📚 Graphing Functions | 函数绘图
In IB Mathematics, graphing functions is more than just plotting points – it is a visual language that reveals the behaviour, roots, and transformations of algebraic relationships. Mastering this skill allows you to interpret real-world situations, solve equations graphically, and justify your reasoning with clear, accurate sketches. This article covers the essential concepts of function graphs, from intercepts and asymptotes to transformations and technology, providing a comprehensive revision resource for both Analysis & Approaches (AA) and Applications & Interpretation (AI) students.
在 IB 数学中，函数绘图不仅仅是描点——它是一种视觉语言，可以揭示代数关系的行为、根和变换。掌握这一技能，你就能解释现实世界的情境、通过图像解方程，并用清晰准确的草图来证明你的推理。本文涵盖函数图像的核心概念，从截距和渐近线到变换与技术，为 AA 和 AI 学生提供全面的复习资源。
1. Introduction to Function Graphs | 函数图像简介
x, f(x)) that satisfy the rule y = f(x). It provides an immediate overview of the function' nd key features, rather than plotting every coordinate. Understanding how to read and create these graphs is foundational to calculus, statistics, and modelling topics.
函数图像是所有满足 y = f(x) 的有序对 (x, f(x)) 的视觉表示。它直观地展示了函数的定义域、值域、增减区间和特殊点。在 IB 考试中，你需要在草图里画出正确的整体形状和关键特征，而不是一一描点。学会读懂并绘制这些图像是微积分、统计和建模主题的基础。
2. Key Features of Graphs | 图像的关键特征
When analysing a function graph, always identify these features: x- and y-intercepts, turning points (maxima and minima), points of inflection, asymptotes, intervals where the function is positive or negative, and where it is increasing or decreasing. These elements tell the story of the function. A well-drawn sketch must label these clearly, often using simple calculations from the equation.
分析函数图像时，务必识别以下特征：x 轴截距和 y 轴截距、转折点（极大值和极小值）、拐点、渐近线、函数为正或负的区间，以及增减区间。这些元素讲述了函数的故事。一张好的草图必须清晰标出这些信息，通常可以利用方程做简单计算得到。
3. Intercepts and Zeros | 截距与零点
e, or several x-intercepts. On the graph, these are the points where the curve crosses or touches the x-axis. Multiplicity of a root affects how the graph behaves near that intercept: a simple root crosses the axis, while a double root touches it and turns.
y 轴截距发生在 x = 0 时；只需计算 f(0) 得到点 (0, f(0))。x 轴截距也称为零点或根，通过解方程 f(x) = 0 得到。一个函数可以有零个、一个或多个 x 截距。在图像上，这些就是曲线穿过或接触 x 轴的点。根的重数会影响图像在该截距附近的行为：单根穿过坐标轴，而二重根则接触坐标轴并转向。
ero (touches and turns). Always factorise polynomial functions to identify multiplicities.
例如，在 f(x) = (x − 1)(x + 2)² 中，x = 1 是单根（穿过），x = −2 是二重根（接触并转向）。请始终对多项式函数进行因式分解以识别重数。
4. Asymptotes | 渐近线
oaches infinity or certain critical values. Vertical asymptotes occur where the function is undefined due to division by zero, provided the numerator isn' are found by comparing degrees of numerator and denominator in rational functions. Oblique (slant) asymptotes appear when the degree of the numerator is exactly one more than the denominator.
渐近线是当 x 趋近于无穷大或某些临界值时图像无限接近但永不相交的直线。垂直渐近线出现在函数因分母为零而无定义的地方（前提是分子不同时为零）。水平渐近线描述的是当 x → ±∞ 时的末端行为，可通过比较有理函数分子与分母的次数来确定。斜渐近线出现在分子的次数恰好比分母大 1 时。
ptotes as dashed lines on your sketch. For logarithmic functions, vertical asymptotes exist at the boundary of the domain.
对于 f(x) = 1/(x − 3)，垂直渐近线为 x = 3，水平渐近线为 y = 0。绘图时务必用虚线标明渐近线。对于对数函数，垂直渐近线存在于定义域的边界处。
5. Symmetry: Even and Odd Functions | 对称性：偶函数与奇函数
nising symmetry reduces the work needed to sketch a graph and helps verify the correctness of your sketch.
如果对于定义域内所有 x 都有 f(−x) = f(x)，则该函数是偶函数；其图像关于 y 轴对称。常见的例子有 f(x) = x² 和 f(x) = cos x。如果 f(−x) = −f(x)，则该函数是奇函数；其图像关于原点具有 180° 旋转对称性。典型奇函数包括 f(x) = x³ 和 f(x) = sin x。识别对称性可以减少绘图工作量，并有助于验证草图的正确性。
6. Transformations of Graphs | 图像的变换
x-axis is y = −f(x); in the y-axis, y = f(−x). Stretches: y = a f(x) stretches vertically by factor a; y = f(bx) compresses horizontally by factor 1/|b| Always apply transformations in the order: stretches/reflections first, then translations.
图像变换让你在掌握母图后可以快速绘制相关函数。主要类型有平移、对称和伸缩。对 y = f(x) + d，图像垂直移动 d 个单位。对 y = f(x + c)，沿水平方向移动 −c 个单位（若 c > 0 左移）。关于 x 轴的反射为 y = −f(x)；关于 y 轴的反射为 y = f(−x)。伸缩：y = a f(x) 垂直伸缩 a 倍；y = f(bx) 水平压缩 1/|b| 倍。应用变换时始终按照先伸缩/反射、再平移的顺序进行。
7. Graphing Polynomial Functions | 多项式函数的图像
h, continuous graphs. The degree n determines the maximum number of turning points (n − 1) and the long-run behaviour: if the leading coefficient aₙ is positive and n is even, the graph rises on both ends; if odd, it falls on the left and rises on the right (with positive aₙ). Factorised form reveals the x-intercepts and their multiplicities. Use the sign of the leading coefficient and multiplicity to sketch the shape between intercepts.
多项式函数形式为 f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀，其图像光滑、连续。次数 n 决定了转折点的最大个数（n − 1）和远端行为：若首项系数 aₙ 为正且 n 为偶数，图像两端均上升；若 n 为奇数，则左端下降、右端上升（当 aₙ 为正时）。因式分解形式可以揭示 x 截距及其重数。利用首项系数符号和重数可以画出截距之间的走向。
t x = −1 means the graph touches the x-axis there, while the simple root at x = 2 crosses.
例如，f(x) = −(x + 1)²(x − 2) 是三次函数（次数 3），首项系数为负，因此左端上升、右端下降。在 x = −1 处的二重根意味着图像在该点接触 x 轴，而在 x = 2 处的单根穿过 x 轴。
8. Graphing Rational Functions | 有理函数的图像
ter cancellation). Determine horizontal or oblique asymptotes based on degrees. Plot x- and y-intercepts, and check values near asymptotes to see whether the graph approaches +∞ or −∞ on each side.
有理函数是两个多项式之比。它们的图像通常包含渐近线，并在有公因式约去时产生可去间断点（洞）。绘图时，先因式分解分子和分母，识别公因式（表示洞），然后令约简后的分母为零求得垂直渐近线。根据次数关系确定水平或斜渐近线。标出 x 轴和 y 轴截距，并在渐近线附近取值以判断图像在每一侧是趋近于 +∞ 还是 −∞。
9. Graphing Exponential and Logarithmic Functions | 指数与对数函数的图像
The basic exponential function is f(x) = aˣ mptote at y = 0, and grows (a > 1) or decays (0 < a < 1). The natural exponential f(x) = eˣ a > 1. Transformations of these graphs follow the standard rules; for example, f(x) = 2e⁻ˣ + 3 shifts the decay curve up by 3 and reflects it in the y-axis.
基本的指数函数为 f(x) = aˣ，其中 a > 0 且 a ≠ 1。它的图像总是经过点 (0,1)，有一条水平渐近线 y = 0，并且当 a > 1 时呈增长型，当 0 < a < 1 时呈衰减型。自然指数函数 f(x) = eˣ 经常使用。对数函数 f(x) = logₐ x 是指数函数的反函数；其图像有垂直渐近线 x = 0，经过点 (1,0)，且当 a > 1 时单调递增。这些图像的变换遵循标准规则；例如，f(x) = 2e⁻ˣ + 3 将衰减曲线向上平移 3 个单位并作 y 轴反射。
10. Graphing Trigonometric Functions | 三角函数的图像
of π/2. When graphing transformations like f(x) = A sin(Bx + C) + D, remember: amplitude = |A|, period = 2π/|B| aw one full wave of the basic shape, then apply the transformations to place it correctly.
正弦、余弦和正切函数是周期函数。f(x) = sin x 和 f(x) = cos x 的周期为 2π，振幅为 1，定义域为全体实数；f(x) = tan x 的周期为 π，并在 π/2 的奇数倍处有垂直渐近线。绘制如 f(x) = A sin(Bx + C) + D 的变换图像时，请记住：振幅 = |A|，周期 = 2π/|B|，相位平移 = −C/B（水平），垂直平移 = D。先画出基本形状的一个完整波，再运用变换放置到正确位置。
11. Using Technology for Graphing | 使用技术绘图
A graphical display calculator (GDC) is a powerful tool in IB examinations. Use it to visualise functions, find intercepts, intersections, and turning points. However, GDC screens can be misleading if the viewing window is poorly chosen. Always adjust the window settings to capture all relevant features. When required to sketch by hand, the GDC helps you check your work. In Paper 2 (AA and AI), you are expected to use technology effectively, but understanding the underlying mathematics remains essential.
图形计算器（GDC）是 IB 考试中的强大工具。你可以用它直观地观察函数、寻找截距、交点和转折点。然而，如果观察窗口选择不当，GDC 屏幕可能会产生误导。请务必调整窗口设置以捕捉所有相关特征。当需要手绘草图时，GDC 也能帮助你检查结果。在 Paper 2（AA 和 AI）中，你需要有效使用技术，但对底层数学原理的理解依然至关重要。
12. Common Mistakes and Tips | 常见错误与提示
her. Always check symmetry first, as it can highlight errors. When transforming, apply stretches before translations. Finally, never simply copy a GDC screen onto paper; you must present a clear, annotated sketch with the correct shape and all features clearly marked.
许多学生因为忘记标注坐标轴、渐近线或关键点而失分。另一些学生容易画错函数形状，特别是在垂直渐近线附近，其中一侧可能趋于 +∞，另一侧趋于 −∞。务必先检查对称性，这能帮你发现错误。进行变换时，先做伸缩再做平移。最后，绝不要把 GDC 屏幕简单抄到纸上；你必须画出一幅清晰、有标注的草图，形状正确，所有特征清楚呈现。
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