📚 IB Mathematics: Key Points for Sketching Function Graphs | IB数学：函数图像绘制要点
Sketching function graphs is a fundamental skill in IB Mathematics, required in both Analysis and Approaches and Applications and Interpretation. A good sketch is not merely a plot of points; it is a summary of a function’s key features: domain, intercepts, symmetry, asymptotes, monotonicity, extrema, concavity, and transformations. This article presents a systematic checklist of these essential points.
绘制函数图像是IB数学中的基本技能，无论是分析与方法（AA）还是应用与解释（AI）课程都要求掌握。一张好的草图并非简单的点图，而是函数核心特征的汇总：定义域、截距、对称性、渐近线、单调性、极值、凹凸性以及变换。本文提供一套系统化的要点清单。
1. Determine the Domain First | 首先确定定义域
Before drawing anything, identify all real x-values for which the function is defined. Exclude values that make denominators zero, radicands of even roots negative, or logarithms of non-positive values. For example, the function f(x) = 1/(x-2) has the domain x ≠ 2, and f(x) = √(x-3) has the domain x ≥ 3.
在动笔之前，先找出函数有定义的所有实数 x 值。排除使分母为零、偶次根号下为负、对数真数非正的取值。例如，函数 f(x) = 1/(x-2) 的定义域为 x ≠ 2，而 f(x) = √(x-3) 的定义域为 x ≥ 3。
If the domain is not explicitly given, assume it is the largest set of real numbers for which the function is meaningful. Writing the domain as an interval or set is an essential part of a complete sketch.
如果题目没有给出定义域，默认取使函数有意义的最大实数集。用区间或集合写出定义域，是完整作图必不可少的部分。
2. Find Intercepts with Axes | 求坐标轴截距
Every graph should clearly show where it meets the coordinate axes. The y-intercept is found by evaluating f(0), provided 0 is in the domain. The x-intercepts (roots or zeros) are found by solving f(x) = 0. For instance, for g(x) = x² – 4, the y-intercept is -4 and the x-intercepts are x = -2 and x = 2.
每张图像都应清楚显示它与坐标轴的交点。y 截距通过计算 f(0) 得到，前提是 0 在定义域内；x 截距（根或零点）则通过解方程 f(x) = 0 得到。例如，对于 g(x) = x² – 4，y 截距为 -4，x 截距为 x = -2 和 x = 2。
If solving f(x) = 0 is difficult, state the approximate roots using a calculator or numerical method. Always label intercepts with their exact or approximate coordinates on the sketch.
如果解 f(x) = 0 较困难，可用计算器或数值方法给出近似根。在草图上务必标注截距的精确或近似坐标。
3. Check Symmetry and Periodicity | 检查对称性与周期性
Symmetry can save half the work of sketching. If f(-x) = f(x) for all x in the domain, the graph is symmetric about the y-axis (even function). If f(-x) = -f(x), the graph has rotational symmetry of 180° about the origin (odd function). For example, y = x² and y = cos x are even; y = x³ and y = sin x are odd.
对称性可以省去一半绘图工作。若对定义域内所有 x 都有 f(-x) = f(x)，则图像关于 y 轴对称（偶函数）。若 f(-x) = -f(x)，则图像关于原点旋转 180° 对称（奇函数）。例如，y = x² 和 y = cos x 为偶函数；y = x³ 和 y = sin x 为奇函数。
For trigonometric functions, identify the period and draw one full cycle before repeating the pattern. A periodic function satisfies f(x + p) = f(x), where p is the fundamental period. For y = tan x, the period is π; for y = sin bx or cos bx, the period is 2π/|b|.
对于三角函数，先确定周期，画出一个完整周期后再重复图形。周期函数满足 f(x + p) = f(x)，其中 p 为最小正周期。对于 y = tan x，周期为 π；对于 y = sin bx 或 cos bx，周期为 2π/|b|。
4. Identify Asymptotes and Unbounded Behaviour | 识别渐近线与无界行为
Vertical asymptotes typically appear where the denominator is zero and the numerator is not zero. For a rational function like f(x) = (2x+1)/(x-1), the line x = 1 is a vertical asymptote. Horizontal asymptotes describe the end behaviour as x → ±∞. If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is y = (leading coefficient ratio).
垂直渐近线通常出现在分母为零而分子不为零的位置。对于有理函数如 f(x) = (2x+1)/(x-1)，直线 x = 1 是一条垂直渐近线。水平渐近线描述 x → ±∞ 时的末端行为。若分子分母次数相等，水平渐近线为 y = 最高次项系数之比。
If the degree of the numerator is one more than that of the denominator, there is an oblique (slant) asymptote. Use polynomial long division to find its equation. Always check both ends of the domain, and mark the side of the approach (above or below the asymptote) using test points.
若分子次数比分母高一次，则存在斜渐近线。用多项式长除法求其方程。始终检查定义域两端，并用测试点判断曲线从渐近线的上方还是下方接近。
5. Use the First Derivative to Find Critical Points | 用一阶导数求关键点
Stationary points occur where f'(x) = 0 or f'(x) is undefined. They may be local maxima, local minima, or stationary inflection points. To classify them, examine the sign of f'(x) on intervals between critical points. If f’ changes from positive to negative, there is a local maximum; if negative to positive, a local minimum.
驻点出现在 f'(x) = 0 或 f'(x) 不存在的点。它们可能是局部最大值、局部最小值或驻点拐点。要判断类型，检查驻点之间区间上 f'(x) 的符号变化：若 f’ 由正变负，则为局部最大值；若由负变正，则为局部最小值。
For example, let f(x) = x³ – 3x² + 2. Then f'(x) = 3x² – 6x = 3x(x-2), so critical points are x = 0 and x = 2. Since f’ changes from positive to negative at x = 0, it is a local maximum with value f(0) = 2. At x = 2, f’ changes from negative to positive, giving a local minimum f(2) = -2.
例如，设 f(x) = x³ – 3x² + 2，则 f'(x) = 3x² – 6x = 3x(x-2)，关键点为 x = 0 和 x = 2。因为 f’ 在 x = 0 处由正变负，所以是局部最大值，f(0) = 2；在 x = 2 处由负变正，所以是局部最小值，f(2) = -2。
6. Determine Intervals of Increase and Decrease | 确定增减区间
The sign of the first derivative tells us where the function is increasing or decreasing. Create a sign table for f'(x) over the domain. If f'(x) > 0, the graph rises from left to right; if f'(x) < 0, it falls. Mark these intervals clearly on your sketch or in your notes.
一阶导数的符号告诉我们函数在哪些区间递增或递减。为 f'(x) 在定义域内建立符号表。若 f'(x) > 0，图像从左到右上升；若 f'(x) < 0，图像下降。在草图上或笔记中清楚地标出这些区间。
Remember that a function can increase even if its derivative is zero at isolated points, such as f(x) = x³ at x = 0. The derivative sign is determined on intervals, not at single points.
注意：函数即使在某些孤立点导数为零也可以整体递增，例如 f(x) = x³ 在 x = 0 处。导数符号由区间决定，而不是由个别点决定。
7. Use the Second Derivative to Analyse Concavity | 用二阶导数分析凹凸性
Concavity describes how the graph bends. If f”(x) > 0, the function is concave up (like y = x²); if f”(x) < 0, it is concave down (like y = -x²). Points where concavity changes are called inflection points, occurring where f''(x) = 0 or is undefined, provided the sign actually changes.
凹凸性描述图像的弯曲方向。若 f”(x) > 0，函数为凹向上（如 y = x²）；若 f”(x) < 0，则为凹向下（如 y = -x²）。凹凸性发生变化的点称为拐点，出现在 f''(x) = 0 或无定义处，且需要符号真正改变。
For f(x) = x³ – 3x² + 2, f”(x) = 6x – 6. Thus f” is negative for x < 1 and positive for x > 1, so the graph is concave down on (-∞, 1) and concave up on (1, ∞), with an inflection point at (1, 0).
对于 f(x) = x³ – 3x² + 2，f”(x) = 6x – 6。因此 x < 1 时 f'' 为负，x > 1 时为正，所以图像在 (-∞, 1) 上凹向下，在 (1, ∞) 上凹向上，拐点为 (1, 0)。
8. Apply Transformations of Basic Graphs | 运用基本图像的变换
Many functions can be seen as transformations of known graphs. Starting from y = f(x), the graph of y = f(x – a) is shifted right by a units; y = f(x) + b is shifted up by b. A vertical stretch by factor k gives y = k f(x), and a horizontal stretch by factor 1/k gives y = f(kx). Reflections include y = -f(x) across the x-axis and y = f(-x) across the y-axis.
许多函数可以视为已知图像的变换。从 y = f(x) 出发：y = f(x – a) 向右平移 a 个单位；y = f(x) + b 向上平移 b 个单位。竖直拉伸 k 倍得到 y = k f(x)；水平伸缩 1/k 倍得到 y = f(kx)。反射变换包括 y = -f(x)（关于 x 轴）和 y = f(-x)（关于 y 轴）。
For the graph of y = |f(x)|, first sketch f(x), then reflect the portions below the x-axis upward. For y = f(|x|), keep the right half of f(x) and mirror it to the left, ignoring the original left half. These modulus transformations are common in IB questions.
对于 y = |f(x)|，先画出 f(x)，然后将 x 轴下方的部分向上翻折。对于 y = f(|x|)，保留 f(x) 的右半部分并镜像到左侧，忽略原来的左半部分。这类绝对值变换在IB题目中很常见。
9. Highlight Special Functions: Exponentials and Logarithms | 突出特殊函数：指数与对数
Exponential functions y = aˣ (a > 1) pass through (0, 1), increase rapidly, and have the x-axis as a horizontal asymptote to the left. The natural exponential y = eˣ is its own derivative. Logarithmic functions y = logₐ x pass through (1, 0), increase slowly, and have the y-axis as a vertical asymptote. They are mirror images of exponentials across the line y = x when the same base is used.
指数函数 y = aˣ（a > 1）过点 (0, 1)，递增迅速，左侧以 x 轴为水平渐近线。自然指数 y = eˣ 的导数等于自身。对数函数 y = logₐ x 过点 (1, 0)，递增缓慢，以 y 轴为垂直渐近线。在相同底数下，它们与指数函数关于直线 y = x 对称。
Pay special attention to the domain: exponential functions have domain ℝ and range (0, ∞), while logarithmic functions have domain (0, ∞) and range ℝ. Also note the effect of the base: for 0 < a < 1, exponential and logarithmic functions are decreasing.
特别注意定义域：指数函数定义域为 ℝ，值域为 (0, ∞)；对数函数定义域为 (0, ∞)，值域为 ℝ。还要注意底数的影响：当 0 < a < 1 时，指数函数和对数函数都是递减的。
10. Solve Equations Graphically by Intersections | 通过交点图像法解方程
To sketch f(x) and g(x) on the same axes, the x-coordinates of their intersection points are the solutions to f(x) = g(x). This graphical method is useful when an algebraic solution is complex. For example, the equation x² = 2ˣ has solutions that can be approximated by finding the intersections of y = x² and y = 2ˣ.
在同一坐标系中画出 f(x) 和 g(x)，它们交点的 x 坐标就是方程 f(x) = g(x) 的解。当代数求解复杂时，这种图像法非常有效。例如，方程 x² = 2ˣ 可以通过找到 y = x² 与 y = 2ˣ 的交点来近似求解。
When using a graphing calculator, set an appropriate viewing window and use the intersection function to obtain accurate values. Always verify that the displayed window includes all relevant asymptotic and turning-point behaviour.
使用图形计算器时，设置合适的显示窗口，并利用交点功能获得精确值。务必确认当前窗口包含了所有相关的渐近线行为和转向点。
11. Produce a Complete Labelled Sketch | 完成带完整标注的草图
A final sketch must include: axes labelled with x and y; intercepts; extrema; inflection points; asymptotes (dashed lines); and any relevant intersection points. Use scales that reflect the important features of the graph. For rational functions, check the behaviour near vertical asymptotes using one-sided limits.
最终草图必须包含：标有 x 和 y 的坐标轴；截距；极值点；拐点；渐近线（虚线）；以及所有相关交点。使用能反映图像重要特征的刻度。对于有理函数，通过单侧极限检查垂直渐近线附近的走势。
It is also helpful to include a small sign table or a short note stating the domain, range, and critical values. A clear, well-organised sketch communicates understanding better than a dense set of disconnected points.
此外，附上简短的符号表或说明定义域、值域和关键值也有帮助。一张清晰、有条理的草图比一堆分散的点更能体现理解程度。
12. Common Pitfalls and Exam Tips | 常见误区与应试技巧
Students often forget to exclude points where the function is undefined, resulting in a graph that passes through a hole or a vertical asymptote. Another common error is drawing a smooth curve through points without checking the derivative, mistaking a cusp for a stationary point. Always differentiate the function before sketching.
学生常常忘记排除无定义点，导致图像穿过空洞或垂直渐近线。另一个常见错误是不检验导数就直接用平滑曲线连接各个点，把尖点误认为驻点。绘图前务必先求导。
In IB exams, marks are awarded for specific features: correct shape, correct intercepts, correct asymptotes, and correct coordinates of turning points. Use a ruler for straight asymptotes, label coordinates precisely, and write the equation of the function near the graph. Practise with past papers to build speed and accuracy.
在IB考试中，分数按具体特征分配：形状正确、截距正确、渐近线正确、转向点坐标正确。画渐近线时使用直尺，精确标注坐标，并在图像旁写下函数方程。通过练习历年真题提高速度和准确度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导