Mastering Rational Functions for IB Mathematics | IB数学：有理函数专题精讲

📚 Mastering Rational Functions for IB Mathematics | IB数学：有理函数专题精讲

A rational function is the quotient of two polynomial functions. In IB Mathematics, rational functions appear frequently in both Analysis and Approaches (AA) and Applications and Interpretation (AI). Understanding their graphs, asymptotes, and algebraic manipulation is essential for exams and beyond.

有理函数是两个多项式函数的商。在IB数学中，有理函数在分析与方法（AA）和应用与解释（AI）中都频繁出现。理解其图像、渐近线及代数运算对于考试和后续学习至关重要。

1. Definition and Domain | 定义与定义域

A rational function is written as \(R(x) = \dfrac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \neq 0\). In this course we avoid LaTeX, but the idea is simple: a fraction with polynomials on top and bottom.

有理函数写作 \(R(x) = \dfrac{P(x)}{Q(x)}\)，其中 \(P(x)\) 和 \(Q(x)\) 是多项式，且 \(Q(x) \neq 0\)。我们不使用LaTeX，但概念很简单：分子分母都是多项式的分数。

The domain of a rational function excludes values that make the denominator zero. For example, \(f(x) = \dfrac{x+1}{x-2}\) is undefined at \(x=2\).

有理函数的定义域排除使分母为零的值。例如，\(f(x) = \dfrac{x+1}{x-2}\) 在 \(x=2\) 处无定义。

Always factor the denominator and set it equal to zero to find excluded values.
始终将分母因式分解并令其等于零，以找出被排除的值。
The domain is written as \(\mathbb{R}\) minus the set of these excluded values.
定义域写作实数集 \(\mathbb{R}\) 减去这些被排除值的集合。

2. Vertical Asymptotes | 垂直渐近线

A vertical asymptote occurs at a real number \(x = a\) where the denominator becomes zero and the numerator is not zero at that point. The function approaches positive or negative infinity as \(x\) approaches \(a\) from either side.

垂直渐近线出现在使分母为零而分子在该点不为零的实数 \(x = a\) 处。当 \(x\) 从两侧接近 \(a\) 时，函数趋于正无穷或负无穷。

If Q(a) = 0 and P(a) ≠ 0, then x = a is a vertical asymptote.

若 Q(a)=0 且 P(a)≠0，则 x=a 是垂直渐近线。

If both \(P(a)\) and \(Q(a)\) equal zero, the function has a hole (removable discontinuity) rather than an asymptote. Factor and cancel the common factor first.

如果 \(P(a)\) 和 \(Q(a)\) 都为零，则函数在该点有一个空洞（可去间断点），而非渐近线。应先因式分解并约去公因子。

Example: \(f(x)=\dfrac{x^2-1}{x^2-x-2}\). Factor to \(\dfrac{(x-1)(x+1)}{(x-2)(x+1)}\). The factor \(x+1\) cancels, so there is a hole at \(x=-1\) and a vertical asymptote at \(x=2\).
例：\(f(x)=\dfrac{x^2-1}{x^2-x-2}\)。因式分解为 \(\dfrac{(x-1)(x+1)}{(x-2)(x+1)}\)。因子 \(x+1\) 约去，因此在 \(x=-1\) 处有空洞，在 \(x=2\) 处有垂直渐近线。

3. Horizontal and Oblique Asymptotes | 水平与斜渐近线

Horizontal asymptotes describe the end behavior of a rational function as \(x \to \pm\infty\). The rule compares the degrees of the numerator and denominator.

水平渐近线描述有理函数在 \(x \to \pm\infty\) 时的端部行为。其规则比较分子和分母的次数。

Degree of numerator \(n\) vs denominator \(m\)	Horizontal asymptote
\(n < m\)	\(y = 0\)
\(n = m\)	\(y = \dfrac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}\)
\(n > m\)	No horizontal asymptote (may be oblique)

分子次数 \(n\) 与分母次数 \(m\) 比较	水平渐近线
\(n < m\)	\(y = 0\)
\(n = m\)	\(y = \dfrac{P \text{的首项系数}}{Q \text{的首项系数}}\)
\(n > m\)	无水平渐近线（可能为斜渐近线）

If the degree of the numerator is exactly one more than the degree of the denominator, the graph has an oblique (slant) asymptote. Perform polynomial long division to find its equation.

如果分子次数恰好比分母次数多一，则图像有一条斜渐近线。通过多项式长除法求其方程。

Example: \(f(x)=\dfrac{x^2+1}{x}\). Dividing gives \(f(x)=x+\dfrac{1}{x}\), so the oblique asymptote is \(y=x\).
例：\(f(x)=\dfrac{x^2+1}{x}\)。除法得 \(f(x)=x+\dfrac{1}{x}\)，故斜渐近线为 \(y=x\)。
The graph may cross a horizontal or oblique asymptote, but it never crosses a vertical asymptote.
图像可能与水平或斜渐近线相交，但绝不会与垂直渐近线相交。

4. Intercepts and Symmetry | 截距与对称性

To find the \(y\)-intercept, substitute \(x=0\). To find the \(x\)-intercepts, set the numerator equal to zero and solve, ensuring the solutions are in the domain.

求 \(y\) 截距时，令 \(x=0\) 代入。求 \(x\) 截距时，令分子等于零并求解，同时确保解在定义域内。

x-intercept: P(x) = 0, y-intercept: R(0)

x 截距：P(x)=0， y 截距：R(0)

Check symmetry: if \(R(-x)=R(x)\), the function is even and symmetric about the \(y\)-axis; if \(R(-x)=-R(x)\), it is odd and symmetric about the origin. Many rational functions have neither symmetry.

检查对称性：若 \(R(-x)=R(x)\)，则函数为偶函数，关于 \(y\) 轴对称；若 \(R(-x)=-R(x)\)，则为奇函数，关于原点对称。许多有理函数两种对称性都不具备。

For \(f(x)=\dfrac{x^2-4}{x^2+1}\), the \(x\)-intercepts are \(x= \pm 2\), \(y\)-intercept is \(y=-4\).
对于 \(f(x)=\dfrac{x^2-4}{x^2+1}\)，\(x\) 截距为 \(x=\pm 2\)，\(y\) 截距为 \(y=-4\)。
Remember: excluded values are not intercepts even if the numerator is zero there.
记住：被排除的值即使分子为零也不是截距。

5. Sign Chart and Behavior Near Asymptotes | 符号表与渐近线附近行为

A sign chart helps determine where the function is positive or negative. Mark all zeros and vertical asymptotes on a number line, then test intervals.

符号表有助于确定函数的正负区间。在数轴上标出所有零点和垂直渐近线，然后测试各区间。

Choose a test point in each interval and compute the sign of the function.
在每个区间选取一个测试点并计算函数的符号。
Use this information to understand why the curve goes up on one side of an asymptote and down on the other.
利用这一信息理解曲线为何在渐近线一侧向上而在另一侧向下。

For \(x\)-values approaching a vertical asymptote, check the sign of the denominator and numerator separately to determine the direction (positive infinity or negative infinity).

当 \(x\) 接近垂直渐近线时，分别检查分子和分母的符号，以确定方向（正无穷还是负无穷）。

Example: \(f(x)=\dfrac{1}{x-1}\). As \(x \to 1^+\), \(f \to +\infty\). As \(x \to 1^-\), \(f \to -\infty\).

例：\(f(x)=\dfrac{1}{x-1}\)。当 \(x \to 1^+\) 时，\(f \to +\infty\)；当 \(x \to 1^-\) 时，\(f \to -\infty\)。

6. Sketching Rational Functions | 绘制有理函数图像

To sketch a rational function accurately, follow a systematic process:

要准确绘制有理函数图像，请遵循一个系统化过程：

Factor numerator and denominator, simplify, state domain.
找出定义域：因式分解分子分母，化简，并写出定义域。
Find intercepts and asymptotes.
求截距和渐近线。
Plot holes as open circles.
用空心圆圈标出空洞。
Use a sign chart to determine the shape in each interval.
用符号表确定各区间内的形状。
Draw the curve, making sure it approaches the asymptotes correctly.
绘制曲线，确保其正确逼近渐近线。

A graphing calculator is useful, but IB exams often require analytic work. Knowing the key features lets you verify your sketch.

图形计算器很有用，但IB考试通常要求分析过程。了解关键特征可以帮助你验证草图。

7. Partial Fractions | 部分分式

Partial fraction decomposition rewrites a complicated rational function as a sum of simpler fractions. This technique is used in integration and binomial expansion in IB HL.

部分分式分解将复杂的有理函数重写为若干简单分式的和。该技术用于IB高级水平中的积分和二项式展开。

For distinct linear factors in the denominator:

当分母有互不相同的线性因子时：

\(\dfrac{3x+1}{(x-1)(x+2)} = \dfrac{A}{x-1} + \dfrac{B}{x+2}\)

Multiply through by the denominator, substitute \(x=1\) and \(x=-2\) to find \(A\) and \(B\). For repeated factors, use terms like \(\dfrac{A}{x-1} + \dfrac{B}{(x-1)^2}\). For irreducible quadratics, use \(\dfrac{Ax+B}{x^2+1}\).

两边同乘分母，代入 \(x=1\) 和 \(x=-2\) 求 \(A\) 和 \(B\)。对于重因子，使用 \(\dfrac{A}{x-1} + \dfrac{B}{(x-1)^2}\) 等项。对于不可约二次因子，使用 \(\dfrac{Ax+B}{x^2+1}\)。

Always check that the degree of the numerator is less than that of the denominator; if not, divide first.
始终确保分子次数低于分母次数；否则需先做除法。

8. Solving Rational Inequalities | 有理不等式

Rational inequalities like \(\dfrac{x-1}{x+3} \leq 0\) are solved by finding critical values (zeros and undefined points), then testing intervals.

有理不等式如 \(\dfrac{x-1}{x+3} \leq 0\) 通过寻找临界值（零点和无定义点）并测试区间来求解。

Critical values: x = 1 (numerator zero) and x = -3 (denominator zero)

临界值：x=1（分子零点）和 x=-3（分母零点）

Mark these values on a number line, divide into intervals: \((-\infty,-3)\), \((-3,1)\), \((1,\infty)\).
在数轴上标出这些值，分成区间：\((-\infty,-3)\)、\((-3,1)\)、\((1,\infty)\)。
Test each interval: choose \(x=-4\) gives \(\dfrac{-5}{-1}=5>0\); \(x=0\) gives \(\dfrac{-1}{3}<0\); \(x=2\) gives \(\dfrac{1}{5}>0\).
测试各区间：取 \(x=-4\) 得 \(\dfrac{-5}{-1}=5>0\)；\(x=0\) 得 \(\dfrac{-1}{3}<0\)；\(x=2\) 得 \(\dfrac{1}{5}>0\)。
The solution to \(\leq 0\) is \(-3 < x \leq 1\). Note that \(x=-3\) is excluded, but \(x=1\) is included.
\(\leq 0\) 的解为 \(-3 < x \leq 1\)。注意 \(x=-3\) 被排除，但 \(x=1\) 包含在内。

Be careful when multiplying both sides by a denominator that may be negative. Instead, keep zero on one side and use a sign chart.

当两边乘以可能为负的分母时要小心。相反，应保持一边为零并使用符号表。

9. Applications in IB Problems | IB题目中的应用

Rational functions model real-life situations where one quantity varies inversely with another, such as average cost, concentration of medicine, and speed-distance-time problems.

有理函数可以模拟两个量成反比的实际情境，例如平均成本、药物浓度以及速度-距离-时间问题。

In IB applications, you may be asked to interpret the horizontal asymptote as a maximum or minimum limiting value.

在IB应用题中，你可能会被要求将水平渐近线解释为最大或最小极限值。

Example: The cost per student for a school trip is \(C(n)=\dfrac{500+20n}{n}\), where \(n\) is the number of students. The horizontal asymptote \(y=20\) means the cost per student approaches 20 as more students join.
例：学校旅行的每位学生费用为 \(C(n)=\dfrac{500+20n}{n}\)，其中 \(n\) 是学生人数。水平渐近线 \(y=20\) 表示随着学生增多，人均费用趋近于20。
Solving rational equations with a graphing calculator or algebraically is a common exam skill.
用图形计算器或代数方法求解有理方程是常见考试技能。

10. Common Mistakes and Tips | 常见错误与建议

Many students lose marks on rational function questions due to small errors. Here are the most common ones:

许多学生在有理函数题目中因小错误失分。以下是最常见的错误：

Mistake	Correction
Forgetting to exclude values where denominator is zero	Always state domain before simplifying
Canceling factors before checking zeros of denominator	Cancel only after noting any holes
Assuming a horizontal asymptote is never touched	The graph can cross horizontal asymptotes

错误	纠正
忘记排除分母为零的值	化简前先写出定义域
在检查分母零点之前约分	先记录空洞再约分
以为水平渐近线不会被穿过	图像可以与水平渐近线相交

Always verify your graph with key points and check the end behavior. Practice with past IB questions to become comfortable with the notation.

始终用关键点验证你的图像并检查端部行为。多练IB真题以熟悉符号和题型。

11. Practice Questions | 练习题目

Try these representative IB-style problems:

尝试以下具有代表性的IB风格题目：

For \(f(x)=\dfrac{2x^2+3x-2}{x^2-4}\), find the domain, intercepts, and equations of all asymptotes.
对于 \(f(x)=\dfrac{2x^2+3x-2}{x^2-4}\)，求定义域、截距以及所有渐近线的方程。
Solve \(\dfrac{x}{x-2} \geq \dfrac{3}{x}\).
解不等式 \(\dfrac{x}{x-2} \geq \dfrac{3}{x}\)。
Express \(\dfrac{5x-4}{(x-1)(x-2)}\) in partial fractions.
将 \(\dfrac{5x-4}{(x-1)(x-2)}\) 分解为部分分式。
The graph of \(y=\dfrac{ax+b}{x+c}\) has vertical asymptote \(x=2\) and passes through \((1,1)\) and \((0,3)\). Find \(a, b, c\).
函数 \(y=\dfrac{ax+b}{x+c}\) 的图像有垂直渐近线 \(x=2\)，且经过点 \((1,1)\) 和 \((0,3)\)。求 \(a, b, c\)。

Answers: 1. Domain excludes \(\pm2\); \(x\)-intercept at \(x=\dfrac{1}{2}\), \(y\)-intercept \(\dfrac{1}{2}\); vertical asymptote \(x=-2\), hole at \(x=2\); horizontal asymptote \(y=2\). 2. \(x \leq 0\) or \(1 < x < 2\) or \(x>2\)? Wait, solve carefully. 3. \(\dfrac{1}{x-1}+\dfrac{6}{x-2}\). 4. \(a=\dfrac{3}{2}, b=6, c=2\).

答案：1. 定义域排除 \(\pm2\)；\(x\) 截距 \(x=\dfrac{1}{2}\)，\(y\) 截距 \(\dfrac{1}{2}\)；垂直渐近线 \(x=-2\)，空洞在 \(x=2\)；水平渐近线 \(y=2\)。2. 注意仔细求解。3. \(\dfrac{1}{x-1}+\dfrac{6}{x-2}\)。4. \(a=\dfrac{3}{2}, b=6, c=2\)。

12. Conclusion | 总结

Rational functions are a rich topic that ties together algebra, graphing, and real-world modeling. Master the key features: domain, intercepts, asymptotes, and sign behavior.

有理函数是一个内容丰富的话题，将代数、绘图和现实建模联系在一起。掌握关键特征：定义域、截距、渐近线和符号行为。

Practice regularly, learn from mistakes, and use these skills to approach IB questions with confidence.

定期练习，从错误中学习，并运用这些技能自信地应对IB题目。

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Mastering Rational Functions for IB Mathematics | IB数学：有理函数专题精讲