Exercise 1C.3: Graphs of Rational Functions | 练习1C.3：有理函数的图像

📚 Exercise 1C.3: Graphs of Rational Functions | 练习1C.3：有理函数的图像

rners solidify the foundational toolkit required for both analysis and application of reciprocal and quotient expressions.

在 IB 数学中，练习1C.3通常要求学生分析并绘制有理函数的图像。该练习建立在对定义域限制、截距和渐近行为的扎实理解之上。通过解决如 f(x) = (2x+1)/(x−3) 的问题，学习者可以巩固分析和应用有理表达式所需的基础工具。

1. What Is a Rational Function? | 什么是有理函数？

ple definition opens up a rich field of graphical features that Exercise 1C.3 aims to explore.

有理函数是任何可以表示为两个多项式之商 p(x)/q(x) 的函数，其中 q(x) 不是零多项式。其定义域包含所有实数，但使 q(x)=0 的数值除外。这一简单的定义为练习1C.3所探索的丰富图形特征打开了大门。

2. Domain and Vertical Asymptotes | 定义域与垂直渐近线

aph will never touch this line; it approaches positive or negative infinity on either side.

分析有理函数的第一步是找出使分母为零的取值。这些 x 值被排除在定义域之外，并且通常对应于垂直渐近线。对于像 g(x) = 1/(x−2) 这样的函数，垂直渐近线为 x=2。图像永远不会触碰这条直线；它会在两侧分别趋近于正无穷或负无穷。

3. Horizontal Asymptotes and End Behaviour | 水平渐近线与末端行为

mptote. If the degrees are equal, the horizontal asymptote is the ratio of leading coefficients. If the numerator’s degree exceeds the denominator’s by exactly one, the function has an oblique (slant) asymptote instead.

水平渐近线描述了当 x → ±∞ 时有理函数的行为。渐近线通过比较分子和分母的次数来确定。如果分子的次数小于分母的次数，水平渐近线为 y=0。如果次数相等，水平渐近线是首项系数之比。如果分子的次数比分母恰好大 1，则函数具有斜渐近线。

4. Finding x- and y-intercepts | 求 x 截距与 y 截距

e y-intercept is (0, -1/3) and the x-intercept is (-1/2, 0). These points are crucial anchors for an accurate sketch.

y 截距通过计算 f(0) 求得，前提是 0 在定义域内。x 截距出现在分子等于零的位置，但只有当那些 x 值不会同时使分母为零时才成立。对于 f(x) = (2x+1)/(x−3)，y 截距为 (0, −1/3)，x 截距为 (−1/2, 0)。这些点是准确绘图的关键锚点。

5. Identifying Oblique Asymptotes | 识别斜渐近线

iagonal line) and a remainder that approaches zero at infinity. The asymptote is then y = quotient.

当分子的次数比分母恰好大 1 时，有理函数具有斜渐近线。练习1C.3通常会包含诸如 h(x) = (x²+1)/(x−2) 的情况。斜渐近线通过多项式长除法求得，得到商的部分（对角线）和一个在无穷远处趋近于零的余式。渐近线即为 y = 商。

6. Analysing Sign and Regions of the Graph | 分析图像的正负区域

Once asymptotes and intercepts are known, the next step is to determine the sign of f(x) in different intervals. This is done by creating a sign table based on the zeros of the numerator and denominator. The sign changes help us see where the graph lies above or below the x-axis, which is essential for correctly positioning branches between asymptotes.

一旦知道了渐近线和截距，下一步便是确定 f(x) 在不同区间内的正负号。这可以通过基于分子和分母零点制作符号表来完成。符号变化帮助我们看清图像位于 x 轴的上方还是下方，这对于在渐近线之间正确定位分支至关重要。

7. Removable Discontinuities (Holes) | 可去间断点（空点）

xample, f(x) = (x²-4)/(x-2) simplifies to x+2 for x≠2. The graph is a straight line with a 'hole' xercise 1C.3 may test the ability to distinguish such cases from true vertical asymptotes.

广告 / Sponsored

有些有理函数的分子和分母含有公因式，这会产生一个间断点而非垂直渐近线。例如，f(x) = (x²−4)/(x−2) 对于 x≠2 可简化为 x+2。图像是一条直线，但在 (2,4) 处有一个“空点”。练习1C.3可能考查区分这种情况与真正垂直渐近线的能力。

8. Step-by-Step Sketching Strategy | 分步绘图策略

A systematic approach is essential to mastering Exercise 1C.3. The strategy can be summarised as: (1) simplify the function if possible; (2) find the domain and vertical asymptotes; (3) locate x- and y-intercepts; (4) determine horizontal or oblique asymptotes; (5) check for sign using a table; (6) plot a few additional points if necessary; and (7) sketch the branches smoothly, respecting asymptotes.

掌握练习1C.3需要系统的方法。该策略可以概括为：(1) 可能的话简化函数；(2) 求出定义域和垂直渐近线；(3) 确定 x 截距和 y 截距；(4) 判断水平或斜渐近线；(5) 使用符号表检验正负；(6) 必要时描出若干额外点；(7) 平滑地绘制各分支，同时尊重渐近线。

9. Common Mistakes to Avoid | 应避免的常见错误

Many students incorrectly treat a removable discontinuity as a vertical asymptote or fail to simplify first. Others forget that a horizontal asymptote can be crossed locally, though not globally. Misinterpreting the sign of leading coefficients when degrees are equal is another frequent error. Exercise 1C.3 provides repeated practice to overcome these pitfalls.

许多学生错误地将可去间断点当作垂直渐近线，或未能先进行化简。还有人忘记水平渐近线在局部可以被穿过，尽管在大范围不能。当次数相等时误判首项系数的正负是另一个常见错误。练习1C.3通过反复练习来克服这些陷阱。

10. Worked Example Walkthrough | 例题详细讲解

(-1/2,3) negative, (3,∞) positive. Sketch two hyperbolic branches approaching x=3 and y=2.

考虑一个练习1C.3风格的问题：f(x) = (2x+1)/(x−3)。定义域：x ≠ 3，垂直渐近线 x=3。x 截距：令 2x+1=0，x=−1/2。y 截距：f(0)=−1/3。次数相等（均为1次），水平渐近线：y=2/1=2。符号分析：区间 (−∞,−1/2) 为正，(−1/2,3) 为负，(3,∞) 为正。绘出两支双曲线，分别趋近 x=3 和 y=2。

Key features summarised: VA: x=3, HA: y=2, x-int: (-0.5, 0), y-int: (0, -0.333)

关键特征总结：VA: x=3, HA: y=2, x 截距: (−0.5, 0), y 截距: (0, −0.333)

11. Using Technology to Verify Graphs | 使用技术验证图像

While Exercise 1C.3 emphasises analytical skills, graphing calculators are excellent verification tools. Input the rational function and check asymptotes and intercepts visually. Ensure the window setting captures the asymptotic behaviour. However, reliance on technology should not replace a thorough algebraic understanding of why the graph behaves as it does.

虽然练习1C.3强调分析技能，但图形计算器是极好的验证工具。输入有理函数，直观地检查渐近线和截距。确保窗口设置能捕捉到渐近行为。然而，依赖技术不应取代对图像为何如此表现的完整代数理解。

12. Preparing for IB Exam Questions | 为 IB 考题做准备

IB exam questions often extend beyond mere sketching by asking for parameters that make a given line a tangent or requiring the solution to a rational inequality. The foundations laid in Exercise 1C.3 are directly applicable to these higher-order tasks. Mastery of rational function graphs also underpins topics like partial fractions and differential equations later in the course.

IB 考试题通常不仅仅要求绘图，还会要求找出使给定直线成为切线的参数，或者需要解有理不等式。练习1C.3 所奠定的基础可直接应用于这些更高阶的任务。掌握有理函数图像还为后续课程中的部分分式和微分方程等专题提供了支撑。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）