📚 Simple Rational Functions | 简单有理函数
Rational functions form a cornerstone of algebra, appearing in contexts from reciprocal relationships to complex curve sketching. In IB Mathematics, students learn to analyse these functions by exploring their asymptotes, intercepts, domain restrictions, and transformations. Understanding simple rational functions builds essential skills for calculus and modelling.
有理函数是代数的基石，出现在倒数关系到复杂曲线绘制的各种情境中。在 IB 数学中，学生通过探索渐近线、截距、定义域限制和变换来学习分析这些函数。理解简单的有理函数为微积分和建模奠定了必要的技能。
1. Definition and Basic Form | 定义与基本形式
x) and Q(x) are polynomials and Q(x) is not the zero polynomial. The simplest rational functions have linear numerator and denominator.
有理函数是可以写成两个多项式之比 f(x) = P(x)/Q(x) 的函数，其中 P(x) 和 Q(x) 是多项式且 Q(x) 不为零多项式。最简单的有理函数具有线性的分子和分母。
f(x) = (ax + b)/(cx + d), c ≠ 0
ctions. Some can be simplified by cancelling common factors, but the domain must still exclude values that make the original denominator zero.
例如，f(x) = (2x + 1)/(x - 3) 和 g(x) = 4/(x² - 1) 都是有理函数。有些可以通过约去公因式简化，但定义域仍必须排除使原分母为零的值。
2. Domain and Excluded Values | 定义域与排除值
denominator zero. These are called excluded values. To find them, set Q(x) = 0 and solve.
有理函数的定义域是所有实数，除了那些使分母为零的 x 值，这些值称为排除值。要求它们，令 Q(x) = 0 并求解。
For f(x) = P(x)/Q(x), the domain is {x ∈ ℝ | iction still stands because the function is undefined at that point before simplification.
对于 f(x) = P(x)/Q(x)，定义域为 {x ∈ ℝ | Q(x) ≠ 0}。如果一个因式被约去，原来的限制仍然成立，因为函数在简化前的那一点无定义。
= (x² - 4)/(x - 2) simplifies to x + 2 for x ≠ 2. The domain excludes x = 2, creating a "hole" at (2, 4). This is a removable discontinuity.
示例：f(x) = (x² - 4)/(x - 2) 可简化为 x + 2，但 x ≠ 2。定义域排除 x = 2，在 (2, 4) 处形成一个 "洞"。这是一个可去间断点。
3. Vertical Asymptotes | 垂直渐近线
he denominator approaches zero while the numerator does not. As x → a, f(x) → ±∞. After cancelling any common factors, the zeros of the simplified denominator give the vertical asymptotes.
当分母趋近于零而分子不趋近于零时，垂直渐近线出现在 x = a 处。当 x → a 时，f(x) → ±∞。约去公因式后，简化分母的零点给出垂直渐近线。
1)], after cancellation we get 1/(x - 1) with vertical asymptote x = 1; x = -1 is a hole.
对于 f(x) = 1/(x - 2)，垂直渐近线为 x = 2。对于 f(x) = (x + 1)/(x² - 1) = (x + 1)/[(x - 1)(x + 1)]，约分后得 1/(x - 1)，垂直渐近线为 x = 1；x = -1 是一个洞。
The behaviour near a vertical asymptote can be determined by checking the sign of f(x) on either side. The graph will shoot upward or downward accordingly.
靠近垂直渐近线的行为可以通过检查两侧的 f(x) 符号来确定。图像相应地向上或向下猛冲。
4. Horizontal Asymptotes | 水平渐近线
The horizontal asymptote describes the end behaviour as x → ±∞. Compare the degrees of P(x) and Q(x):
水平渐近线描述当 x → ±∞ 时的末端行为。比较 P(x) 和 Q(x) 的次数：
If deg(P) < deg(Q), the horizontal asymptote is y = 0.
如果 deg(P) < deg(Q)，水平渐近线为 y = 0。
deg(P) = deg(Q), the asymptote is y = leading coefficient of P / leading coefficient of Q.
如果 deg(P) = deg(Q)，渐近线为 y = P 的领先系数 / Q 的领先系数。
If deg(P) > deg(Q), there is no horizontal asymptote (an oblique asymptote may exist).
如果 deg(P) > deg(Q)，没有水平渐近线（可能存在斜渐近线）。
g(Q) = 2, asymptote y = 3/1 = 3. For f(x) = 5x/(x³ + 2), deg(P) < deg(Q), asymptote y = 0.
示例：f(x) = (3x² + 1)/(x² - 4)。deg(P) = deg(Q) = 2，渐近线 y = 3/1 = 3。对于 f(x) = 5x/(x³ + 2)，deg(P) < deg(Q)，渐近线 y = 0。
5. Oblique (Slant) Asymptotes | 斜渐近线
±∞, the remainder term → 0, so the graph approaches the line y = mx + b.
当分子的次数恰好比分母次数大 1 时，函数有斜渐近线。使用多项式长除法：P(x) 除以 Q(x) 得到商式 mx + b 和余式。于是 f(x) = mx + b + 余式/Q(x)。当 x → ±∞，余项 → 0，因此图像趋近直线 y = mx + b。
f(x) = (x² + 3x + 2)/(x + 1) = x + 2, x ≠ -1
que asymptote appears in f(x) = (x² + 1)/x = x + 1/x, giving y = x as the slant asymptote.
这个简化形式表明图像是一条带有洞的直线；然而，典型的斜渐近线出现在 f(x) = (x² + 1)/x = x + 1/x 中，斜渐近线为 y = x。
6. Intercepts with Axes | 与坐标轴的截距
To find the y-intercept, evaluate f(0), provided 0 is in the domain. To find x-intercepts, set the numerator equal to zero and solve, then check that the solutions are not excluded by the domain (they must not make the denominator zero).
要求 y 截距，计算 f(0)，前提是 0 在定义域内。要求 x 截距，令分子等于零并求解，然后检查解是否被定义域排除（它们不能使分母为零）。
2/1 = -2, point (0, -2). Both are valid because x = -1 is excluded only for the asymptote.
对于 f(x) = (x - 2)/(x + 1)，x 截距：令 x - 2 = 0 得 x = 2，点 (2, 0)。y 截距：f(0) = -2/1 = -2，点 (0, -2)。两者都有效，因为只有 x = -1 被排除作为渐近线。
7. Sign Analysis and Inequalities | 符号分析与不等式
Sign analysis helps solve rational inequalities. Mark the zeros of the numerator and denominator on a number line. These values split the line into intervals. Choose a test point in each interval and determine the sign of f(x).
符号分析有助于求解有理不等式。在数轴上标记分子和分母的零点。这些值将数轴分为若干区间。在每个区间选取一个测试点，确定 f(x) 的符号。
ge; or ≤), only include points where f(x) = 0 (x-intercepts) if they are in the domain.
要解 f(x) > 0 或 f(x) < 0，使用符号表。记住函数在垂直渐近线处无定义，因此这些点绝不包含在解中。对于包含等号的不等式 (≥ 或 ≤)，只包含 f(x) = 0 的点（x 截距），如果它们在定义域内。
o of numerator), x = -2 (zero of denominator). Test intervals: x < -2, (x - 1)/(x + 2) > 0? (-)/(+) = -, false. x > 1, test x = 2 gives (+)/(+) = +, true. Solution: x < -2 or x > 1.
示例：解 (x - 1)/(x + 2) > 0。关键数：x = 1（分子零点），x = -2（分母零点）。测试区间：x < -2，如 x = -3 得 (-)/(-) = +，成立。-2 < x < 1，x = 0 得 (-)/(+) = -，不成立。x > 1，x = 2 得 (+)/(+) = +，成立。解：x < -2 或 x > 1。
8. Transformations of Rational Functions | 有理函数的变换
n f(x) = 1/x serves as the parent function. A transformed version is f(x) = a/(x - h) + k.
简单的有理函数可以通过平移、伸缩和反射进行变换。倒数函数 f(x) = 1/x 用作母函数。变换后的形式为 f(x) = a/(x - h) + k。
k. The graph maintains the same hyperbolic shape with branches in quadrants depending on a.
这里，h 给出水平移动，k 垂直移动，a 是垂直伸缩（若 a < 0 则反射）。垂直渐近线移至 x = h，水平渐近线变为 y = k。图像保持相同的双曲线形状，分支所在象限取决于 a。
tched. The two branches lie in the second and fourth quadrants relative to the asymptotes.
示例：f(x) = -2/(x + 3) + 1 的垂直渐近线为 x = -3，水平渐近线为 y = 1，并进行了反射和伸缩。两个分支相对于渐近线位于第二和第四象限。
9. The Reciprocal Function | 倒数函数
The parent reciprocal function f(x) = 1/x has domain {x | x ≠ 0}, range {y | nate axes as asymptotes. It is an odd function: f(-x) = -f(x), symmetric about the origin.
母函数倒数函数 f(x) = 1/x 的定义域为 {x | x ≠ 0}，值域为 {y | y ≠ 0}。其图像是直角双曲线，以坐标轴为渐近线。它是一个奇函数：f(-x) = -f(x)，关于原点对称。
The function decreases on intervals (-∞, 0) and (0, ∞), but is not decreasing over its entire domain because of the discontinuity. Key points include (1, 1), (-1, -1), and using scale factors assists sketching.
该函数在区间 (-∞, 0) 和 (0, ∞) 上递减，但由于间断点，并非在整个定义域上递减。关键点包括 (1, 1) 和 (-1, -1)，使用比例因子有助于绘图。
The reciprocal function serves as a building block for more complicated rational functions through arithmetic operations and transformations.
倒数函数通过算术运算和变换作为构建更复杂有理函数的基石。
10. Linear Fractional Functions | 线性分式函数
; 0, and ad - bc ≠ 0 ensures it is not a constant. By performing division, we can rewrite it in transformational form.
线性分式函数形如 f(x) = (ax + b)/(cx + d)，c ≠ 0，且 ad - bc ≠ 0 保证它不是常数。通过除法，我们可以将其重写为变换形式。
(ax + b)/(cx + d) = a/c + (bc - ad)/[c(cx + d)]
e constant term a/c is the horizontal asymptote value. The remainder term determines the direction and scale of the branches.
这给出垂直渐近线 x = -d/c 和水平渐近线 y = a/c。常数项 a/c
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