有有理函数的方程时，首先找到所有分式的最小公分母 (LCD)，并用它乘以每一项，注意排除使任何分母为零的值。化简为多项式方程并求解。务必根据原始定义域限制检查解，以丢弃使分母为零的增根。IB 考官希望你明确限制条件并验证解。
8. Solving Rational Inequalities | 解有理不等式
Solving inequalities such as (x - 1)/(x + 2) > 2 requires bringing all terms to one side, combining into a single rational expression, and finding critical points where the numerator or denominator is zero. Create a sign table with intervals separated by these critical values. Determine the sign of the expression in each interval by testing a value. Finally, write the solution set in interval notation, excluding points where the denominator is zero.
解诸如 (x - 1)/(x + 2) > 2 的不等式需要将所有项移到一边，合并成一个有理表达式，并找到分子或分母为零的临界点。用这些临界值划分区间，创建一个符号表。通过代入测试值确定每个区间内表达式的符号。最后，用区间记号写出解集，排除分母为零的点。
9. Partial Fractions | 部分分式
constants A and B by clearing denominators and equating coefficients or substituting convenient x-values. Repeated linear factors and irreducible quadratic factors may also appear in HL content.
部分分式分解是一种将复杂有理表达式拆分为更简单分式的技术，便于积分或级数展开。在 IB 中，你主要分解具有线性因子的真分式，例如 (3x + 5)/[(x - 1)(x + 2)] = A/(x - 1) + B/(x + 2)。通过去分母并比较系数或代入方便的 x 值来求解常数 A 和 B。重线性因子和不可约二次因子也可能出现在 HL 内容中。
10. Applications and Modelling | 应用与建模
(fixed cost + variable cost * x)/x is rational. IB questions often ask you to interpret asymptotes in context: a horizontal asymptote might represent a long-term limiting value, while a vertical asymptote indicates a breakdown in the model.
有理函数可模拟许多现实场景，如平均成本、浓度随时间变化或强度与距离的关系（平方反比定律）。例如，平均成本函数 A(x) = (固定成本 + 可变成本 * x)/x 就是有理函数。IB 问题常要求你在上下文中解释渐近线：水平渐近线可能代表长期极限值，而垂直渐近线则表明模型失效。
11. Graphical Analysis with GDC | 用图形计算器分析
Your graphical display calculator (GDC) is a powerful tool for exploring rational functions. Use it to visualise asymptotes that are not immediately obvious, find intersection points with other graphs, and verify your algebraic work. Be mindful that GDC screens sometimes connect branches across vertical asymptotes due to resolution; you must use your mathematical knowledge to interpret the display correctly. The "zoom" and "trace" features are particularly useful for sign analysis and intercept approximation.
你的图形显示计算器 (GDC) 是探索有理函数的强大工具。用它来可视化不明显的渐近线，找到与其他图像的交点，并验证代数运算。要注意，由于分辨率原因，GDC 屏幕有时会连接垂直渐近线两侧的分支；你必须用数学知识正确解读显示内容。“zoom” 和 “trace” 功能对于符号分析和截距逼近特别有用。
12. Common Pitfalls and Tips | 常见错误与技巧
5). When simplifying, keep the original function in mind. A final tip: practise linking algebraic features to graphical behaviour — this holistic understanding is exactly what IB examination questions reward.
常见错误包括忘记检查洞、未进行长除法就声称存在斜渐近线，以及在乘以负分母时错误处理不等号。务必明确写出定义域，并将渐近线标记为直线方程（例如 x = 5，而非 5）。化简时要牢记原始函数。最后一个建议：练习将代数特征与图形行为联系起来——这种整体理解正是 IB 考题所奖励的。
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