📚 FC Paper 156: Curve Sketching Techniques for Rational Functions | 曲线草图技巧：有理函数与渐近线
A solid command of curve sketching is essential for success in Edexcel A‑Level Mathematics. This article explores the systematic approach to sketching rational functions, focusing on asymptotes, intercepts, stationary points and concavity. By mastering these techniques, you will be able to handle any rational function that appears in Paper 1 or Paper 2 with confidence.
熟练掌握曲线草图对 Edexcel A‑Level 数学的考试成功至关重要。本文探讨绘制有理函数图像的系统方法，重点关注渐近线、截距、驻点与凹凸性。掌握这些技巧后，你将能信心十足地应对试卷一或试卷二中出现的任何有理函数。
1. Introduction to Rational Functions | 有理函数简介
ational function is any function that can be written as the ratio of two polynomials. In A‑ , where both P and Q are polynomials. Understanding the behaviour of these functions requires a combination of algebraic manipulation, limits and calculus.
有理函数是指可以写成两个多项式之比的任意函数。在 A‑Level 考试中，你经常会遇到形如 f(x) = P(x)/Q(x) 的表达式，其中 P 和 Q 都是多项式。要理解这类函数的性态，需要综合运用代数变换、极限和微积分知识。
The graph of a rational function can exhibit rich features: breaks (discontinuities), approaches to lines at infinity, turning points and inflections. Our goal is to construct a reliable sketch that displays all intercepts, asymptotes and critical points.
有理函数的图像可能展现丰富的特征：间断点、无穷远处的趋近直线、转折点与拐点。我们的目标是画出一张可靠的草图，标明所有截距、渐近线和关键点。
2. Domain and Intercepts | 定义域与截距
Before drawing anything, restrict the domain by excluding any x‑values that make the denominator zero. These are potential locations of vertical asymptotes or holes. Write the domain in set notation, e.g. { x ∈ ℝ : x ≠ 1, x ≠ −2 }.
动笔之前，通过排除所有使分母为零的 x 值来限定定义域。这些往往是垂直渐近线或可去间断点所在的位置。用集合记号写出定义域，例如 { x ∈ ℝ : x ≠ 1, x ≠ −2 }。
Next, find the y‑intercept by evaluating f(0), provided 0 is in the domain. To find x‑intercepts, set the numerator equal to zero and solve, being careful to reject any solutions that also make the denominator zero (these would be holes, not intercepts).
接下来，若 0 在定义域内，通过计算 f(0) 求出 y 截距。要求 x 截距，则令分子等于零并求解，同时要剔除那些也使分母为零的解（这类解代表可去间断点，而非截距）。
3. Vertical Asymptotes | 垂直渐近线
vanishes at x = 2 and x = −3. Since neither factor cancels, both give vertical asymptotes.
垂直渐近线出现在分母的孤立零点上，且这些零点不与分子相约。假设 f(x) = (x+1)/(x−2)(x+3)，分母在 x = 2 和 x = −3 处为零。由于这两个因式均未约去，它们都产生垂直渐近线。
; for x → 2⁺, use 2.1. This tells you whether the function tends to +∞ or −∞ on each side.
利用邻近检测值判断每个渐近线左侧和右侧的符号。对于 x → 2⁻，代入 x = 1.9；对于 x → 2⁺，代入 2.1。由此可知函数在该侧是趋向 +∞ 还是 −∞。
4. Horizontal Asymptotes | 水平渐近线
e horizontal asymptote is y = 0. If n = d, the asymptote is y = leading coefficient ratio.
水平渐近线可通过考察 x → ±∞ 时 f(x) 的末端性态来确定。对于有理函数，比较分子次数 n 与分母次数 d。若 n < d，水平渐近线为 y = 0。若 n = d，渐近线为 y = 首项系数之比。
If n > d, there is no horizontal asymptote; instead the function may have a slant asymptote or grow without bound. In such cases, proceed to the next section to check for an oblique asymptote.
若 n > d，则不存在水平渐近线；此时函数可能具有斜渐近线，或者无界增长。这种情况下，请进入下一节检查是否存在斜渐近线。
5. Oblique (Slant) Asymptotes | 斜渐近线
ynomial long division to express f(x) = mx + b + R(x)/Q(x), where R(x)/Q(x) → 0 as x → ±∞.
当分子次数恰好比分母次数高一次时，图像具有一条斜渐近线。进行多项式长除法，将 f(x) 写成 mx + b + R(x)/Q(x) 的形式，其中当 x → ±∞ 时 R(x)/Q(x) → 0。
ifies to x − 2 + 0/(x−1) after division, giving the line y = x − 2 as the slant asymptote.
直线 y = mx + b 便是斜渐近线。当 x 的绝对值很大时，曲线紧贴该直线。例如，f(x) = (x² − 3x + 2)/(x − 1) 经除法可化简为 x − 2 + 0/(x−1)，于是得到斜渐近线 y = x − 2。
6. First Derivative and Stationary Points | 一阶导数与驻点
To identify turning points and regions where the function is increasing or decreasing, differentiate the rational function. Use the quotient rule or rewrite the function in a more manageable form before differentiating.
要确定转折点以及函数的增减区间，需对有理函数求导。利用商法则，或在求导前将函数改写为更易处理的形式。
If f(x) = u(x)/v(x), then f '(x) = (u'v − uv') / v²
若 f(x) = u(x)/v(x)，则 f '(x) = (u'v − uv') / v²
Set f ' ionary points. Determine their nature using the first derivative test: check the sign of f ' er side of the stationary point, or proceed to the second derivative test when convenient.
令 f '(x) = 0 求得驻点。利用一阶导数检定法判断其性质：检查驻点两侧 f ' 的符号，或在方便时转而使用二阶导数检定法。
7. Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative f ''(x) reveals the curvature of the graph. Where f ''(x) > 0, the curve is concave up (∪); where f ''(x) < 0, it is concave down (∩). Points where f '' changes sign are points of inflection, provided f is continuous at that point.
二阶导数 f ''(x) 揭示图像的弯曲方向。当 f ''(x) > 0 时，曲线凹向上（∪）；当 f ''(x) < 0 时，曲线凹向下（∩）。在 f 连续的前提下，f '' 改变符号的点即为拐点。
For rational functions, compute f '' by differentiating f ' carefully. Although the algebra can be lengthy, concavity information often helps confirm the shape between asymptotes.
对于有理函数，可通过仔细求 f ' 的导数来得到 f ''。尽管代数运算可能较繁琐，凹凸性信息通常有助于确认渐近线之间图像的形状。
8. Step-by-Step Sketching Example | 逐步草图绘制示例
(x+2) to notice a hole at x = −2, but keep the simplified form g(x) = (x−2)/(x−3), x ≠ −2.
让我们将完整策略应用于函数 f(x) = (x² − 4)/(x² − x − 6)。首先，对分子分母进行因式分解：f(x) = (x−2)(x+2) / (x−3)(x+2)。约去公因式 (x+2) 后可知 x = −2 处存在可去间断点，但在定义域中保留简化形式 g(x) = (x−2)/(x−3)，x ≠ −2。
Domain: x ∈ ℝ, x ≠ 3, x ≠ −2. y‑intercept: g(0) = (0−2)/(0−3) = 2/3. x‑intercept: set numerator = 0 ⇒ te at x = 3. Horizontal asymptote: degrees equal (1 and 1), so y = 1/1 = 1. Derivatives: g'(x) = −1/(x−3)², always negative; no stationary points. g'' ow plot the hole, intercepts, asymptotes and indicate the curve decreasing on each branch.
定义域：x ∈ ℝ, x ≠ 3, x ≠ −2。y 截距：g(0) = (0−2)/(0−3) = 2/3。x 截距：令分子为零 ⇒ x = 2。垂直渐近线 x = 3。水平渐近线：分子分母次数相等（均为 1 次），故 y = 1/1 = 1。导数：g'(x) = −1/(x−3)²，恒为负值，无驻点。g''(x) = 2/(x−3)³，在 x = 3 处变号。现在标出可去间断点、截距、渐近线，并表明曲线在每一支上递减。
9. Sketching with Multiple Asymptotes | 含有多个渐近线的草图
egrees are equal, so horizontal asymptote is y = 1. The numerator is always positive, so y‑intercept is 0.25? Actually f(0) = 1/(−4) = −0.25, negative. No x‑intercepts because numerator never zero.
考虑 f(x) = (x² + 1)/(x² − 4)。这里分母为 (x−2)(x+2)，在 x = 2 和 x = −2 处存在垂直渐近线。分子分母次数相等，故水平渐近线为 y = 1。分子恒正，因此 y 截距为 f(0) = 1/(−4) = −0.25，为负值。由于分子永不为零，无 x 截距。
Compute f '(x) = −10x/(x²−4)². Critical point at x = 0. Use sign analysis: f ' om below as x → ±∞, with branches in the left and right intervals separated by asymptotes.
计算 f '(x) = −10x/(x²−4)²。临界点位于 x = 0。通过符号分析：x < 0 时 f ' 为正，x > 0 时 f ' 为负，因此在 (0, −0.25) 处取得局部极大值。当 x → ±∞ 时，图像从下方趋近 y = 1，左右区间内的分支被渐近线隔开。
10. Common Pitfalls and How to Avoid Them | 常见误区与避坑指南
Many candidates forget to check for holes after simplification. Always re‑state the domain with the original denominator’s zeros excluded, even if cancellation occurs. A hole should be represented by a small open circle on your sketch.
许多考生在化简后忘记检查可去间断点。即使发生约分，也务必用原分母的零点所排除的值重新表述定义域。可去间断点在草图上应用一个小空心圆圈表示。
Another trap is misinterpreting the end behaviour when the degree condition gives a slant asymptote. Avoid simply writing “no horizontal asymptote” and guessing the shape. Always perform long division to obtain the exact slant line.
另一个陷阱是当次数条件表明存在斜渐近线时，错误解读末端性态。不要仅仅写上“无水平渐近线”然后靠猜测画图。务必执行长除法以获得精确的斜渐近线方程。
Failing to check the sign on both sides of vertical asymptotes can lead to an incorrect branch direction. Use a number line and test points systematically.
未能检查垂直渐近线两侧的符号可能导致分支方向错误。请使用数轴并有条理地选取测试点。
11. Exam-Style Practice Problem | 考试风格练习题
), clearly indicating any asymptotes, intercepts, stationary points and points of inflection. Show all algebraic steps, including domain restrictions and the nature of any critical points.
绘制 y = (2x² − 8)/(x² − 1) 的图像，清楚标注所有渐近线、截距、驻点和拐点。给出全部代数步骤，包括定义域限制与各临界点的性质。
Solution outline: Factor to 2(x−2)(x+2)/(x−1)(x+1). Domain: x ≠ 1, −1. y‑intercept: (0, 8). x‑intercepts: (2,0), (−2,0). V.A.: x = 1, x = −1. H.A.: y = 2. f '(x) = 12x/(x²−1)² ⇒ critical point at (0, 8). f '' confirms concavity. Use symmetry (even function) to help the sketch.
解题提纲：因式分解得 2(x−2)(x+2)/(x−1)(x+1)。定义域：x ≠ 1, −1。y 截距：(0, 8)。x 截距：(2,0), (−2,0)。垂直渐近线：x = 1, x = −1。水平渐近线：y = 2。f '(x) = 12x/(x²−1)² ⇒ 临界点 (0, 8)。f '' 确认凹凸性。利用对称性（偶函数）辅助作图。
12. Final Tips for the Exam | 考试终极提示
Always label your sketch with the coordinates of intercepts, stationary points and the equations of asymptotes. Use a pencil and ruler for asymptotes; draw the curve smoothly with two or three well‑chosen points in each region.
务必在草图上标明截距坐标、驻点坐标以及渐近线方程。用铅笔和直尺画出渐近线；在每个区域选取两到三个恰当的点，光滑地连接出曲线。
If time permits, verify your sketch with a graphic calculator or by evaluating f(x) at a couple of strategic x‑values. A quick numerical check can prevent sign errors and ensure your asymptote branches are on the correct sides.
若时间允许，用图形计算器或通过代入几个关键 x 值检验草图。快速的数值检验能避免符号错误，并确保渐近线分支位于正确的一侧。
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