📚 Sketching Graphs of Functions | 函数图像的绘制
Sketching graphs of functions is a fundamental skill in IB Mathematics, bridging algebraic manipulation, calculus and geometric intuition. A well-drawn sketch does not need to be perfectly to scale, but it must accurately show the key features of a curve: domain, intercepts, symmetry, asymptotes, intervals of increase and decrease, local extrema, concavity and points of inflection. This article provides a comprehensive guide to mastering these techniques, so that you can interpret and construct function graphs with confidence.
绘制函数图像是IB数学的一项基本技能，它连接了代数运算、微积分和几何直观。一幅良好的草图不必完全按比例，但必须准确展示曲线的关键特征：定义域、截距、对称性、渐近线、单调区间、局部极值点、凹凸性和拐点。本文将提供一份全面指南，帮助你掌握这些技巧，从而能够自信地解读和构建函数图形。
1. What a Sketch Must Show | 草图必须展示什么
A sketch is not a plot. A plot is produced by a graphing calculator or by plotting hundreds of points; a sketch is a small, deliberate diagram that a marker can read in three seconds. Examiners look for shape, position and labelled coordinates rather than perfect curvature. If your curve passes through the right intercepts, approaches the correct asymptotes and bends the right way, it earns full credit even if the scale is approximate.
草图不等于精确作图。精确作图依靠图形计算器或描出成百上千个点；草图则是一幅小而精炼的示意图，阅卷老师能在三秒内读出信息。评分关注的是形状、位置和标注好的坐标，而不是完美的弧度。只要曲线经过正确的截距、趋近正确的渐近线、弯曲方向正确，即使比例不完全精确也能拿满分。
The features you must always consider are: the domain, the coordinates of any intercepts, symmetry (even, odd or periodic), all asymptotes, the coordinates of stationary points and their nature, the intervals where the function increases or decreases, and the concavity together with any point of inflection. A quick check of end behaviour ties everything together and prevents the common error of drawing a curve that drifts off in the wrong direction.
你必须始终考虑的特征包括：定义域、截距坐标、对称性（偶函数、奇函数或周期函数）、所有渐近线、驻点坐标及其性质、函数的单调增区间与减区间，以及凹凸性与拐点。最后用端点趋势做一次快速检验，可以把所有信息串联起来，避免曲线"飘向错误方向"这一常见错误。
2. Understanding the Domain and Range | 理解定义域与值域
Before plotting any points, determine the set of all possible input values (the domain) and the corresponding output values (the range). The domain may be restricted by denominators (which cannot be zero), even roots (the radicand must be non-negative), logarithmic arguments (which must be positive), inverse trigonometric functions, and the context of the problem. The range is found by examining the behaviour of the function or by using calculus to locate its maximum and minimum values.
在描点之前，首先要确定所有可能的输入值集合（定义域）和相应的输出值集合（值域）。定义域可能受到分母（不能为零）、偶次根式（被开方数非负）、对数函数的真数（必须为正）、反三角函数以及问题背景的限制。值域可通过考察函数的行为，或利用微积分求出最大值与最小值来确定。
For example, consider the function f(x) = √x − 2 e domain is x ≥ 2. The output of a square root is always non-negative, and as x increases, √x − 2 ng the domain early prevents drawing parts of the graph where the function does not exist.
例如，考虑函数 f(x) = √x − 2。被开方数 x − 2 必须大于等于 0，因此定义域为 x ≥ 2。平方根的输出总是非负的，并且随着 x 增大，√x − 2 可取遍所有非负值，故值域为 y ≥ 0。及早识别定义域可避免画出函数不存在的图形部分。
l real y; for f(x) = tan x the domain excludes x = π/2 + kπ for integers k. Writing the domain and range beside your sketch is a cheap way to secure method marks.
值得记住的典型限制：f(x) = 1/(x − 3) 的定义域为 x ≠ 3，值域为 y ≠ 0；f(x) = ln x 的定义域为 x > 0，值域为全体实数；f(x) = tan x 的定义域排除 x = π/2 + kπ（k 为整数）。在草图旁写出定义域与值域，是稳拿方法分的低成本做法。
3. Intercepts: Where the Graph Meets the Axes | 截距：图形与坐标轴的交点
rcept and you should state this explicitly. To find the x-intercepts, also called roots or zeros, solve the equation f(x) = 0. Factorising is usually the fastest route for polynomials, while for rational functions you need only set the numerator equal to zero, provided the resulting x is still in the domain.
y 轴截距为 f(0)，前提是 0 在定义域内。如果定义域排除 0（例如因为分母或对数），则不存在 y 轴截距，应当明确说明这一点。求 x 轴截距（也称根或零点）就是解方程 f(x) = 0。对多项式而言，因式分解通常最快；对有理函数，只需令分子等于零，但要检查求得的 x 是否仍在定义域内。
(multiplicity 2) touches the axis and turns back, producing a local extremum; a triple root (multiplicity 3) crosses but flattens out, like the origin for y = x³. Recognising this pattern turns a list of numbers into a shape.
根的重数决定了曲线在该点的行为。单根（重数 1）以非零夹角穿过 x 轴；二重根（重数 2）与 x 轴相切后折返，形成一个局部极值；三重根（重数 3）穿过 x 轴但在此变平，正如 y = x³ 在原点那样。识别这一规律，就能把一串数字变成一幅形状。
y-intercept: x = 0 → y = f(0) | x-intercept: y = 0 → f(x) = 0
4. Symmetry: Even, Odd and Periodic Functions | 对称性：偶函数、奇函数与周期函数
half and reflect it. A function is odd if f(−x) = −f(x); its graph has rotational symmetry of order 2 about the origin, so sketching the first quadrant gives you the third quadrant for free. Polynomials built only from even powers of x are even, those built from odd powers are odd, and any function containing both is neither.
对称性可以让工作量减半。若对定义域内所有 x 都有 f(−x) = f(x)，则函数为偶函数，其图像关于 y 轴对称，因此只需画右半部分再作镜像。若 f(−x) = −f(x)，则函数为奇函数，其图像关于原点有 2 阶旋转对称，画出第一象限即可免费得到第三象限。仅由 x 的偶次幂构成的多项式是偶函数，仅由奇次幂构成的是奇函数，两者兼有的则不具备对称性。
) for some constant T > 0, the smallest such T being the period. Sine curves with period 2π, tangents with period π, and the modulus of a trigonometric function halving the period are all standard cases. Once you identify the period, sketch one full cycle and translate it by multiples of T. Never draw more cycles than the question requires; unnecessary repetition wastes time and can hide the labelled features.
若存在常数 T > 0 使 f(x + T) = f(x) 对一切 x 成立，则函数为周期函数，最小的这样的 T 即为周期。周期为 2π 的正弦曲线、周期为 π 的正切函数，以及三角函数取绝对值后周期减半的情况，都是标准例子。识别周期后，只需画出一个完整周期，再按 T 的整数倍平移复制。不要画出超过题目要求的周期数，多余的重复既浪费时间，也可能掩盖那些需要标注的特征。
5. Asymptotes: Vertical, Horizontal and Oblique | 渐近线：垂直、水平与斜渐近线
y of a logarithmic domain such as x = 0 for ln x. A horizontal asymptote is found from the limits as x → +∞ and x → −∞; the graph approaches the line y = L without necessarily touching it. Note that a graph may cross a horizontal asymptote, but never a vertical one.
当函数值趋于 ±∞ 时出现垂直渐近线，最常见的情形是分母为零而分子不为零，或者对数定义域的边界（例如 ln x 在 x = 0 处）。水平渐近线由 x → +∞ 与 x → −∞ 的极限求得；图形会趋近直线 y = L，但未必与之相交。注意图形可以与水平渐近线相交，但绝不会与垂直渐近线相交。
tal asymptote is y = 0. If the degrees are equal, the asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator). If the numerator degree exceeds the denominator degree by exactly 1, there is instead an oblique (slant) asymptote found by polynomial long division.
对有理函数，比较次数可得到快速判据。若分子次数小于分母次数，水平渐近线为 y = 0。若两者次数相等，渐近线为 y =（分子首项系数）/（分母首项系数）。若分子次数恰好比分母次数大 1，则不存在水平渐近线，而要用多项式长除法求斜渐近线。
f(x) = (x² + 1)/x can be written as x + 1/x, so y = x is an oblique asymptote and x = 0 is a vertical asymptote. Always mark asymptotes with dashed lines and label them with their equations.
例如，f(x) = (2x + 1)/(x − 3) 在 x = 3 处有垂直渐近线，由于分子分母同为一次，水平渐近线为 y = 2。而 f(x) = (x² + 1)/x 可写成 x + 1/x，故 y = x 是斜渐近线，x = 0 是垂直渐近线。切记用虚线画出渐近线，并标注其方程。
6. The First Derivative: Increasing, Decreasing and Stationary Points | 一阶导数：单调性与驻点
lt; 0, f is strictly decreasing; where f′(x) = 0, the tangent is horizontal and the point is stationary. Setting f′(x) = 0 and solving gives the x-coordinates of all stationary points, and substituting back into f gives their y-coordinates.
一阶导数的符号决定曲线的走向。在区间上若 f′(x) > 0，则 f 在此严格递增；若 f′(x) < 0，则严格递减；若 f′(x) = 0，则切线水平，该点为驻点。令 f′(x) = 0 并求解，可得到所有驻点的横坐标；代回 f 即得相应纵坐标。
ediately to the left and right: negative then positive means a local minimum, positive then negative means a local maximum, and no sign change means a stationary point of inflection. The second derivative test is quicker when f″ is easy to compute: f″(x₀) > 0 gives a local minimum, f″(x₀) < 0 gives a local maximum, and f″(x₀) = 0 is inconclusive, so you must fall back on the sign-change test.
判断驻点性质可用一阶导数判别法或二阶导数判别法。一阶导数判别法是考察 f′ 在驻点左右两侧的符号：先负后正为局部极小，先正后负为局部极大，符号不变则为驻点型拐点。当 f″ 容易计算时，二阶导数判别法更快：f″(x₀) > 0 为局部极小，f″(x₀) < 0 为局部极大，而 f″(x₀) = 0 无法判定，必须回到符号变化法。
f′(x) > 0 → increasing | f′(x) < 0 → decreasing | f′(x) = 0 → stationary point
7. The Second Derivative: Concavity and Points of Inflection | 二阶导数：凹凸性与拐点
Concavity describes the way a curve bends. If f″(x) > 0 on an interval, the gradient is increasing and the curve is concave up (convex), like the bowl of a cup; the graph lies above its tangent lines. If f″(x) < 0, the gradient is decreasing and the curve is concave down, like a dome; the graph lies below its tangent lines. Marking concavity correctly is what makes a sketch look convincing rather than piecewise linear.
凹凸性描述曲线的弯曲方式。若在区间上 f″(x) > 0，则斜率递增，曲线开口向上（下凸），形如杯底，图像位于其切线上方；若 f″(x) < 0，则斜率递减，曲线开口向下（上凸），形如穹顶，图像位于其切线下方。正确标出凹凸性，才能使草图看起来可信，而不是由若干直线段拼成。
Do not assume every solution of f″(x) = 0 is an inflection point: for f(x) = x⁴, f″(x) = 12x² which is zero at x = 0 but never negative, so the origin is a minimum, not an inflection point. For a stationary point of inflection, both f′ and f″ vanish, as with y = x³ at the origin.
拐点是凹凸性发生改变的点，它要求 f″(x) = 0（或 f″ 不存在），并且 f″ 在两侧真正变号。不要以为 f″(x) = 0 的每一个解都是拐点：对 f(x) = x⁴，f″(x) = 12x² 在 x = 0 处为零但永不为负，故原点是极小值点而非拐点。若拐点同时是驻点，则 f′ 与 f″ 同时为零，例如 y = x³ 在原点的情况。
8. End Behaviour and Extreme Values | 端点行为与极端值
overall direction of your sketch. For polynomials, only the leading term matters: an even degree with a positive leading coefficient rises at both ends, an odd degree with a positive leading coefficient falls to the left and rises to the right. For functions defined on a closed interval, remember to evaluate f at the endpoints, since an absolute maximum or minimum may occur there rather than at a stationary point.
端点行为描述当 x → +∞ 与 x → −∞ 时 y 的变化趋势，它决定了草图的整体走向。对多项式而言，只有首项起作用：偶次且首项系数为正者两端上升；奇次且首项系数为正者左端下降、右端上升。对于在闭区间上定义的函数，要记得计算区间端点处的函数值，因为绝对最大值或最小值可能出现在端点而非驻点。
tends to +∞ at one end, there is no global maximum at all; if it tends to a finite limit L, that limit may be approached but never attained. Writing a short sentence such as "as x → ∞, f(x) → 2, so y = 2 is a horizontal asymptote" earns communication marks and clarifies your diagram.
在无界定义域上，局部最大值未必是全局最大值。要作出判断，需把所有局部极值与无穷远处的极限进行比较。若函数在某一端趋于 +∞，则根本不存在全局最大值；若趋于有限极限 L，则该值可能只是趋近而永远取不到。写上一句"当 x → ∞ 时 f(x) → 2，故 y = 2 为水平渐近线"，既能拿到表达分，也让你自己的图形更清晰。
9. A Worked Example and a Feature Table | 完整示例与特征表
−f(x); indeed f(x) = x³ − 3x satisfies f(−x) = −f(x), so the graph has rotational symmetry about the origin. The y-intercept is f(0) = 0. Setting x³ − 3x = 0 gives x(x² − 3) = 0, so the x-intercepts are x = 0 and x = ±√3 ≈ ±1.73.
考虑 f(x) = x³ − 3x。定义域为全体实数；由于 f(−x) = −f(x)，它是奇函数，图像关于原点中心对称。y 轴截距为 f(0) = 0。令 x³ − 3x = 0，得 x(x² − 3) = 0，故 x 轴截距为 x = 0 与 x = ±√3 ≈ ±1.73。
) and a local minimum at (1, −2). The second derivative is f″(x) = 6x, which is zero at x = 0 and changes sign there, so (0, 0) is a point of inflection: concave down for x < 0, concave up for x > 0.
求导得 f′(x) = 3x² − 3 = 3(x − 1)(x + 1)，故驻点出现在 x = −1 与 x = 1。由 f(−1) = 2、f(1) = −2，得局部极大点 (−1, 2) 与局部极小点 (1, −2)。二阶导数为 f″(x) = 6x，在 x = 0 处为零且变号，故 (0, 0) 为拐点：x < 0 时开口向下，x > 0 时开口向上。
| Feature | 特征 | Result for f(x) = x³ − 3x | 结果 |
| Domain | 定义域 | x ∈ ℝ |
| x-intercepts | x 轴截距 | x = −√3, 0, √3 |
| y-intercept | y 轴截距 | (0, 0) |
| Stationary points | 驻点 | (−1, 2) local max; (1, −2) local min |
| Inflection | 拐点 | (0, 0) |
| End behaviour | 端点行为 | f(x) → −∞ as x → −∞; f(x) → +∞ as x → +∞ |
asymptote at y = 1, x-intercept (1, 0) and y-intercept (0, −0.5). Since f′(x) = 3/(x + 2)² > 0, the function is increasing on each branch, and since f″(x) = −6/(x + 2)³, it is concave up for x < −2 and concave down for x > −2, with no point of inflection.
另一个极常见的考题是 f(x) = (x − 1)/(x + 2)，可改写为 f(x) = 1 − 3/(x + 2)。它有垂直渐近线 x = −2、水平渐近线 y = 1，x 轴截距 (1, 0)，y 轴截距 (0, −0.5)。由 f′(x) = 3/(x + 2)² > 0 可知函数在每一支上均递增；由 f″(x) = −6/(x + 2)³ 可知 x < −2 时开口向上，x > −2 时开口向下，且没有拐点。
10. Standard Function Families and Their Shapes | 常见函数族及其形状
Recognising
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