Rational Functions: Graphs and Properties | 有理函数的图像与性质

📚 Rational Functions: Graphs and Properties | 有理函数的图像与性质

A rational function is a function of the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials, and Q(x) is not the zero polynomial. Understanding its graph and properties is a core topic in IB Mathematics, especially for Analysis and Approaches.

有理函数是形如 f(x) = P(x)/Q(x) 的函数，其中 P(x) 和 Q(x) 都是多项式，且 Q(x) 不恒为零。理解它的图像与性质是 IB 数学（尤其是分析与方法）的核心内容。

1. Definition and Domain | 定义与定义域

The domain of a rational function consists of all real numbers except those that make the denominator zero. For example, f(x) = 1/(x − 3) is defined for all x except x = 3. In interval notation, the domain is (−∞, 3) ∪ (3, ∞).

有理函数的定义域包含所有使分母不为零的实数。例如，f(x) = 1/(x − 3) 对除 x = 3 之外的所有 x 都有定义。用区间表示，定义域为 (−∞, 3) ∪ (3, ∞)。

When finding the domain, factor both numerator and denominator first. Any value that makes Q(x) = 0 is excluded, but if the same factor cancels, a “hole” may occur instead of a vertical asymptote — this will be explained in Section 2.

求定义域时，先对分子、分母因式分解。任何使 Q(x) = 0 的值都要排除；但如果某因式可以约分，则可能产生“空洞”而非垂直渐近线——这一点将在第 2 节解释。

If f(x) = P(x)/Q(x), then domain = {x ∈ ℝ | Q(x) ≠ 0}.

若 f(x) = P(x)/Q(x)，则定义域 = {x ∈ ℝ | Q(x) ≠ 0}。
Always simplify the function before stating the domain? No — the domain is determined by the original expression, not the simplified one.

是否一定要先化简再写出定义域？不——定义域由原表达式决定，而不是化简后的式子。

2. Vertical Asymptotes and Holes | 垂直渐近线与空洞

After simplifying the rational function, any factor remaining in the denominator that is zero at x = a gives a vertical asymptote: the line x = a. Near a vertical asymptote, the y-values tend to +∞ or −∞.

化简有理函数后，分母中仍保留的因式若在 x = a 处为零，则会产生垂直渐近线 x = a。在垂直渐近线附近，y 值趋向 +∞ 或 −∞。

If a factor (x − a) appears in both numerator and denominator and cancels completely, then the graph has a hole at x = a, not a vertical asymptote. The y-coordinate of the hole is found by evaluating the simplified function at x = a.

如果因式 (x − a) 同时出现在分子和分母中并被完全约去，那么图像在 x = a 处有一个空洞，而不是垂直渐近线。空洞的 y 坐标要通过代入化简后的函数来求。

f(x) = (x² − 1)/(x − 1) = x + 1, x ≠ 1 → hole at (1, 2)

Vertical asymptote: denominator factor does not cancel.

垂直渐近线：分母因式没有约去。
Hole: denominator factor cancels with numerator factor.

空洞：分母因式与分子因式约去。

3. Horizontal Asymptotes | 水平渐近线

Horizontal asymptotes describe the behaviour of f(x) as x → +∞ or x → −∞. The rules depend on the degrees of the numerator, n, and denominator, m:

水平渐近线描述当 x → +∞ 或 x → −∞ 时 f(x) 的走势。判断规则取决于分子次数 n 与分母次数 m：

Degree condition 次数条件	Horizontal asymptote 水平渐近线
n < m	y = 0
n = m	y = aₙ / bₘ (ratio of leading coefficients)
n > m	No horizontal asymptote (may have oblique asymptote)

For example, f(x) = (2x² + 1)/(3x² − 5) has horizontal asymptote y = 2/3, because the highest powers are both x².

例如，f(x) = (2x² + 1)/(3x² − 5) 的水平渐近线为 y = 2/3，因为分子分母的最高次都是 x²。

4. Oblique (Slant) Asymptotes | 斜渐近线

When the degree of the numerator is exactly one more than the degree of the denominator, the rational function has an oblique asymptote. The asymptote is the quotient obtained by polynomial long division of P(x) by Q(x).

当分子次数比分母恰好高 1 次时，有理函数存在斜渐近线。该渐近线是 P(x) 除以 Q(x) 的多项式长除法所得的商式。

For instance, f(x) = (x² + 2)/(x − 1). Dividing gives x + 1 with remainder 3, so the oblique asymptote is y = x + 1. As x → ∞, the graph approaches this line.

例如，f(x) = (x² + 2)/(x − 1)。长除法得商 x + 1，余数 3，因此斜渐近线为 y = x + 1。当 x → ∞ 时，图像趋向这条直线。

If deg(P) = deg(Q) + 1, then f(x) = q(x) + r(x)/Q(x), with q(x) oblique asymptote.

若 deg(P) = deg(Q) + 1，则 f(x) = q(x) + r(x)/Q(x)，其中 q(x) 为斜渐近线。

5. Intercepts and Symmetry | 截距与对称性

The y-intercept is found by setting x = 0 in the original function. The x-intercepts are found by setting the numerator equal to zero (provided those values are in the domain).

y 截距通过令 x = 0 代入原函数求得。x 截距通过令分子等于零求得（前提是这些值在定义域内）。

To check symmetry: if f(−x) = f(x), the graph is even and symmetric about the y-axis. If f(−x) = −f(x), it is odd and symmetric about the origin. Most rational functions have neither symmetry.

判断对称性：若 f(−x) = f(x)，则函数为偶函数，图像关于 y 轴对称；若 f(−x) = −f(x)，则为奇函数，图像关于原点对称。大多数有理函数不具备这两种对称性。

x-intercepts: solve P(x) = 0, then check Q(x) ≠ 0.

x 截距：解 P(x) = 0，再确认 Q(x) ≠ 0。
y-intercept: compute f(0) if 0 is in the domain.

y 截距：若 0 在定义域内，计算 f(0)。

6. Sign Analysis and Behaviour Near Asymptotes | 符号分析与渐近线附近行为

To sketch accurately, determine the sign of f(x) on each interval separated by vertical asymptotes and x-intercepts. Choose a test point in each interval and evaluate the sign of the function.

为准确作图，需确定在由垂直渐近线和 x 截距划分的每个区间内 f(x) 的符号。在每个区间内选取测试点，判断函数值的正负。

Near a vertical asymptote, the graph goes to +∞ on one side and −∞ on the other, or sometimes both sides go to the same infinity if the denominator factor has even multiplicity. For example, 1/x² approaches +∞ from both sides of x = 0.

在垂直渐近线附近，图像一侧趋向 +∞，另一侧趋向 −∞；若分母因式的重数为偶数，则两侧可能同向趋向无穷。例如，1/x² 在 x = 0 两侧都趋向 +∞。

Use a sign table: intervals → test point → sign → sketch.

使用符号表：区间 → 测试点 → 符号 → 绘图。

7. Sketching Graphs Step-by-Step | 逐步作图法

A systematic approach to graphing rational functions ensures accuracy. Follow these steps:

系统作图法能够保证准确性。请按以下步骤进行：

Factor numerator and denominator completely.

将分子、分母完全因式分解。
State the domain; identify holes and vertical asymptotes.

写出定义域；识别空洞与垂直渐近线。
Find horizontal or oblique asymptotes.

求水平或斜渐近线。
Find intercepts with the axes.

求坐标轴截距。
Perform a sign analysis on each interval.

对每个区间进行符号分析。
Plot asymptotes, intercepts, holes, and enough points to draw a smooth curve.

标出渐近线、截距、空洞以及足够多的点，再画出平滑曲线。

Always remember that the graph cannot cross a vertical asymptote, but it may cross a horizontal or oblique asymptote.

切记：图像不能穿过垂直渐近线，但可能穿过水平或斜渐近线。

8. Transformations of Rational Functions | 有理函数的变换

Rational functions can be transformed using the same rules as other functions. If f(x) = 1/x, then:

有理函数与其他函数一样可以施加变换。若 f(x) = 1/x，则：

f(x − a) + b shifts the graph right by a and up by b; the asymptotes shift accordingly.

f(x − a) + b 将图像向右平移 a 个单位、向上平移 b 个单位；渐近线随之平移。
−f(x) reflects across the x-axis, while f(−x) reflects across the y-axis.

−f(x) 关于 x 轴翻转，而 f(−x) 关于 y 轴翻转。
k·f(x) vertically stretches/compresses by factor k.

k·f(x) 在纵向上拉伸/压缩 k 倍。

For a general rational function, a vertical translation changes the horizontal asymptote, while a horizontal translation changes the vertical asymptote.

对一般有理函数，纵向平移改变水平渐近线，横向平移改变垂直渐近线。

9. Solving Equations and Inequalities | 解方程与不等式

To solve f(x) = k, set P(x)/Q(x) = k and multiply by Q(x), but note that any solution causing Q(x) = 0 is invalid.

解方程 f(x) = k 时，令 P(x)/Q(x) = k，然后两边乘以 Q(x)，但注意任何使 Q(x) = 0 的解都要舍去。

For inequalities, do not simply multiply by Q(x) because its sign may be unknown. Use a sign table or express the inequality with a common denominator, then analyze intervals.

解不等式时，不能直接乘以 Q(x)，因为其符号未知。应使用符号表，或通分后按区间分析符号。

(x − 2)/(x + 1) ≥ 0 → critical values: x = 2, x = −1

(x − 2)/(x + 1) ≥ 0 → 关键值：x = 2, x = −1

Test intervals: (−∞, −1): f > 0; (−1, 2): f < 0; (2, ∞): f > 0. Thus the solution is (−∞, −1) ∪ [2, ∞).

测试区间：(−∞, −1) 内 f > 0；(−1, 2) 内 f < 0；(2, ∞) 内 f > 0。因此解集为 (−∞, −1) ∪ [2, ∞)。

10. Applications and Modelling | 应用与建模

Rational functions appear in physics (inverse square law), economics (average cost), and chemistry (concentration over time). For example, the average cost C(x) = (ax + b)/x often has a horizontal asymptote representing minimum average cost.

有理函数出现在物理学（平方反比定律）、经济学（平均成本）和化学（浓度随时间变化）中。例如，平均成本 C(x) = (ax + b)/x 常具有代表平均成本下限的水平渐近线。

In IB exam contexts, word problems may ask you to interpret the meaning of the asymptote or the maximum/minimum value from the graph.

在 IB 考试中，应用题可能要求你解释渐近线的实际意义，或根据图像求出最大值/最小值。

11. Common Errors and Exam Tips | 常见错误与考试建议

Students often confuse holes and vertical asymptotes, forget to exclude restricted domain values when solving, or incorrectly identify oblique asymptotes.

学生常混淆空洞与垂直渐近线，在解方程时忘记排除定义域限制值，或错误判断斜渐近线。

Always reduce the fraction first, but use the original denominator for domain.

化简分数前先用原分母确定定义域。
Check the degree of numerator vs. denominator before deciding which asymptote exists.

先比较分子与分母的次数，再判断存在哪种渐近线。
When sketching, mark all asymptotes with dashed lines and label significant points.

作图时用虚线标出所有渐近线，并标注关键点。

12. Summary and Revision Checklist | 总结与复习清单

Mastering rational functions requires practice. Use this checklist before your exam:

掌握有理函数需要练习。考试前请使用以下清单自查：

Can I find the domain and holes instantly?

我能快速找出定义域和空洞吗？
Do I know the three cases for horizontal asymptotes?

我清楚水平渐近线的三种情况吗？
Can I perform long division to find oblique asymptotes?

我能用长除法求斜渐近线吗？
Can I sketch a complete graph from scratch?

我能从零开始画出完整图像吗？
Am I comfortable solving rational inequalities with sign tables?

我能熟练使用符号表解有理不等式吗？

Once you can answer yes to all, rational functions will be one of your strongest topics.

当以上问题你都能回答“是”时，有理函数将成为你最拿手的主题之一。

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Rational Functions: Graphs and Properties | 有理函数的图像与性质