📚 Exercise 9A: Rational Functions and Asymptotes | 练习 9A：有理函数与渐近线
In IB Mathematics, Exercise 9A typically serves as an introduction to rational functions, focusing on identifying asymptotes, intercepts, and the overall shape of their graphs. Mastering this exercise is crucial for success in both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses. This article breaks down every concept tested in Exercise 9A, providing clear explanations, worked examples, and tips to avoid common pitfalls.
在 IB 数学中，练习 9A 通常是对有理函数的入门练习，重点在于识别渐近线、截距以及函数图像的整体形状。掌握这部分内容对于在分析与方法 (AA) 以及应用与解释 (AI) 课程中取得成功至关重要。本文将逐一解析练习 9A 所考查的每个概念，提供清晰的讲解、实例解析以及避免常见错误的技巧。
1. What Is a Rational Function? | 什么是有理函数？
q(x) is not the zero polynomial. In Exercise 9A you will meet simple examples such as f(x) = 1 / (x − 2) or f(x) = (x + 3) / (x − 1), and later more complicated ones where both p and q are quadratics.
有理函数是指可以写成两个多项式之比的任何函数，形式为 f(x) = p(x) / q(x)，其中 p 与 q 都是多项式，且 q(x) 不是零多项式。在练习 9A 中，你会遇到诸如 f(x) = 1 / (x − 2) 或 f(x) = (x + 3) / (x − 1) 的简单形式，之后还会遇到分子分母都是二次式的复杂形式。
f(x) = p(x) / q(x), q(x) ≠ 0
udes every x-value that makes q(x) = 0. This excluded value often leads to a vertical asymptote, a core feature explored in Exercise 9A.
关键的限制是分母不能为零，因为除以零是无定义的。因此，有理函数的定义域排除了所有使 q(x) = 0 的 x 值。这个被排除的值通常会导致一条垂直渐近线，这是练习 9A 中要探究的核心特征之一。
Note the difference between a polynomial and a rational function: a polynomial is defined for every real number, while a rational function almost always has at least one gap in its domain. Recognising that gap is the first step of every Exercise 9A question.
请注意多项式与有理函数的区别：多项式对所有实数都有定义，而有理函数几乎总会在定义域中至少留出一个"缺口"。识别这个缺口，就是每一道练习 9A 题目的第一步。
2. Domain and Excluded Values | 定义域与排除值
tor is zero when x = 2, so the domain is all real numbers except x = 2, often written as x ∈ ℝ, x ≠ 2, or in interval notation as (−∞, 2) ∪ (2, ∞).
求定义域的方法是令分母等于零并求解。对于 f(x) = 1 / (x − 2)，当 x = 2 时分母为零，因此定义域为除 x = 2 外的所有实数，通常写作 x ∈ ℝ, x ≠ 2，或用区间符号写作 (−∞, 2) ∪ (2, ∞)。
(x + 3), so the domain excludes both x = 3 and x = −3. When a factor appears in both numerator and denominator, the excluded value may produce a hole rather than an asymptote, which is discussed in Section 4.
若分母可以因式分解，就要列出分母的每一个根。例如 f(x) = (x + 1) / (x² − 9) 的分母为 (x − 3)(x + 3)，因此定义域同时排除 x = 3 与 x = −3。当某个因式同时出现在分子与分母中时，该排除值可能产生一个"洞"而不是渐近线，这将在第 4 节讨论。
In AI-style questions the domain is usually stated in words, such as "the function is not defined at x = 2" In AA-style questions you should also be comfortable with set notation and with the phrase "x tends to 2 from the left" written as x → 2⁻.
在 AI 风格的题目中，定义域通常用文字表述，如"函数在 x = 2 处无定义"。在 AA 风格的题目中，你还应熟练掌握集合符号，以及"x 从左侧趋近于 2"这种写法 x → 2⁻。
3. Vertical Asymptotes | 垂直渐近线
f the graph of f if the function values grow without bound as x approaches a from the left or from the right — ver crosses a vertical asymptote, and the curve on either side of it shoots off the top or bottom of the page.
如果当 x 从左侧或右侧趋近于 a 时函数值无限增大（即 f(x) → ∞ 或 f(x) → −∞），则称直线 x = a 为 f 图像的垂直渐近线。图像永远不会穿过垂直渐近线，且曲线在其两侧会冲向纸面的上方或下方。
efore: set q(x) = 0, solve for x, then check that these values do not also make p(x) = 0. If both are zero, the factor cancels and you have a hole instead.
对于有理函数，垂直渐近线出现在分母为零且分子在同一位置不为零的地方。方法因此是：令 q(x) = 0，解出 x，然后确认这些值不会同时使 p(x) = 0。如果两者都为零，因式会被约去，你得到的是一个"洞"。
Exercise 9A often asks: "Find the equations of any vertical asymptotes for f(x) = (2x + 1) / (x − 3)." ince the numerator at x = 3 is 2(3) + 1 = 7 ≠ 0, we confirm a vertical asymptote at x = 3.
练习 9A 经常问道："求函数 f(x) = (2x + 1) / (x − 3) 的所有垂直渐近线方程。" 这里由分母 x − 3 = 0 得 x = 3；由于当 x = 3 时分子为 2(3) + 1 = 7 ≠ 0，我们可以确认 x = 3 是一条垂直渐近线。
3⁻, the numerator is positive and the denominator is a small negative number, so f(x) → −∞; as x → 3⁺, f(x) → +∞. These one-sided statements are worth full marks in AA papers.
要正确描述渐近线附近的行为，需分别判断 x = 3 左侧与右侧 f 的符号。对 f(x) = (2x + 1) / (x − 3)：当 x → 3⁻ 时，分子为正而分母是绝对值很小的负数，所以 f(x) → −∞；当 x → 3⁺ 时，f(x) → +∞。这些单侧极限的表述在 AA 试卷中可以直接拿满分。
4. Holes and Removable Discontinuities | 可去间断点（洞）
A hole (removable discontinuity) appears when the same factor occurs in both the numerator and the denominator. Cancelling that factor simplifies the function, but the original function is still undefined at the cancelled value, so the graph has a single missing point.
当同一个因式同时出现在分子与分母中时，就会出现一个"洞"（可去间断点）。约去该因式可以化简函数，但原函数在约去的那个值处仍然无定义，因此图像上有一个孤立的空缺点。
ight line y = x + 2 with a single hole at x = 2. The coordinates of the hole are found by substituting x = 2 into the simplified expression: y = 2 + 2 = 4, so the hole is at (2, 4).
考虑 f(x) = (x² − 4) / (x − 2)。因式分解得 f(x) = (x + 2)(x − 2) / (x − 2)，约去后当 x ≠ 2 时 f(x) = x + 2。因此图像就是直线 y = x + 2，但在 x = 2 处有一个洞。把 x = 2 代入化简后的表达式即可求洞的坐标：y = 2 + 2 = 4，所以洞位于点 (2, 4)。
An important exam point: always state the domain before cancelling. Writing "f(x) = x + 2" t suggests the function is defined at x = 2. The correct statement is f(x) = x + 2, x ≠ 2.
一个重要的考点：务必先写出定义域再约分。只写"f(x) = x + 2"是不完整的，因为这会暗示函数在 x = 2 处有定义。正确的表述是 f(x) = x + 2, x ≠ 2。
5. Horizontal Asymptotes | 水平渐近线
ed not touch, and the value of L depends only on the degrees of the numerator p(x) and denominator q(x).
水平渐近线描述的是有理函数在 x → ∞ 或 x → −∞ 时的末端趋势。它是一条水平直线 y = L，图像会无限接近它（但不必碰到它）。L 的值只取决于分子 p(x) 与分母 q(x) 的次数。
A crucial subtlety: unlike a vertical asymptote, a graph may cross its horizontal asymptote at finite values of x. The asymptote only governs what happens at the far left and far right of the graph.
一个关键细节：与垂直渐近线不同，图像在有限的 x 值处是可以穿过水平渐近线的。水平渐近线只决定图像在最左端与最右端的行为。
as x → ±∞, f(x) → L
ote y = 3. Meanwhile f(x) = 1 / (x − 2) has a numerator of lower degree, so the horizontal asymptote is y = 0, the x-axis.
例如，f(x) = (3x² + 2) / (x² − 1) 的分子与分母次数相同，首项系数 3 与 1 给出水平渐近线 y = 3。而 f(x) = 1 / (x − 2) 的分子次数更低，因此水平渐近线为 y = 0，即 x 轴。
6. Degree Comparison Rules | 次数比较规则
All horizontal and oblique asymptote questions in Exercise 9A reduce to comparing the degree of the numerator with the degree of the denominator. Let n be the degree of p(x) and m the degree of q(x); the three cases are summarised in the table below.
练习 9A 中所有水平与斜渐近线的问题，最终都归结为比较分子与分母的次数。设 p(x) 的次数为 n，q(x) 的次数为 m，三种情形总结如下表。
| Case | 情形 | Asymptote | 渐近线 | Example | 例子 |
|---|---|---|
| n < m | y = 0 (the x-axis) | 1 / (x − 2) → y = 0 |
| n = m | y = aₙ / bₘ (ratio of leading coefficients) | (2x + 1) / (x − 3) → y = 2 |
| n = m + 1 | Oblique asymptote from division | (x² + 1) / x → y = x |
| n > m + 1 | No horizontal or oblique asymptote | x³ / (x − 1) → none |
A common mistake is to assume that every rational function has a horizontal asymptote. If the numerator degree exceeds the denominator degree by two or more, the graph has no horizontal asymptote at all — it simply grows without bound in both directions, as x³ / (x − 1) does.
常见错误是以为每个有理函数都有水平渐近线。如果分子次数比分母次数高出两个或更多，图像根本没有水平渐近线，它会向两个方向无限增长，例如 x³ / (x − 1)。
7. Oblique (Slant) Asymptotes | 斜渐近线
When the degree of the numerator is exactly one more than the degree of the denominator, the graph eventually behaves like a straight line with a non-zero gradient. That line is called an oblique asymptote, or slant asymptote, and it is found by polynomial long division.
当分子次数恰好比分母次数多 1 时，图像的大范围行为就像一条斜率非零的直线。这条直线称为斜渐近线，可通过多项式长除法求得。
(x + 1) tends to zero, leaving f(x) → x + 2. The oblique asymptote is therefore y = x + 2.
取 f(x) = (x² + 3x + 5) / (x + 1)。做除法得 x² + 3x + 5 = (x + 1)(x + 2) + 3，所以 f(x) = x + 2 + 3 / (x + 1)。当 x → ±∞ 时，分式 3 / (x + 1) 趋于零，于是 f(x) → x + 2。因此斜渐近线为 y = x + 2。
f(x) = (quotient) + (remainder) / q(x), remainder / q(x) → 0
The remainder term is what tells you which side of the oblique asymptote the graph lies: a positive remainder means the curve sits above the line for large positive x, a negative remainder means it sits below.
余项告诉你图像位于斜渐近线的哪一侧：余项为正表示当 x 很大时曲线在直线之上，余项为负则表示曲线在直线之下。
8. Intercepts and Sign Analysis | 截距与符号分析
nstant terms. If q(0) = 0, the y-axis is a vertical asymptote and there is no y-intercept.
求 y 轴截距的方法是代入 x = 0（前提是 0 在定义域内）；对 f(x) = p(x) / q(x) 而言就是 p(0) / q(0)，即常数项之比。若 q(0) = 0，则 y 轴是垂直渐近线，函数没有 y 轴截距。
, solve, then discard any solution that makes q(x) = 0, because those correspond to holes.
x 轴截距是分子的根中那些不是分母的根的值。令 p(x) = 0 求解，再剔除使 q(x) = 0 的解，因为它们对应的是"洞"。
A sign table makes the shape of the graph clear. Mark all the critical x-values — the roots and the vertical asymptotes — on a number line, test one value in each interval, and record whether f is positive or negative. This tells you which side of the x-axis each branch of the curve lies on.
符号表能让图像的形状一目了然。把所有临界 x 值——即零点与垂直渐近线——标在数轴上，在每个区间内取一个测试值，记录 f 的正负。这样你就知道曲线的每一支位于 x 轴的哪一侧。
9. Step-by-Step Graphing Workflow | 分步作图流程
Exercise 9A almost always asks for a sketch, so it helps to use the same sequence of steps every time. Mark each step on your working so the examiner can follow your reasoning.
练习 9A 几乎总会要求作图，因此每次都采用同一套步骤会很有帮助。把每一步写清楚，阅卷老师才能跟上你的思路。
- Step 1: Factorise the numerator and denominator completely. | 第一步：把分子与分母彻底因式分解。
- Step 2: State the domain by setting q(x) = 0. | 第二步：令 q(x) = 0，写出定义域。
- Step 3: Cancel any common factor and record the hole. | 第三步：约去公因式，并记录"洞"的坐标。
- Step 4: Solve q(x) = 0 on the simplified expression to find vertical asymptotes. | 第四步：在化简后的表达式上解 q(x) = 0，求垂直渐近线。
- Step 5: Compare degrees to find the horizontal or oblique asymptote. | 第五步：比较次数，求水平或斜渐近线。
- Step 6: Find the x- and y-intercepts. | 第六步：求 x 轴与 y 轴截距。
- Step 7: Build a sign table and test the behaviour on each side of every vertical asymptote. | 第七步：列出符号表，检验每条垂直渐近线两侧的行为。
- Step 8: Plot the asymptotes as dashed lines, plot the intercepts and the hole, then join the points with smooth branches. | 第八步：用虚线画出渐近线，标出截距与"洞"，再用光滑曲线连接各分支。
Sketching asymptotes as dashed lines before drawing the curve saves a great deal of time and prevents the branches from being drawn in the wrong region.
先画虚线渐近线再画曲线，可以节省大量时间，也能避免把分支画到错误的区域。
10. Worked Example | 典型例题
2) / (x² − 4). Factorising gives f(x) = (x − 2)(x + 1) / [(x − 2)(x + 2)]. The domain is x ∈ ℝ, x ≠ 2, x ≠ −2.
设 f(x) = (x² − x − 2) / (x² − 4)。因式分解得 f(x) = (x − 2)(x + 1) / [(x − 2)(x + 2)]。定义域为 x ∈ ℝ
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