📚 Sign Diagrams of Functions: Construction and Applications | 函数符号图的绘制与应用
A sign diagram (or sign chart) is a powerful visual tool used in IB Mathematics to summarise the intervals where a function is positive, negative, or zero. It compactly represents the algebraic sign of a function or its derivative over its domain, and it forms the backbone for solving inequalities, locating extrema, and understanding the shape of a curve.
符号图（或称正负号图）是IB数学中一个功能强大的可视化工具，用于概括函数在哪些区间为正、为负或为零。它紧凑地表示了函数或其导数在其定义域上的代数符号，是求解不等式、定位极值以及理解曲线形状的基础。
1. What Is a Sign Diagram? | 什么是符号图？
A sign diagram is a number line that shows the sign of a function \(f(x)\) in each interval determined by its critical points (where the function is zero or undefined). The critical points are marked on the line, and the sign (‘+’ or ‘−’) is written in each interval between them. Open circles are used where the function is undefined, while filled circles indicate zeros.
符号图是一条数轴，它显示函数 \(f(x)\) 在其临界点（函数为零或未定义的点）所划分的各个区间内的符号。临界点标记在数轴上，并在每个区间内写出符号（’+’ 或 ‘−’）。函数未定义处用空心圆表示，零点用实心圆表示。
f(x) = (x − 1)(x + 2), x ∈ ℝ
For this quadratic, the zeros are x = 1 and x = −2. A sign diagram shows ‘+’ on (−∞, −2), ‘−’ on (−2, 1), and ‘+’ on (1, ∞).
对于这个二次函数，零点是 x = 1 和 x = −2。符号图显示在 (−∞, −2) 上为 ‘+’，在 (−2, 1) 上为 ‘−’，在 (1, ∞) 上为 ‘+’。
2. Steps to Construct a Sign Diagram | 绘制符号图的步骤
The standard procedure involves four clear steps. First, find all critical values by setting the numerator equal to zero and identifying where the function is undefined. Second, place these values on a number line in increasing order. Third, choose a test point in each interval and evaluate the sign of the function at that point. Fourth, record the sign in each interval, using filled or open circles as appropriate.
标准绘制流程包含四个清晰的步骤。第一，通过令分子为零并确定函数无定义的点来找出所有临界值。第二，将这些值按递增顺序放置在数轴上。第三，在每个区间内选取一个测试点，计算该点处函数的符号。第四，在每个区间内记录符号，并相应使用实心或空心圆。
It is essential to remember that the sign of a function changes at a zero of odd multiplicity, but not at a zero of even multiplicity. This greatly speeds up the construction of a sign diagram.
务必记住，函数在奇重零点的两侧符号会改变，而在偶重零点的两侧符号不变。这能大大加快符号图的构建。
| Zero multiplicity | Sign change? | Graph behaviour |
| Odd (1, 3, 5…) | Yes | Crosses the x-axis |
| Even (2, 4, 6…) | No | Touches and rebounds |
For example, for f(x) = (x − 1)²(x + 2), the zero at x = 1 has multiplicity 2, so the sign does not change there. The function is negative on (−∞, −2), positive on (−2, 1), and positive on (1, ∞).
例如，对于 f(x) = (x − 1)²(x + 2)，x = 1 处的零点重数为 2，因此符号在该处不改变。函数在 (−∞, −2) 上为负，在 (−2, 1) 上为正，在 (1, ∞) 上为正。
3. Sign Diagrams for Polynomial Functions | 多项式函数的符号图
Polynomial functions are continuous everywhere, so their sign diagrams only involve zeros. To construct the diagram, factor the polynomial completely, find all real roots, mark them on the number line, and test one point in each interval.
多项式函数处处连续，因此其符号图仅涉及零点。构建符号图时，需要将多项式完全因式分解，找出所有实根，将它们标记在数轴上，并在每个区间内测试一个点。
Consider P(x) = x³ − 3x² − 4x. Factoring gives P(x) = x(x − 4)(x + 1). The roots are x = −1, 0, 4. Testing intervals gives: (−∞, −1): −, (−1, 0): +, (0, 4): −, (4, ∞): +.
考虑 P(x) = x³ − 3x² − 4x。因式分解得 P(x) = x(x − 4)(x + 1)。根为 x = −1, 0, 4。测试区间得：(−∞, −1): −, (−1, 0): +, (0, 4): −, (4, ∞): +。
It is important to note that the degree of the polynomial determines the end behaviour, which must be consistent with the outermost sign intervals. For an odd-degree polynomial with a positive leading coefficient, the sign is negative as x → −∞ and positive as x → ∞.
需要注意，多项式的次数决定其端部行为，这必须与最外侧区间的符号一致。对于首项系数为正的奇次多项式，当 x → −∞ 时符号为负，当 x → ∞ 时符号为正。
x³ − 3x² − 4x = x(x − 4)(x + 1)
4. Sign Diagrams for Rational Functions | 有理函数的符号图
For rational functions of the form f(x) = N(x)/D(x), the sign diagram must include both zeros of N(x) and zeros of D(x), because the function is undefined at the latter. At a vertical asymptote (zero of D of odd multiplicity), the sign changes; at a zero of D of even multiplicity, it does not.
对于形如 f(x) = N(x)/D(x) 的有理函数，符号图必须同时包含 N(x) 的零点和 D(x) 的零点，因为函数在 D(x) 的零点处无定义。在奇重分母零点（垂直渐近线）处，符号改变；在偶重分母零点处，符号不变。
Take f(x) = x/(x² − 1). The numerator zero is x = 0; the denominator zeros are x = 1 and x = −1. The sign diagram has open circles at x = ±1 and a filled circle at x = 0. Testing yields: (−∞, −1): −, (−1, 0): +, (0, 1): −, (1, ∞): +.
以 f(x) = x/(x² − 1) 为例。分子零点为 x = 0；分母零点为 x = 1 和 x = −1。符号图在 x = ±1 处为空心圆，在 x = 0 处为实心圆。测试得到：(−∞, −1): −, (−1, 0): +, (0, 1): −, (1, ∞): +。
Care must be taken with removable discontinuities. If a factor cancels between N and D, the point is a hole, not a vertical asymptote. It should be marked as an open circle, but the sign of the function around it may or may not change depending on the multiplicity of the cancelled factor.
必须注意可去间断点。如果 N 和 D 之间有一个因子相消，则该点是空洞而非垂直渐近线。它应标记为空心圆，但周围的符号是否改变取决于被消因子的重数。
f(x) = x/(x² − 1) ⇒ sign: − + − + on (−∞,−1), (−1,0), (0,1), (1,∞)
5. Using Sign Diagrams to Solve Inequalities | 利用符号图求解不等式
The most common application of a sign diagram is solving inequalities such as f(x) > 0, f(x) ≤ 0, etc. Once the sign diagram is drawn, the solution is simply the union of intervals where the inequality condition holds, with careful attention to whether the endpoints are included.
符号图最常见的应用是求解形如 f(x) > 0、f(x) ≤ 0 等不等式。一旦作出符号图，解就是满足不等式条件的区间并集，并需仔细注意端点是否包含在内。
For example, solve (x + 1)/(x − 3) ≤ 0. The critical values are x = −1 (zero) and x = 3 (undefined). A sign diagram shows: (−∞, −1): +, (−1, 3): −, (3, ∞): +. Since we need ≤ 0, the solution is x ∈ [−1, 3), excluding x = 3 because the function is undefined.
例如，解不等式 (x + 1)/(x − 3) ≤ 0。临界值为 x = −1（零点）和 x = 3（无定义）。符号图显示：(−∞, −1): +, (−1, 3): −, (3, ∞): +。由于需要 ≤ 0，解为 x ∈ [−1, 3)，排除 x = 3 因为函数无定义。
When solving rational inequalities, never multiply both sides by an expression whose sign depends on x. Always bring all terms to one side, combine into a single fraction, factor, and then use a sign diagram.
求解有理不等式时，切勿将两边同时乘以符号依赖于 x 的表达式。应始终将各项移到一边，合并成一个分式，因式分解，然后使用符号图。
6. Sign Diagram of f′(x) and Locating Extrema | 一阶导数的符号图与极值定位
Sign diagrams are essential in calculus. The sign of the first derivative f′(x) tells us where the original function f(x) is increasing or decreasing. If f′(x) > 0, f is increasing; if f′(x) < 0, f is decreasing.
符号图在微积分中至关重要。一阶导数 f′(x) 的符号告诉我们原函数 f(x) 在何处递增或递减。若 f′(x) > 0，则 f 递增；若 f′(x) < 0，则 f 递减。
At a point where f′(x) changes from positive to negative, f has a local maximum. Where f′(x) changes from negative to positive, f has a local minimum. If f′(x) does not change sign, the critical point is neither a maximum nor a minimum.
在 f′(x) 由正变负的点处，f 具有局部极大值。在 f′(x) 由负变正的点处，f 具有局部极小值。如果 f′(x) 符号不改变，则该临界点既不是极大值也不是极小值。
Let f(x) = x³ − 3x² + 1. Then f′(x) = 3x² − 6x = 3x(x − 2). The sign diagram of f′ shows ‘+’ on (−∞, 0), ‘−’ on (0, 2), and ‘+’ on (2, ∞). Hence f has a local maximum at x = 0 and a local minimum at x = 2.
设 f(x) = x³ − 3x² + 1。则 f′(x) = 3x² − 6x = 3x(x − 2)。f′ 的符号图显示在 (−∞, 0) 上为 ‘+’，在 (0, 2) 上为 ‘−’，在 (2, ∞) 上为 ‘+’。因此 f 在 x = 0 处有局部极大值，在 x = 2 处有局部极小值。
f′(x) = 3x(x − 2) ⇒ local max at x = 0, local min at x = 2
7. Sign Diagram of f″(x) and Points of Inflection | 二阶导数的符号图与拐点
The second derivative f″(x) measures the concavity of f. When f″(x) > 0, the graph is concave up; when f″(x) < 0, it is concave down. A point of inflection occurs where f″(x) changes sign, provided the function is continuous at that point.
二阶导数 f″(x) 衡量 f 的凹凸性。当 f″(x) > 0 时，图像凹向上；当 f″(x) < 0 时，图像凹向下。拐点出现在 f″(x) 改变符号处，前提是函数在该点连续。
For f(x) = x⁴ − 6x², we have f″(x) = 12x² − 12 = 12(x − 1)(x + 1). The sign diagram of f″ shows ‘+’ on (−∞, −1), ‘−’ on (−1, 1), and ‘+’ on (1, ∞). Therefore, there are points of inflection at x = −1 and x = 1.
对于 f(x) = x⁴ − 6x²，有 f″(x) = 12x² − 12 = 12(x − 1)(x + 1)。f″ 的符号图显示在 (−∞, −1) 上为 ‘+’，在 (−1, 1) 上为 ‘−’，在 (1, ∞) 上为 ‘+’。因此，在 x = −1 和 x = 1 处存在拐点。
Note that f″(x) = 0 alone is not sufficient for an inflection point; the sign must actually change. For example, f(x) = x⁴ has f″(0) = 0 but no inflection point at x = 0 because f″(x) ≥ 0 on both sides.
注意，仅有 f″(x) = 0 并不足以判定拐点；符号必须真正改变。例如，f(x) = x⁴ 在 x = 0 处有 f″(0) = 0，但该处不是拐点，因为两侧 f″(x) ≥ 0。
8. Sign Diagrams and Vertical Asymptotes | 符号图与垂直渐近线
In rational functions, vertical asymptotes correspond to real zeros of the denominator that do not cancel. Sign diagrams must represent these as open circles. The sign of the function changes at a vertical asymptote if the denominator factor has odd multiplicity, but not if it has even multiplicity.
在有理函数中，垂直渐近线对应于分母中未相消的实零点。符号图必须将这些点表示为空心圆。如果分母因子的重数为奇数，则函数符号在垂直渐近线处改变；如果重数为偶数，则符号不变。
Consider f(x) = 1/(x² + 1). The denominator has no real zeros, so there are no vertical asymptotes and the sign is always positive. A sign diagram of a function with no critical values is simply a single interval with the constant sign.
考虑 f(x) = 1/(x² + 1)。分母没有实零点，因此没有垂直渐近线，符号始终为正。没有临界值的函数的符号图就是一个单一区间，符号恒定。
On the other hand, f(x) = 1/(x² − 4) has vertical asymptotes at x = ±2. The sign diagram shows ‘+’ on (−∞, −2), ‘−’ on (−2, 2), and ‘+’ on (2, ∞).
另一方面，f(x) = 1/(x² − 4) 在 x = ±2 处有垂直渐近线。符号图显示在 (−∞, −2) 上为 ‘+’，在 (−2, 2) 上为 ‘−’，在 (2, ∞) 上为 ‘+’。
9. Common Mistakes When Drawing Sign Diagrams | 绘制符号图时的常见错误
One common error is ignoring the domain of the function. For example, a radical function like f(x) = √(x − 2) has sign only for x ≥ 2. A sign diagram drawn for all real x would be misleading.
一个常见错误是忽略函数的定义域。例如，根式函数 f(x) = √(x − 2) 仅对 x ≥ 2 有符号。如果画出整个实数轴上的符号图，会产生误导。
Another error is forgetting to check multiplicity. A sign change does not occur at a double root. Many students incorrectly alternate signs across every zero, which leads to wrong inequality solutions.
另一个错误是忘记检查重数。在二重根处符号不改变。许多学生错误地在每个零点两侧改变符号，从而导致不等式解出错。
Also, when a function has a horizontal asymptote, the end-behavior sign must be checked separately. For instance, f(x) = (2x + 1)/(x − 3) approaches 2 as x → ∞; its sign near x = ∞ is positive, but a simple alternation from the critical points might suggest otherwise if the end sign is not verified.
此外，当函数具有水平渐近线时，必须单独检验端部行为的符号。例如，f(x) = (2x + 1)/(x − 3) 当 x → ∞ 时趋近于 2；其在 x = ∞ 附近的符号为正，但如果未验证端部符号，仅从临界点交替推导可能得出错误结论。
10. Worked Example: Complete Sign Diagram Analysis | 例题演练：完整的符号图分析
Let us analyse the function f(x) = (x² − 1)/(x² − 4) completely using sign diagrams.
让我们用符号图完整分析函数 f(x) = (x² − 1)/(x² − 4)。
Factor: f(x) = ((x − 1)(x + 1))/((x − 2)(x + 2)). The zeros are x = ±1; the vertical asymptotes are x = ±2. Mark −2, −1, 1, 2 on the number line with open circles at ±2 and filled circles at ±1.
因式分解：f(x) = ((x − 1)(x + 1))/((x − 2)(x + 2))。零点为 x = ±1；垂直渐近线为 x = ±2。在数轴上标记 −2, −1, 1, 2，其中 ±2 用空心圆，±1 用实心圆。
Choose test points: x = −3 → positive; x = −1.5 → negative; x = 0 → positive; x = 1.5 → negative; x = 3 → positive. Thus the sign diagram is: + − + − +.
选取测试点：x = −3 → 正；x = −1.5 → 负；x = 0 → 正；x = 1.5 → 负；x = 3 → 正。因此符号图为：+ − + − +。
From this diagram, f(x) > 0 on (−∞, −2) ∪ (−1, 1) ∪ (2, ∞), and f(x) ≤ 0 on (−2, −1] ∪ [1, 2). This illustrates how one compact diagram answers multiple questions about the function.
由此图，f(x) > 0 在 (−∞, −2) ∪ (−1, 1) ∪ (2, ∞) 上成立，而 f(x) ≤ 0 在 (−2, −1] ∪ [1, 2) 上成立。这展示了一张简洁的符号图如何回答关于该函数的多个问题。
(x² − 1)/(x² − 4) > 0 ⇔ x ∈ (−∞, −2) ∪ (−1, 1) ∪ (2, ∞)
11. Applications in Curve Sketching | 符号图在曲线绘制中的应用
Combining the sign diagrams of f, f′, and f″ allows a complete sketch of the curve without plotting many points. The sign of f gives the region above/below the x-axis; the sign of f′ gives the monotonic intervals; the sign of f″ gives the concavity.
结合 f、f′ 和 f″ 的符号图，无需描出大量点即可完整绘制曲线。f 的符号给出图像在 x 轴上方/下方的区域；f′ 的符号给出单调区间；f″ 的符号给出凹凸性。
For example, to sketch y = x e⁻ˣ, first note the domain is ℝ and f(0) = 0. Then f′(x) = e⁻ˣ(1 − x), so f is increasing on (−∞, 1) and decreasing on (1, ∞). Also f″(x) = e⁻ˣ(x − 2), giving an inflection point at x = 2. Together with the limit as x → ∞ being 0, the sketch is accurate.
例如，要绘制 y = x e⁻ˣ，首先注意定义域为 ℝ 且 f(0) = 0。然后 f′(x) = e⁻ˣ(1 − x)，因此 f 在 (−∞, 1) 上递增，在 (1, ∞) 上递减。又有 f″(x) = e⁻ˣ(x − 2)，在 x = 2 处有拐点。结合 x → ∞ 时极限为 0，即可准确作图。
In IB examinations, sketching a graph using sign diagrams is often worth several marks. It is wise to clearly present each sign diagram separately before drawing the final curve.
在IB考试中，使用符号图绘制图像通常占多分。明智的做法是在绘制最终曲线前，分别清晰展示每个符号图。
12. Summary | 总结
A sign diagram is not merely a homework exercise; it is a compact mathematical tool that encodes the behaviour of a function across its domain. Mastering its construction and interpretation is essential for solving inequalities, analyzing derivatives, and producing accurate curve sketches in IB Mathematics.
符号图不仅仅是课后练习；它是一种紧凑的数学工具，编码了函数在其定义域上的行为。掌握其构造与解读，对于在IB数学中求解不等式、分析导数以及精确绘制曲线至关重要。
Remember the key rules: find all critical values, respect the domain, check multiplicities, and test each interval. With practice, sign diagrams become an intuitive and quick way to unlock the properties of any function.
记住关键规则：找到所有临界值、尊重定义域、检查重数、测试每个区间。通过练习，符号图将成为解锁任何函数性质的直观而快速的方法。
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