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

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

A rational function is defined as the quotient of two polynomial functions, \(f(x) = \frac{P(x)}{Q(x)}\), where \(Q(x) \neq 0\). Understanding its graph involves analysing domain, asymptotes, intercepts, and behaviour near undefined points.

有理函数定义为两个多项式函数的商，即 \(f(x) = \frac{P(x)}{Q(x)}\)，其中 \(Q(x) \neq 0\)。理解其图像需要分析定义域、渐近线、截距以及未定义点附近的行为。

1. Definition and Standard Form | 定义与标准形式

A rational function can be written in the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials with real coefficients. The domain excludes all values of \(x\) for which \(Q(x) = 0\).

有理函数可以写成 \(f(x) = \frac{P(x)}{Q(x)}\) 的形式，其中 \(P(x)\) 和 \(Q(x)\) 是实系数多项式。定义域排除所有使 \(Q(x) = 0\) 的 \(x\) 值。

Example: \(f(x) = \frac{2x + 1}{x – 3}\) has domain \(x \in \mathbb{R}, x \neq 3\).
示例：\(f(x) = \frac{2x + 1}{x – 3}\) 的定义域为 \(x \in \mathbb{R}, x \neq 3\)。
The degree of \(P\) and \(Q\) determines the type and existence of horizontal or oblique asymptotes.
\(P\) 与 \(Q\) 的次数决定了水平或斜渐近线的类型与存在性。

Standard form: \(f(x) = \frac{a_n x^n + \dots + a_0}{b_m x^m + \dots + b_0}\)

标准形式：\(f(x) = \frac{a_n x^n + \dots + a_0}{b_m x^m + \dots + b_0}\)

2. Domain and Zeros | 定义域与零点

The domain of a rational function is all real numbers except the roots of the denominator. The zeros of the function are the roots of the numerator that are not also roots of the denominator.

有理函数的定义域是除去分母根之外的所有实数。函数的零点是分子的根，且这些根不能同时是分母的根。

If \(P(a) = 0\) and \(Q(a) \neq 0\), then \(x = a\) is a zero of \(f\).
若 \(P(a) = 0\) 且 \(Q(a) \neq 0\)，则 \(x = a\) 是 \(f\) 的零点。
If both \(P(a) = 0\) and \(Q(a) = 0\), then there is a hole (removable discontinuity) at \(x = a\).
若 \(P(a) = 0\) 且 \(Q(a) = 0\)，则在 \(x = a\) 处存在空洞（可去间断点）。

3. Vertical Asymptotes | 垂直渐近线

A vertical asymptote occurs at \(x = a\) if \(Q(a) = 0\) and \(P(a) \neq 0\). The graph approaches \(+\infty\) or \(-\infty\) as \(x\) approaches \(a\) from either side.

当 \(Q(a) = 0\) 且 \(P(a) \neq 0\) 时，在 \(x = a\) 处出现垂直渐近线。当 \(x\) 从任一侧接近 \(a\) 时，图像趋向 \(+\infty\) 或 \(-\infty\)。

Example: \(f(x) = \frac{1}{x – 2}\) has vertical asymptote \(x = 2\).

示例：\(f(x) = \frac{1}{x – 2}\) 有垂直渐近线 \(x = 2\)。

To determine the direction, test values slightly less and greater than \(a\).
要判断方向，可测试略小于和略大于 \(a\) 的值。
If the multiplicity of the factor in \(Q\) is even, both sides go to the same infinity; if odd, they go to opposite infinities.
若分母中因式的重数为偶数，两侧趋向同一个无穷；若为奇数，则两侧趋向相反无穷。

4. Horizontal Asymptotes | 水平渐近线

Horizontal asymptotes describe the behaviour of \(f(x)\) as \(x \to \pm \infty\). They are determined by comparing the degrees of \(P\) and \(Q\).

水平渐近线描述 \(f(x)\) 当 \(x \to \pm \infty\) 时的行为。它们由 \(P\) 和 \(Q\) 的次数比较决定。

Degree of \(P\) vs \(Q\)	Horizontal asymptote
\(\deg P < \deg Q\)	\(y = 0\)
\(\deg P = \deg Q\)	\(y = \frac{a_n}{b_m}\)
\(\deg P > \deg Q\)	No horizontal asymptote (possibly oblique)

If \(\deg P = \deg Q\), then \(y = \frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}\).

若 \(\deg P = \deg Q\)，则 \(y = \frac{P\text{的首项系数}}{Q\text{的首项系数}}\)。

5. Oblique Asymptotes | 斜渐近线

When \(\deg P = \deg Q + 1\), the rational function has an oblique (slant) asymptote. The asymptote is the quotient polynomial obtained by long division of \(P\) by \(Q\).

当 \(\deg P = \deg Q + 1\) 时，有理函数具有斜渐近线。该渐近线是 \(P\) 除以 \(Q\) 所得的多项式商。

Example: \(f(x) = \frac{x^2 + 1}{x} = x + \frac{1}{x}\) has oblique asymptote \(y = x\).

示例：\(f(x) = \frac{x^2 + 1}{x} = x + \frac{1}{x}\) 有斜渐近线 \(y = x\)。

The distance between the curve and the oblique asymptote tends to zero as \(x \to \pm \infty\).
当 \(x \to \pm \infty\) 时，曲线与斜渐近线之间的距离趋于零。

6. Intercepts | 截距

The \(x\)-intercepts are found by setting \(P(x) = 0\) (provided these values are in the domain). The \(y\)-intercept is found by evaluating \(f(0)\), if \(0\) is in the domain.

\(x\) 截距通过令 \(P(x) = 0\) 求得（前提是这些值在定义域内）。\(y\) 截距通过计算 \(f(0)\) 求得，若 \(0\) 在定义域内。

For \(f(x) = \frac{x^2 – 4}{x – 1}\), the \(x\)-intercepts are \(x = \pm 2\), and the \(y\)-intercept is \(y = 4\).

对于 \(f(x) = \frac{x^2 – 4}{x – 1}\)，\(x\) 截距为 \(x = \pm 2\)，\(y\) 截距为 \(y = 4\)。

7. Holes and Removable Discontinuities | 空洞与可去间断点

If a factor \((x – a)\) appears in both \(P(x)\) and \(Q(x)\), the graph has a hole at \(x = a\). The simplified function can be used to find the \(y\)-coordinate of the hole.

如果因子 \((x – a)\) 同时出现在 \(P(x)\) 和 \(Q(x)\) 中，则图像在 \(x = a\) 处有一个空洞。可使用简化后的函数求空洞的 \(y\) 坐标。

Example: \(f(x) = \frac{(x – 1)(x + 2)}{(x – 1)}\) has a hole at \(x = 1\), and the simplified form is \(g(x) = x + 2\).
示例：\(f(x) = \frac{(x – 1)(x + 2)}{(x – 1)}\) 在 \(x = 1\) 处有空洞，简化形式为 \(g(x) = x + 2\)。
The hole’s y-coordinate is \(g(1) = 3\), so the point \((1, 3)\) is excluded from the graph.
空洞的 \(y\) 坐标为 \(g(1) = 3\)，因此点 \((1, 3)\) 不在图像上。

8. Sign Chart and Behaviour Near Asymptotes | 符号表与渐近线附近的行为

To sketch the graph accurately, create a sign chart using the zeros of the numerator and the vertical asymptotes. These values divide the \(x\)-axis into intervals where \(f(x)\) is positive or negative.

为了准确绘制图像，使用分子的零点和垂直渐近线制作符号表。这些值将 \(x\) 轴分成若干区间，在每个区间内 \(f(x)\) 为正或为负。

Test points: choose one \(x\)-value in each interval and evaluate the sign of \(f(x)\).

测试点：在每个区间内选取一个 \(x\) 值，判断 \(f(x)\) 的符号。

Near a vertical asymptote, determine whether \(f(x)\) tends to \(+\infty\) or \(-\infty\) by examining the sign of the function as \(x\) approaches the asymptote from the left and right.

在垂直渐近线附近，通过考察函数从左侧和右侧接近渐近线时的符号，判断 \(f(x)\) 趋向 \(+\infty\) 还是 \(-\infty\)。

9. Graph Sketching Steps | 绘图步骤

A systematic approach ensures an accurate graph. Follow these steps for any rational function.

系统化的方法可以确保图像准确。对任何有理函数，请遵循以下步骤。

Find the domain and note any excluded \(x\)-values.
确定定义域并记录被排除的 \(x\) 值。
Simplify the function and identify holes.
简化函数并识别空洞。
Find \(x\)- and \(y\)-intercepts.
求 \(x\) 截距和 \(y\) 截距。
Find vertical, horizontal, and oblique asymptotes.
求垂直、水平和斜渐近线。
Create a sign chart and analyse behaviour near asymptotes.
制作符号表并分析渐近线附近的行为。
Sketch the graph using all gathered information.
利用所有收集的信息绘制图像。

10. Common Exam Questions | 常见考试题型

IB exams often ask for the equation of asymptotes, coordinates of intercepts, and a sketch showing all key features. Some questions require interpreting a graph or deriving the function from its graph.

IB 考试常要求写出渐近线方程、截点坐标，以及展示所有关键特征的图像。有些题目要求解读图像或从图像推导函数。

Type 1: Find the domain, asymptotes, and intercepts of \(f(x) = \frac{2x – 1}{x + 3}\).
类型 1：求 \(f(x) = \frac{2x – 1}{x + 3}\) 的定义域、渐近线和截距。
Type 2: Determine the values of \(k\) such that the equation \(\frac{x^2 + kx + 1}{x – 1} = 0\) has exactly one solution.
类型 2：确定使得方程 \(\frac{x^2 + kx + 1}{x – 1} = 0\) 恰有一个解的 \(k\) 值。
Type 3: Sketch the graph of \(f(x) = \frac{x^2 – 1}{x^2 – 4}\), labelling all asymptotes and intercepts.
类型 3：绘制 \(f(x) = \frac{x^2 – 1}{x^2 – 4}\) 的图像，标注所有渐近线和截距。

11. Applications and Modelling | 应用与建模

Rational functions are used to model real-world situations such as average cost, concentration of a drug in the bloodstream, and electrical resistance in parallel circuits.

有理函数用于模拟现实情境，如平均成本、血液中药物浓度以及并联电路中的电阻。

Example: Average cost \(C(x) = \frac{5000 + 20x}{x}\), where \(x\) is the number of units. As \(x \to \infty\), \(C(x) \to 20\), meaning the average cost approaches the marginal cost.

示例：平均成本 \(C(x) = \frac{5000 + 20x}{x}\)，其中 \(x\) 为产量。当 \(x \to \infty\) 时，\(C(x) \to 20\)，即平均成本趋向边际成本。

In such models, the horizontal asymptote represents the long-term minimum or maximum value.
在此类模型中，水平渐近线代表长期最小值或最大值。

12. Key Pitfalls and Exam Tips | 常见错误与考试提示

Students often confuse holes with vertical asymptotes or forget to exclude common factors. Remember that a common factor creates a hole, not an asymptote.

学生常混淆空洞和垂直渐近线，或忘记排除公因式。请记住，公因式产生空洞，而非渐近线。

Always check if the numerator and denominator share a common factor before finding asymptotes.
在求渐近线之前，务必检查分子和分母是否有公因式。
When \(\deg P = \deg Q + 1\), write the oblique asymptote using polynomial division.
当 \(\deg P = \deg Q + 1\) 时，使用多项式除法写出斜渐近线。
Use a graphical calculator to verify your sketch, but show all algebraic steps in your written answer.
使用图形计算器验证草图，但书面答案中要展示所有代数步骤。
Remember that the graph may cross a horizontal asymptote for finite values of \(x\); the asymptote only describes the end behaviour.
记住，图像可能在有限的 \(x\) 值处穿过水平渐近线；渐近线只描述末端行为。

Mastering rational functions requires practice with sign charts, long division, and careful graph sketching. These skills are essential for the IB Analysis and Approaches HL exam.

掌握有理函数需要练习符号表、长除法和仔细的绘图技巧。这些技能对 IB 数学分析与方法 HL 考试至关重要。

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