Curve Sketching | 曲线描绘

📚 Curve Sketching | 曲线描绘

In AQA A-Level Mathematics, curve sketching is the process of drawing a clear, labelled diagram that shows the main features of a function y = f(x). A good sketch does not need an accurate scale, but it must show roots, intercepts, turning points, asymptotes, symmetry and end behaviour clearly.

在 AQA A-Level 数学中，曲线描绘是为函数 y = f(x) 画出一张清晰、带标注的示意图的过程。一张好的草图不要求精确比例，但必须清楚展示根、截距、驻点、渐近线、对称性和端部行为。

1. What is Curve Sketching? | 什么是曲线描绘？

A curve sketch is a visual summary of the most important information about a function. Before drawing, you should identify the key features: where the curve crosses the axes, where it turns or flattens, where it approaches an asymptote, and what happens as x becomes very large or very small.

曲线草图是函数最重要信息的直观总结。在绘图前，你应当识别关键特征：曲线在哪里穿过坐标轴、在哪里转向或变平、在哪里接近渐近线，以及当 x 非常大或非常小时函数的表现。

You are not expected to plot points one by one. Instead, use calculus and algebra to find the features that define the shape, then join them with a smooth curve.

你不需要一点一点地描点。相反，应使用微积分和代数找出决定曲线形状的特征，然后用平滑曲线将它们连接起来。

2. Domain and Intercepts | 定义域与截距

Start with the domain: the set of x-values for which f(x) is defined. For rational functions, exclude values that make the denominator zero; for logarithmic functions, the argument must be positive; for square roots, the radicand must be non-negative.

从定义域开始：即使得 f(x) 有定义的所有 x 值。对于有理函数，要排除使分母为零的值；对于对数函数，真数必须为正；对于平方根，被开方数必须非负。

The x-intercepts are found by solving f(x) = 0. The y-intercept is found by evaluating f(0), provided x = 0 is in the domain.

令 f(x) = 0 可以求出 x 轴截距；若 x = 0 在定义域内，计算 f(0) 可以得到 y 轴截距。

x-intercepts: solve f(x) = 0; y-intercept: y = f(0)

3. Symmetry | 对称性

Symmetry can reduce the work. If f(-x) = f(x), the function is even and its graph is symmetric about the y-axis. If f(-x) = -f(x), the function is odd and its graph has 180° rotational symmetry about the origin.

对称性可以减少绘图工作量。如果 f(-x) = f(x)，函数是偶函数，图像关于 y 轴对称。如果 f(-x) = -f(x)，函数是奇函数，图像关于原点具有 180° 旋转对称性。

Check symmetry early, for example for polynomial or trigonometric functions. A curve that is even only needs to be sketched on one side and then reflected.

对于多项式或三角函数，尽早检查对称性。偶函数只需要画出一侧的曲线，再反射到另一侧即可。

4. Asymptotes: Vertical, Horizontal and Oblique | 渐近线：垂直、水平与斜渐近线

Vertical asymptotes occur where the function tends to ±∞ as x approaches a finite value, often from a zero denominator in a rational expression. Check whether the factor cancels first; if it cancels, there may be a hole rather than an asymptote.

垂直渐近线出现在当 x 趋于某个有限值时函数趋于 ±∞ 的位置，通常由有理式中分母为零引起。先检查该因式是否可约；如果可以约去，可能是一个空心点而不是渐近线。

Horizontal asymptotes describe end behaviour when f(x) tends to a finite constant as x → ±∞. For a rational function, compare the degrees of the numerator and denominator: if the degrees are equal, y = leading coefficient ratio; if the denominator has higher degree, y = 0.

水平渐近线描述当 x → ±∞ 时 f(x) 趋于某个有限常数的端部行为。对于有理函数，比较分子与分母的次数：如果次数相等，y 等于首项系数之比；如果分母次数更高，则 y = 0。

If the degree of the numerator is exactly one more than the degree of the denominator, there is an oblique asymptote. Divide the numerator by the denominator; the quotient gives the line y = mx + c as the asymptote.

如果分子的次数恰好比分母次数高一次，则存在斜渐近线。用分子除以分母，商式给出的直线 y = mx + c 就是斜渐近线。

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）