📚 Sketching and Transforming Function Graphs | 函数图像的绘制与变换
A graph is a visual representation of a function. In A-level mathematics, you are expected not only to read graphs, but also to sketch them accurately, showing key features such as intercepts, asymptotes, turning points and end behaviour. Transformations allow you to move, reflect or stretch a known base graph so that you can sketch a new function quickly without plotting many points.
函数图像是函数的可视化表现。在 A-level 数学中，你不仅要会读图，更要会准确地绘草图，标出截距、渐近线、极值点和端点行为等关键特征。利用变换，你可以在已知基本图的基础上平移、反射或伸缩，从而快速画出新函数的图像，无需逐个描点。
1. Why Sketching Matters | 为什么绘制函数图像很重要
In exams, sketch graphs are used to solve inequalities, locate roots, and understand limits. A well-drawn sketch shows the general shape and all essential features, but it does not have to be perfectly scaled. The goal is to communicate the behaviour of the function at key points.
在考试中，草图常用于解不等式、找出根和理解极限。一张好的草图只需展示函数的大致形状和所有关键特征，并不需要精确缩放。目标是在关键点处表达出函数的行为。
2. Basic Graphs You Must Know | 必须掌握的基本函数图像
You should be able to recall and sketch the following graphs quickly: y = x, y = x², y = x³, y = 1/x, y = √x, y = eˣ, y = ln x, y = sin x, y = cos x and y = tan x. Each has a unique shape and a set of important coordinates or asymptotes.
你应该能快速回忆并画出以下函数的草图：y = x、y = x²、y = x³、y = 1/x、y = √x、y = eˣ、y = ln x、y = sin x、y = cos x 和 y = tan x。它们各自有独特的形状以及一组重要的坐标或渐近线。
| Function | 函数 | Key features | 关键特征 |
|---|---|
| y = x² | Parabola, vertex at (0,0), symmetric about y-axis | 抛物线，顶点(0,0)，关于 y 轴对称 |
| y = 1/x | Hyperbola, asymptotes x=0 and y=0 | 双曲线，渐近线 x=0 和 y=0 |
| y = √x | Half parabola, domain x ≥ 0 | 半个抛物线，定义域 x ≥ 0 |
| y = eˣ | Exponential growth, horizontal asymptote y=0, passes through (0,1) | 指数增长，水平渐近线 y=0，过点(0,1) |
| y = sin x | Periodic wave, range [-1,1], period 2π | 周期波动，值域 [-1,1]，周期 2π |
3. Vertical and Horizontal Translations | 平移变换：上下与左右平移
For y = f(x), the graph of y = f(x) + a is a vertical translation by a units: if a > 0 it moves up, if a < 0 it moves down. The graph of y = f(x + a) is a horizontal translation by -a units: y = f(x + 2) moves the graph 2 units to the left, while y = f(x - 2) moves it 2 units to the right.
对于 y = f(x)，y = f(x) + a 的图像是沿垂直方向平移 a 个单位：a > 0 时上移，a < 0 时下移。y = f(x + a) 的图像是沿水平方向平移 -a 个单位：y = f(x + 2) 向左移动 2 个单位，而 y = f(x - 2) 向右移动 2 个单位。
- y = f(x) + k: shift up by k (k > 0) | 上移 k 个单位（k>0）
- y = f(x + h): shift left by h (h > 0) | 向左平移 h 个单位（h>0）
4. Reflections | 反射变换
Replace f(x) by -f(x) to reflect the graph in the x-axis. Replace x by -x inside the function to reflect the graph in the y-axis. For example, y = -√x is the reflection of y = √x across the x-axis, and y = e⁻ˣ is the reflection of y = eˣ across the y-axis.
将 f(x) 替换为 -f(x) 会把图像沿 x 轴反射；将函数内部的 x 替换为 -x 会把图像沿 y 轴反射。例如，y = -√x 是 y = √x 关于 x 轴的反射，y = e⁻ˣ 是 y = eˣ 关于 y 轴的反射。
5. Stretches and Compressions | 伸缩变换
y = a f(x) with a > 0 gives a vertical stretch by factor a. If 0 < a < 1, the graph is vertically compressed. y = f(ax) gives a horizontal compression by factor 1/a. For example, y = 2 sin x is twice as tall as y = sin x, and y = sin 2x completes two full cycles in the interval 0 to 2π.
y = a f(x) 且 a > 0 表示沿垂直方向拉伸 a 倍；若 0 < a < 1，则图像在垂直方向被压缩。y = f(ax) 表示沿水平方向压缩到原来的 1/a。例如，y = 2 sin x 的高度是 y = sin x 的两倍，而 y = sin 2x 在 0 到 2π 内完成两个完整周期。
6. Combining Transformations | 组合变换
When several transformations are applied, the order is important. As a general rule, work on the x-coordinates first (inside the brackets), then the y-coordinates (outside the brackets). For example, to sketch y = -2f(x – 3), start with f(x), translate right by 3, stretch vertically by factor 2, then reflect in the x-axis.
当多个变换叠加时，顺序非常重要。一般原则是：先处理括号内的 x 方向变化，再处理括号外的 y 方向变化。例如，要画 y = -2f(x – 3)，先从 f(x) 开始，向右平移 3 个单位，再垂直拉伸 2 倍，最后沿 x 轴反射。
7. Transforming y = f(x) to y = a f(bx + c) + d | 从 y =
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