📚 Sign Diagrams | 符号图
A sign diagram is a visual tool used in calculus and algebra to determine where a function is positive, negative, or zero. By outlining the sign behaviour of factors across real number intervals, it provides a clear foundation for sketching graphs and solving inequalities efficiently.
符号图是用来判断函数在实数范围内何时为正、何时为负、何时为零的可视化工具。通过梳理各个因子在不同区间内的符号行为，它为函数图像的绘制和不等式的求解提供了清晰的基础。
1. What Is a Sign Diagram? | 什么是符号图？
A sign diagram displays a real number line on which we mark critical points — zeros of the function and locations where the function is undefined. Between these points we record whether the function values are positive (+) or negative (−).
符号图展示了一根实数轴，上面标出关键点——函数的零点以及函数无定义的位置。在这些点之间的区间里，我们记录函数值是正还是负。
Instead of evaluating the function at countless points, a sign diagram uses the sign of each factor to deduce the sign of the whole expression. The method is systematic, visual, and nearly error‑proof when applied correctly.
符号图不需要在无数个点上求值，而是利用每个因子的符号来推断整个表达式的符号。这个方法系统、直观，只要正确使用几乎不会出错。
2. The Foundation: Zeros and Discontinuities | 基础：零点与间断点
Every sign diagram begins by solving f(x)=0 to find zeros and determining any x‑values that make the denominator zero. Together these form the boundary points that split the number line into test intervals.
每个符号图的起点都是求解方程 f(x)=0 找出零点，并找出使分母为零的所有 x 值。这些点共同构成边界点，把数轴分割成一个个检测区间。
For a polynomial, discontinuities do not exist, so only zeros matter. For a rational function, both zeros of the numerator and zeros of the denominator (vertical asymptotes) must be placed on the number line. The function cannot change sign inside an interval that contains no zero or asymptote.
对于多项式函数，没有间断点，所以只需考虑零点。有理函数则必须同时把分子的零点和分母的零点（垂直渐近线）在数轴上标出。在既不含零点也不含渐近线的区间内，函数不会改变符号。
- Zeros from f(x)=0 — where the graph crosses or touches the x‑axis. / 来自 f(x)=0 的零点 —— 图像与 x 轴相交或相切的位置。
- Discontinuities from denominator = 0 — where the function is undefined and often has a vertical asymptote. / 来自分母为零的间断点 —— 函数无定义且通常有垂直渐近线。
3. Step‑by‑Step Construction for Polynomials | 多项式符号图的逐步构造
We will break down the process using a fully factorised polynomial. The steps stay the same regardless of the degree.
我们以一个完全分解的多项式为例，逐步拆解整个过程。无论多项式次数多高，步骤都是相同的。
Step 1 Factorise the polynomial completely into linear and irreducible quadratic factors. / 第一步 将多项式彻底分解为一次因子和不可约二次因子。
Step 2 Solve each factor equal to zero to obtain all distinct x‑intercepts (zeros). Write them in increasing order. / 第二步 令每个因子等于零，解出所有不同的 x 截距（零点），并将它们从小到大排序。
Step 3 Draw a horizontal line representing the real number line, mark the zeros, and label the intervals. / 第三步 画一条水平线表示实数轴，标出零点，并在下方写出区间。
Step 4 Choose a test value inside each interval and substitute it into each factor, recording only the sign. Alternatively, use the multiplicity rule: a factor (x−a)ⁿ with odd n changes sign at x=a; with even n it does not. / 第四步 在每个区间内选取一个测试值代入每个因子，仅记录符号。或者利用重数规则：因子 (x−a)ⁿ 若 n 为奇数则在 x=a 处变号，n 为偶数则不变号。
Step 5 Multiply the signs of all factors to obtain the sign of f(x) in that interval and write + or − below the line. / 第五步 将所有因子的符号相乘，得到该区间内 f(x) 的符号，并在数轴下方写上 + 或 −。
Consider f(x) = (x+2)(x−1)²(x−4) tors each have odd multiplicity. Testing intervals gives the sign pattern −, +, +, − from left to right.
考虑 f(x) = (x+2)(x−1)²(x−4)。零点在 x = −2, 1, 4。因式 (x−1)² 的重数为偶数，所以在 x=1 处符号不变。其余因子重数为奇数。自左向右测试区间得到符号模式为 −, +, +, −。
4. The Multiplicity Rule in Detail | 重数规则的详细解读
Multiplicity describes the power to which a factor is raised. It controls whether the function crosses the x‑axis or merely touches it and turns back. The sign behaviour is entirely determined by whether the multiplicity is odd or even.
重数指的是因式的指数大小。它决定了函数是穿过 x 轴，还是仅仅相切并返回。符号行为完全取决于重数是奇数还是偶数。
If a factor such as (x−a) has odd multiplicity, the function changes sign at x=a. Graphically, the curve crosses the x‑axis. For even multiplicity, the function keeps the same sign on both sides of x=a, producing a touch point.
若因式如 (x−a) 的 重数为奇数，函数在 x=a 处变号。图像穿过 x 轴。若 重数为偶数，函数在 x=a 两侧同号，形成一个触碰点。
This rule dramatically shortens the construction of a sign diagram: once zeros and their multiplicities are known, the sign to the far right can be found from the leading coefficient, and the pattern is built backwards by toggling the sign only at odd‑multiplicity zeros.
这一规则极大地简化了符号图的绘制：一旦知道零点及其重数，最右端的符号可由首项系数的正负决定，然后仅需在每个奇重数零点处翻转符号，就能向左依次推出全部符号。
- Odd multiplicity: sign flips (e.g. (x−3)¹, (x−3)³). / 奇数重数：符号翻转（例如 (x−3)¹、(x−3)³）。
- Even multiplicity: sign stays (e.g. (x−3)², (x−3)⁴). / 偶数重数：符号不变（例如 (x−3)²、(x−3)⁴）。
- Far right sign: determined by the sign of the leading coefficient. / 最右端符号：由首项系数的正负决定。
5. Sign Diagrams for Rational Functions | 有理函数的符号图
Rational functions of the form f(x) = p(x)/q(x) rom q(x)=0) on the number line. The function can change sign at a zero or at an asymptote.
形如 f(x) = p(x)/q(x) 的有理函数需要在数轴上同时标出零点（来自 p(x)=0）和垂直渐近线（来自 q(x)=0）。函数可能在零点或渐近线处改变符号。
The process remains identical: list all critical values in increasing order, choose test values in each interval, and determine the sign of p(x) and q(x) separately before dividing the signs. A factor appearing in both numerator and denominator needs careful handling: it removes a zero and creates a hole, but the hole does not change sign unless the factor has odd multiplicity in the net expression.
构建过程完全一致：将所有临界值从小到大排列，在每个区间选取测试点，分别确定 p(x) 和 q(x) 的符号，再相除。若某个因子同时出现在分子和分母中，需要谨慎处理：它会消去零点并产生一个空洞，但空洞不改变符号，除非该因子在净表达式中具有奇数次幂。
For f(x) = (x−1)/(x+2)(x−3) 2 and x=3 mean f(x) is undefined there; the diagram shows a vertical dashed line or a gap.
对于 f(x) = (x−1)/(x+2)(x−3)，临界点为 x = −2, 1, 3。有理函数的符号会越过每个点而交替改变，除非重数导致跳过。x=−2 和 x=3 处的垂直渐近线意味着此处无定义；符号图中常用竖直虚线或留空表示。
6. Solving Inequalities Using a Sign Diagram | 利用符号图解不等式
One of the most powerful applications of sign diagrams is solving polynomial and rational inequalities. Once the sign pattern is established, we simply read off the intervals where the expression satisfies the required inequality.
符号图最强大的应用之一就是解多项式不等式和分式不等式。一旦确定了符号模式，只需读出满足所求不等式的区间即可。
Example: Solve (x−2)(x+1)(x−5) ≤ 0 . Working leftwards, signs are +, −, +, −. Therefore (x−2)(x+1)(x−5) ≤ 0 holds on (−∞, −1] ∪ [2, 5]. Brackets include the zeros because the inequality allows equality.
例题： 解不等式 (x−2)(x+1)(x−5) ≤ 0。先作符号图：零点在 −1, 2, 5。每个因子均为奇数次，符号交替改变：当 x>5 时乘积为正（因首项 x³>0）。左推可得符号依次为 +, −, +, −。因此不等式成立的范围是 (−∞, −1] ∪ [2, 5]。方括号包含零点，因为不等式允许等号。
For rational inequalities such as (x+3)/(x−2) ≥ 2, rearrange to have zero on one side, combine into a single fraction, and then build the sign diagram for the new expression. Always exclude values that make a denominator zero, even if the inequality is non‑strict.
对于 (x+3)/(x−2) ≥ 2 这类分式不等式，需先移项使一边为零，合并成一个分式，再为新的表达式构建符号图。务必排除使分母为零的值，即使不等式为“≥”也要用开区间。
7. From Sign Diagram to Graph Sketching | 从符号图到函数草图
A sign diagram reveals where the graph lies above or below the x‑axis. Combined with zeros, asymptotes, and end behaviour, we can produce an accurate sketch without plotting many points.
符号图揭示了图像位于 x 轴上方还是下方的区域。结合零点、渐近线和首尾走势，我们无需描许多点就能画出较为准确的草图。
For each interval with a positive sign, the curve is drawn above the x‑axis; for a negative sign asymptotes, the curve shoots up to +∞ or down to −∞ depending on the sign on either side.
在每个 正号区间，曲线画在 x 轴上方；在 负号区间，曲线在 x 轴下方。靠近奇重数零点时，图像穿过坐标轴；靠近偶重数零点时，图像触碰并反弹。靠近垂直渐近线时，曲线根据两侧符号朝 +∞ 或 −∞ 方向无限延伸。
This approach is heavily emphasised in IB Mathematics; exam questions often ask for a sketch of a function given its factorised form or ask to determine the number of solutions to an equation graphically.
IB 数学十分强调这种方法；考试中常要求根据因式分解形式画出函数草图，或根据图像判定方程解的个数。
8. Handling Absolute Value and Composite Factors | 处理绝对值和复合因子
When a function involves |f(x)|, the sign diagram of the inner expression helps identify where the absolute value changes the sign. For piecewise definition, we split the domain at the zeros of f(x).
当函数包含 |f(x)| 时，内部表达式的符号图有助于确定绝对值在哪里改变符号。为了进行分段定义，我们需要在 f(x) 的零点处拆分定义域。
For example, g(x) = |x²−4| t x=−2 and x=2. A sign diagram for x²−4 shows negative values between −2 and 2. Therefore, |x²−4| uals x²−4 elsewhere. This is essential when differentiating or integrating such functions.
例如 g(x) = |x²−4|，内部表达式的零点在 x=−2 和 x=2。x²−4 的符号图显示在 −2 与 2 之间取负值。因此 |x²−4| 在 (−2, 2) 上等于 −(x²−4)，而在其他区间等于 x²−4。这在微分或积分此类函数时非常重要。
Factors such as √x−a or ln(x−b) impose domain restrictions as well. The sign diagram must first consider where the function exists; outside the domain, no sign is given.
√x−a 或 ln(x−b) 这类因子还会施加定义域限制。符号图必须优先考虑函数存在的范围；在定义域之外不标记任何符号。
9. Summary Table of Sign Diagram Rules | 符号图规则总结表
| Aspect / 方面 | Rule / 规则 | Notes / 备注 |
|---|---|---|
| Finding critical points / 确定临界点 | Set numerator = 0 and denominator = 0 | Include zeros and asymptotes |
| Multiplicity effect / 重数效应 | Odd → sign changes; Even → sign unchanged | Applies to each factor crossing |
| Rightmost sign / 最右端符号 | Sign of leading coefficient for large x | Works for fully expanded form |
| Inequalities / 不等式 | Read intervals from diagram; include zeros if ≤ or ≥ | Always exclude denominator zeros |
| Rational functions / 有理函数 | Treat vertical asymptotes as sign-change barriers | But check multiplicity in denominator |
10. Avoiding Common Mistakes | 避免常见错误
Many students lose marks because they forget to consider the domain before drawing the diagram, or they misapply the multiplicity rule. Always check that all factors are written with a positive coefficient for the variable.
许多学生丢分是因为在画图前忘了考虑定义域，或者错误地应用了重数规则。一定要确保所有因子中变量的系数为正。
If a factor is (a−x), rewrite it as −(x−a). The negative sign can be factored out and treated as a constant multiplier; it does not alter the location of the zero but may flip the overall sign of the function.
如果因子是 (a−x)，要将其改写为 −(x−a)。负号可以提出来作为常数因子处理；它不改变零点的位置，但可能使整个函数的符号翻转。
Another common error is treating asymptotes as zeros. An asymptote makes the function undefined; it cannot be included in a solution set for a non‑strict inequality. Marking a hollow circle on the number line is a good habit.
另一个常见错误是把渐近线当成零点。渐近线使函数无定义，即使是“≤”或“≥”的不等式也不能将其包含在解集中。在数轴上画空心圆是个好习惯。
11. Worked Example: Polynomial and Rational Sign Diagrams | 例题详解：多项式与有理式的符号图
Problem 1: Construct a sign diagram for h(x) = −(x+3)²(x−1)(x−4)³.
问题1： 为函数 h(x) = −(x+3)²(x−1)(x−4)³ 构造符号图。
would be −(x²)(x)(x³) = −x⁶, so for large positive x the sign is negative. Thus the right‑ s to +, at x=1 flips to −, at x=−3 stays − (even multiplicity). Final pattern: −, −, +, −.
零点在 x=−3（偶重数 2）、x=1（奇重数）、x=4（奇重数）。展开后的首项是 −(x²)(x)(x³) = −x⁶，因此 x 很大的正数时符号为负。所以最右端区间 (x>4) 是 −。向左推：在 x=4 处变号为 +，在 x=1 处变号为 −，在 x=−3 处符号不变。最终模式为 −, −, +, −。
Problem 2: Solve (2x−5)/(x²−9) ≤ 0.
问题2： 解不等式 (2x−5)/(x²−9) ≤ 0。
enominator: (x−3)(x+3). Critical points: x=−3, 5/2, 3. All multiplicities are 1. The right‑ se open intervals at asymptotes x=−3 and x=3; include x=5/2 because numerator can be zero.
分母分解为 (x−3)(x+3)。临界点：x=−3、5/2、3。重数均为 1。最右端符号：x 很大时分子正分母正 → +。交替符号：+, −, +, −。不等式 ≤0 的解集为 (−∞, −3) 和 [5/2, 3)。渐近线 x=−3 和 x=3 处用开区间；x=5/2 可取等号因为分子可以为零。
12. Key Takeaways and Exam Tips | 核心要点与应试技巧
A sign diagram turns a complicated function into a simple map of positive and negative regions. It underpins curve sketching, inequality solving, and understanding function behaviour without technology.
符号图将复杂的函数转化为一张简洁的正负区域图。它支撑着曲线绘制、不等式求解以及在无计算器辅助下对函数性质的理解。
For IB examinations, always present your sign diagram neatly with a labelled number line, critical points in order, and clear +/− signs. Connect the diagram directly to the final answer — whether it is an interval notation, a sketch, or a justification of the number of roots.
在 IB 考试中，始终要整洁地画出符号图，包括标注清晰的数轴、按顺序排列的临界点以及明确的 +/− 符号。要将符号图与最终答案直接联系起来——无论是区间记号、函数草图还是关于根的个数的论证。
Practice constructing sign diagrams from both factorised and non‑factorised forms, and always double‑check multiplicities and domain restrictions. With strong sign diagram skills, many calculus and algebra problems become straightforward.
多做从因式分解形式和非分解形式出发构建符号图的练习，并始终复核重数和定义域的限制。掌握了符号图技巧之后，许多微积分与代数问题都会变得举重若轻。
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