📚 3B Sketching Graphs | 3B 函数图形素描技巧
Graph sketching is a fundamental skill in IB Mathematics, allowing you to quickly visualise the behaviour of functions without plotting an exhaustive number of points. By identifying key features such as intercepts, asymptotes, turning points and symmetry, you can produce an accurate freehand sketch that reveals the essential character of the function.
图形素描是IB数学中的一项基本技能，让你在无需绘制大量坐标点的情况下快速直观地了解函数的行为。通过确定截距、渐近线、转折点以及对称性等关键特征，你就能画出一幅准确的手绘草图，揭示函数的本质特性。
1. Introduction to Graph Sketching | 图形素描简介
Sketching a graph is not the same as plotting a precise graph using a table of values. Instead, it relies on understanding the structure of a function and applying transformations to known parent graphs. The aim is to produce a rough but informative picture, showing where the graph crosses axes, how it behaves at extremes, and any special points like maxima or minima. In IB examinations, you are often asked to sketch a function after performing calculus operations, making this a skill that integrates several areas of the syllabus.
画草图与利用数值表绘制精确图形不同。它依赖于理解函数的结构，并将变换应用于已知的基本图形。目的是产出一幅粗略但信息丰富的图像，显示图形与坐标轴的交点、在极端值处的行为，以及极大值或极小值等特殊点。在IB考试中，你经常需要在完成微积分运算后素描函数，这使得该技能整合了课程大纲中的多个领域。
2. Recognising Parent Functions | 识别基本函数
Every graph sketch begins with a set of parent functions whose shapes you must recognise instantly. These include linear (y = x), quadratic (y = x²), cubic (y = x³), reciprocal (y = 1/x), square root (y = √x), exponential (y = eˣ), logarithmic (y = ln x), and the trigonometric functions y = sin x, y = cos x and y = tan x. The table below summarises their key characteristics.
每一幅图形素描都始于一组你必须能迅速识别的基本函数。这些包括线性函数 y = x，二次函数 y = x²，三次函数 y = x³，倒数函数 y = 1/x，平方根函数 y = √x，指数函数 y = eˣ，对数函数 y = ln x，以及三角函数 y = sin x、y = cos x 和 y = tan x。下表总结了它们的关键特征。
| Parent function | 基本函数 | Shape & domain | 形状与定义域 | Symmetry & asymptotes | 对称性与渐近线 |
|---|---|---|
| y = x² | U-shaped parabola, all real x | Even function; minimum at (0,0) |
| y = 1/x | Hyperbola, x ≠ 0 | Odd; V.A. x=0, H.A. y=0 |
| y = √x | Half-parabola, x ≥ 0 | No symmetry; starts at (0,0) |
| y = eˣ | Increasing exponential, all real x | H.A. y=0 as x → -∞ |
| y = ln x | Logarithmic growth, x > 0 | V.A. x=0 |
| y = sin x | Wave, all real x | Odd, period 2π, amplitude 1 |
Being able to recall these graphs instantly gives you a starting point for applying transformations and analysing more complicated functions. In IB, you may also encounter absolute value, rational and composite functions, all of which derive from these basic forms.
能够即刻回忆起这些图形，为你应用变换和分析更复杂的函数提供了起点。在IB考试中，你还可能遇到绝对值函数、有理函数和复合函数，它们全都源于这些基本形式。
3. Transformations: Translations | 变换：平移
A translation shifts a graph horizontally, vertically or both without changing its shape. The general form is y = f(x − h) + k, where h represents the horizontal shift (right if h > 0, left if h < 0) and k the vertical shift (up if k > 0, down if k < 0). For example, starting from y = x², the graph of y = (x − 3)² + 2 is a parabola shifted 3 units to the right and 2 units up. The vertex moves from (0,0) to (3,2).
平移是将图形水平、垂直或同时两个方向移动而不改变其形状。一般形式为y = f(x − h) + k，其中 h 表示水平移动（若 h > 0 向右，若 h < 0 向左），k 表示垂直移动（若 k > 0 向上，若 k < 0 向下）。例如，从 y = x² 出发，y = (x − 3)² + 2 的图像是一条向右平移 3 个单位、向上平移 2 个单位的抛物线。其顶点从 (0,0) 移动到 (3,2)。
4. Transformations: Stretches and Reflections | 变换：拉伸与反射
Stretches and reflections are controlled by the parameters a and b in y = a f(bx). The factor a causes a vertical stretch by factor |a|; if a is negative, the graph is also reflected in the x-axis. The factor b causes a horizontal stretch by factor 1/|b|; if b is negative, the graph is reflected in the y-axis. For instance, y = 2 sin x doubles the amplitude of the sine wave, while y = sin(3x) compresses the period to 2π/3. Combining these, y = −0.5 cos(2x) reflects the cosine graph across the x-axis, compresses it horizontally (period = π) and halves the amplitude.
拉伸和反射由y = a f(bx) 中的参数 a 和 b 控制。a 产生垂直拉伸，拉伸倍数为 |a|；若 a 为负，图形还会关于 x 轴反射。b 产生水平拉伸，拉伸倍数为 1/|b|；若 b 为负，图形关于 y 轴反射。例如，y = 2 sin x 将正弦波的振幅加倍，而 y = sin(3x) 将周期压缩至 2π/3。将这些组合起来，y = −0.5 cos(2x) 将余弦图形先关于 x 轴反射，再水平压缩（周期 = π），并将振幅减半。
5. Composite Transformations | 复合变换
When a function contains multiple transformations, the correct order matters. The recommended sequence is: first deal with horizontal shifts inside the bracket, then horizontal stretches/reflections (b), then vertical stretches/reflections (a), and finally vertical shifts (k). Written in the form y = a f(b(x − h)) + k, you should apply the horizontal translation (h) using x − h, then apply b to the argument, then multiply the output by a, and finally add k. Many mistakes occur when students apply transformations in the wrong order, especially when a negative sign is involved.
当一个函数包含多种变换时，正确的顺序十分重要。推荐的顺序是：先处理括号内的水平平移，然后处理水平拉伸/反射（b），接着是垂直拉伸/反射（a），最后是垂直平移（k）。写成 y = a f(b(x − h)) + k 的形式后，你应该先用 x − h 完成水平平移，再对自变量应用 b，然后将输出乘以 a，最后加上 k。许多错误都源于学生以错误的顺序施加变换，尤其当涉及负号时。
6. Finding Intercepts and Asymptotes | 求截距与渐近线
Intercepts are where the graph meets the axes. Set x = 0 to find the y-intercept, and solve f(x) = 0 to find x-intercepts (roots). Asymptotes are lines the graph approaches but never touches. Vertical asymptotes occur where the denominator of a rational function is zero (and the numerator is non-zero). Horizontal asymptotes are found by considering the limit of f(x) as x → ±∞. For rational functions, if the degree of the numerator is less than the degree of the denominator, y = 0 is the horizontal asymptote; if degrees are equal, it is y = a/b (leading coefficients); if the numerator’s degree is one greater, there is a slant asymptote found by polynomial division. Exponential and logarithmic functions also have horizontal and vertical asymptotes respectively.
截距是图形与坐标轴的交点。令 x = 0 求得 y-截距，解 f(x) = 0 求得 x-截距（根）。渐近线是图形趋近但永不相交的直线。垂直渐近线出现在有理函数中分母为零（且分子非零）的位置。水平渐近线通过考虑 x → ±∞ 时 f(x) 的极限求得。对于有理函数，若分子的次数低于分母的次数，则水平渐近线为 y = 0；若次数相等，则为 y = a/b（首项系数之比）；若分子的次数比分母大 1，则存在斜渐近线，可通过多项式除法求得。指数函数与对数函数也分别具有水平渐近线和垂直渐近线。
7. Using Calculus: Derivatives and Critical Points | 使用微积分：导数与临界点
Calculus provides powerful tools for precise sketching. The first derivative f ‘(x) gives the gradient of the tangent. Solve f ‘(x) = 0 to locate stationary points (maxima, minima or horizontal points of inflection). Determine the nature of these points by using the first derivative test (sign changes) or the second derivative test. Also find intervals where f ‘(x) > 0 (increasing) and f ‘(x) < 0 (decreasing). For instance, for f(x) = x³ − 3x, f '(x) = 3x² − 3 = 0 gives x = ±1. Testing signs reveals a local maximum at x = −1 and a local minimum at x = 1.
微积分为精确素描提供了强大工具。一阶导数 f ‘(x) 给出切线的斜率。解 f ‘(x) = 0 可找到驻点（极大值、极小值或水平拐点）。通过一阶导数检验（符号变化）或二阶导数检验判断这些点的性质。还要找出 f ‘(x) > 0（递增）和 f ‘(x) < 0（递减）的区间。例如，对于 f(x) = x³ − 3x，f '(x) = 3x² − 3 = 0 给出 x = ±1。检验符号表明在 x = −1 处有局部极大值，在 x = 1 处有局部极小值。
8. Concavity and Points of Inflection | 凹凸性与拐点
The second derivative f ”(x) describes the curvature of the graph. If f ”(x) > 0, the graph is concave up (cup-shaped); if f ”(x) < 0, it is concave down (cap-shaped). Points where f ''(x) = 0 or is undefined and the concavity changes are points of inflection. These may be stationary (if f '(x) = 0) or non-stationary. Identifying intervals of concavity helps you refine the shape between stationary points and approach asymptotes smoothly. In the example f(x) = x³ − 3x, f ''(x) = 6x, so concavity changes at x = 0, giving a non-stationary inflection point at (0,0).
二阶导数 f ”(x) 描述图形的弯曲方向。若 f ”(x) > 0，图形为凹向上（杯状）；若 f ”(x) < 0，则为凹向下（帽状）。使得 f ''(x) = 0 或无定义且凹凸性发生变化的点是拐点。这些拐点可以是驻点（当 f '(x) = 0）或非驻点。确定凹凸区间有助于你细化驻点之间的形状并平滑地趋近渐近线。在例子 f(x) = x³ − 3x 中，f ''(x) = 6x，因此凹凸性在 x = 0 处改变，在 (0,0) 处给出非驻点拐点。
9. Sketching Rational Functions | 有理函数素描
Rational functions are quotients of polynomials. To sketch f(x) = (x + 1)/(x − 2), first find intercepts: y-intercept at (0, −1/2), x-intercept at x = −1. Vertical asymptote at x = 2 (denominator zero). Horizontal asymptote: degrees equal (both 1), so y = 1/1 = 1. Determine behaviour near asymptotes: as x → 2⁺, f(x) → +∞; as x → 2⁻, f(x) → −∞. Check if graph crosses horizontal asymptote: solve (x+1)/(x−2) = 1 gives 1 = 1, no solution, so it does not cross. Use signs of f(x) in intervals divided by vertical asymptote and x-intercept to sketch the curve. The resulting graph is a hyperbola with two branches.
有理函数是多项式的商。要素描 f(x) = (x + 1)/(x − 2)，首先求截距：y-截距在 (0, −1/2)，x-截距在 x = −1。垂直渐近线在 x = 2（分母为零）。水平渐近线：分子和分母次数相同（均为 1 次），因此 y = 1/1 = 1。确定渐近线附近的行为：当 x → 2⁺ 时 f(x) → +∞；当 x → 2⁻ 时 f(x) → −∞。检查图形是否穿过水平渐近线：解方程 (x+1)/(x−2) = 1 得到恒等式 1 = 1，无解，故不穿过。利用被垂直渐近线与 x-截距划分的区间内 f(x) 的符号来素描曲线。最终图形为具有两支的双曲线。
10. Sketching Trigonometric Graphs | 三角函数图形素描
Trigonometric functions are periodic, so you need to identify amplitude, period, phase shift and vertical shift. For a function like y = A sin(B(x − C)) + D, the amplitude is |A|, the period is 2π/|B|, the phase shift is C (right if C > 0), and the vertical shift is D. Plot key points over one period, often using quarter-period intervals (e.g. 0, period/4, period/2, 3·period/4, period). For y = 3 sin(2x − π/2) + 1, rewrite as y = 3 sin(2(x − π/4)) + 1. Amplitude = 3, period = π, phase shift π/4 to the right, vertical shift 1 up. Sketch the sinusoidal wave with these parameters.
三角函数是周期性的，因此你需要确定振幅、周期、相移和垂直移动。对于形如 y = A sin(B(x − C)) + D 的函数，振幅为 |A|，周期为 2π/|B|，相移为 C（若 C > 0 则向右），垂直移动为 D。在一个周期内绘制关键点，通常采用四分之一周期的间隔（例如 0、周期/4、周期/2、3·周期/4、周期）。对于 y = 3 sin(2x − π/2) + 1，重写为 y = 3 sin(2(x − π/4)) + 1。振幅 = 3，周期 = π，相移向右 π/4，垂直向上移动 1。利用这些参数素描正弦波。
11. Common Mistakes and Tips | 常见错误与技巧
One frequent error is confusing horizontal transformations: remember that f(x − 2) moves the graph 2 units to the right, not left, because the substitution x → x−2 requires a larger x to achieve the same output. Another mistake is forgetting to show intercepts clearly or omitting asymptotes. Always label axes and key coordinates in IB exams to gain full marks. When using calculus, ensure you check both stationarity and concavity; a point where f ‘(x) = 0 could be a point of inflection rather than an extremum. Finally, practise sketching without a calculator to build intuition, then verify using your GDC.
一个常见错误是混淆水平变换：记住，f(x − 2) 将图形向右移动 2 个单位，而非向左，因为代入 x → x−2 需要一个更大的 x 来获得相同的输出。另一个错误是忘记清晰地显示截距或遗漏渐近线。在 IB 考试中务必标注坐标轴和关键坐标以获得满分。使用微积分时，确保你同时检验了驻点的稳定性和凹凸性；f ‘(x) = 0 的点可能是拐点而非极值点。最后，练习在不使用计算器的情况下素描以培养直觉，然后用你的图形计算器验证。
12. Summary | 总结
Graph sketching in IB Mathematics blends algebraic manipulation, function transformations and calculus. Start by recognising the parent function, apply transformations in the correct order, find intercepts and asymptotes, use derivatives to locate critical points and concavity, then combine all features into a neat diagram. With practice, you can sketch even complex functions confidently, saving time and gaining deeper insight into the behaviour of mathematical models. Always present your sketch with labelled axes, intercepts, turning points and asymptotes for a complete answer.
IB 数学中的图形素描融合了代数操作、函数变换和微积分。首先识别基本函数，按正确顺序施加变换，求出截距和渐近线，利用导数确定临界点和凹凸性，然后将所有特征整合成一幅简明的图形。通过练习，你甚至可以自信地素描复杂函数，节省时间并对数学模型的行为获得更深刻的洞见。务必在答题时呈现标有坐标轴、截距、转折点和渐近线的完整草图。
Published by TutorHao | IB 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