📚 IGCSE Edexcel Maths 068: Graphs of Functions and Transformations | IGCSE 爱德思数学 068：函数图像与变换
Graphs of functions form a cornerstone of the Edexcel IGCSE Mathematics syllabus. In this topic, you learn to interpret, sketch and transform a variety of functions, including quadratic, cubic, reciprocal and exponential curves. Mastering these transformation rules — translations, stretches and reflections — will help you tackle graph sketching questions, solve equations graphically and understand the behaviour of functions.
函数图像是 Edexcel IGCSE 数学大纲的核心内容。在本专题中，你将学习如何解读、绘制和变换各种函数图像，包括二次、三次、倒数和指数曲线。掌握平移、拉伸和反射等变换规律，将帮助你应对函数草图绘制、利用图像解方程以及理解函数行为等问题。
1. What Is a Function? | 什么是函数？
ecking whether a graph represents a function can be done using the vertical line test: if a vertical line intersects the graph more than once, the relation is not a function.
函数是一种规则，它为每一个输入（x 值）指定唯一的输出（y 值）。常用 f(x)、g(x) 或 h(x) 表示。例如，f(x) = x² - 3x + 2 是一个二次函数。判断图像是否表示一个函数可用垂直线检验：如果一条垂直线与图像不止一个交点，那么这个关系就不是函数。
The domain of a function is the set of all possible x-values, while the range is the set of all possible y-values the function can produce. In IGCSE, you will often work with unrestricted parabolic, cubic or exponential graphs, whose domains are all real numbers.
函数的定义域是所有可能的 x 值集合，值域是函数能产生的所有可能 y 值集合。在 IGCSE 中，经常处理不受限制的抛物线、三次曲线或指数曲线，它们的定义域都是全体实数。
2. Drawing Graphs of Basic Functions | 基本函数的图像绘制
You must be able to recognise and sketch the shapes of fundamental functions. These include:
你需要能够识别并绘制基本函数的形状。包括：
- Linear: y = mx + c — a straight line.
- Quadratic: y = x² — a smooth U-shaped parabola, vertex at (0,0).
- Cubic: y = x³ — an S-shaped curve passing through the origin.
- Reciprocal: y = 1/x — a hyperbola with asymptotes x = 0 and y = 0.
- Exponential growth: y = aˣ (a > 1) — rising curve, horizontal asymptote y = 0.
- 线性函数：y = mx + c —— 一条直线。
- 二次函数：y = x² —— 平滑的 U 形抛物线，顶点在 (0,0)。
- 三次函数：y = x³ —— S 形曲线，经过原点。
- 倒数函数：y = 1/x —— 双曲线，渐近线为 x = 0 和 y = 0。
- 指数增长：y = aˣ (a > 1) —— 上升曲线，水平渐近线 y = 0。
Always plot a few key points when sketching. For quadratics, find the vertex; for cubics, note the inflection at the origin; for reciprocals, remember that the curve approaches but never touches the axes.
绘制草图时务必标出几个关键点。对于二次函数，找出顶点；对于三次函数，注意原点处的拐点；对于倒数函数，记住曲线趋近坐标轴但从不接触。
3. Understanding f(x) Notation | 理解 f(x) 记号
- 1, then f(2) = 5. You also need to interpret f(x) = 0 as the x-intercepts of the graph.
函数记号用于清楚地表示代入求值和变换。f(2) 表示将 x = 2 代入函数式。例如，若 f(x) = 3x - 1，则 f(2) = 5。你还需要将 f(x) = 0 理解为图像与 x 轴的交点。
When a question states “the graph of y = f(x) is shown” (a) or solve f(x) = k. Always relate the algebraic expression to coordinates on the curve.
当题目给出“y = f(x) 的图像如图所示”时，可能会要求找出 f(a) 或解方程 f(x) = k。始终将代数表达式与曲线上的坐标对应起来。
4. Vertical Translations: y = f(x) + a | 垂直平移：y = f(x) + a
graph vertically. For y = f(x) + a, the graph moves up by a units if a > 0, and down by |a| f(x) = x², then y = x² + 3 is the parabola moved 3 units upward; its vertex becomes (0,3).
在 f(x) 上加上常数 a 将使整个图像垂直移动。对于 y = f(x) + a，如果 a > 0，图像向上移动 a 个单位；如果 a < 0，图像向下移动 |a| 个单位。例如，若 f(x) = x²，那么 y = x² + 3 是抛物线向上平移 3 个单位；其顶点变为 (0,3)。
Key features such as intercepts and turning points also shift vertically by the same amount. Remember that the shape of the graph does not change — only its vertical position.
截距、转折点等关键特征同样垂直平移相同的量。记住图像的形状不会改变——只有纵向位置发生变化。
5. Horizontal Translations: y = f(x + a) | 水平平移：y = f(x + a)
y. y = f(x + a) shifts the curve to the left by a units when a > 0, and to the right by |a| lt; 0. Be cautious: the sign inside the bracket is opposite to the direction of the shift.
用 (x + a) 替换 x 会使图像水平移动。y = f(x + a) 当 a > 0 时将曲线向左平移 a 个单位，当 a < 0 时向右平移 |a| 个单位。注意：括号内的符号与移动方向相反。
Example: f(x) = √x, then f(x - 4) = √x - 4 moves the graph 4 units to the right. The starting point shifts from (0,0) to (4,0). Practice this carefully, as it is a common examination pitfall.
例子：f(x) = √x，那么 f(x - 4) = √x - 4 将图像向右平移 4 个单位。起点从 (0,0) 移到 (4,0)。仔细练习这一变换，这是考试中的常见陷阱。
6. Vertical Stretches: y = a f(x) | 垂直拉伸：y = a f(x)
essed towards the x-axis. For example, y = 2 sin x doubles the amplitude of the sine wave.
将整个函数乘以常数 a 将产生垂直拉伸。如果 a > 1，图像沿 y 轴方向拉伸，远离 x 轴；如果 0 < a < 1，则向 x 轴压缩。例如，y = 2 sin x 将正弦波的振幅加倍。
Points on the x-axis remain fixed, because any y-coordinate of zero multiplied by a is still zero. This transformation also affects maximum and minimum values and any intercept with the y-axis.
x 轴上的点保持不变，因为任何零纵坐标乘以 a 仍为零。这个变换还会影响函数的最大值、最小值以及与 y 轴的截距。
7. Horizontal Stretches: y = f(ax) | 水平拉伸：y = f(ax)
mation is inversely proportional: y = f(2x) squeezes the graph to half its original width.
用 ax 替换 x 将水平拉伸图像。如果 a > 1，图像沿 x 轴方向压缩，向 y 轴靠拢；如果 0 < a < 1，则远离 y 轴方向拉伸。注意这个变换是反比的：y = f(2x) 将图像宽度压缩为原来的一半。
lved. When dealing with horizontal stretches, it often helps to examine how the x-intercepts change.
例如，y = sin 2x 的频率是 y = sin x 的两倍，因此周期减半。处理水平拉伸时，考察 x 轴截距如何变化往往很有帮助。
8. Reflections: y = -f(x) and y = f(-x) | 反射：y = -f(x) 与 y = f(-x)
required at IGCSE level. The graph of y = -f(x) is a reflection of y = f(x) in the x-axis — ign. The graph of y = f(-x) is a reflection in the y-axis, where all x-values change sign.
IGCSE 阶段要求掌握两种简单的反射。y = -f(x) 的图像是 y = f(x) 关于 x 轴的反射——所有 y 值改变符号。y = f(-x) 的图像是关于 y 轴的反射，所有 x 值改变符号。
ip it left to right. Combinations of reflections can produce rotations, so always apply step by step.
实用方法：绘制 y = -f(x) 时，将原曲线上下颠倒。绘制 y = f(-x) 时，将其左右翻转。反射的组合可以产生旋转效果，因此务必逐步应用变换。
9. Combining Transformations | 组合变换
In many Edexcel IGCSE questions, you must apply more than one transformation. A typical task might be: “Describe the transformation that maps f(x) to 2f(x - 1) + 3”. Always work from the inside out: first horizontal shift right by 1, then vertical stretch by factor 2, finally vertical shift up by 3.
在许多 Edexcel IGCSE 题目中，你需要应用多个变换。典型题目可能是：“描述将 f(x) 映射为 2f(x - 1) + 3 的变换。”始终由内向外进行：先向右水平平移 1，然后以因子 2 垂直拉伸，最后向上垂直平移 3。
A useful checklist:
| Order | Transformation |
|---|---|
| 1 | Horizontal translation inside the bracket |
| 2 | Horizontal stretch/reflection in y-axis |
| 3 | Vertical stretch/reflection in x-axis |
| 4 | Vertical translation |
实用顺序清单：
| 顺序 | 变换 |
|---|---|
| 1 | 括号内的水平平移 |
| 2 | 水平拉伸 / 关于 y 轴反射 |
| 3 | 垂直拉伸 / 关于 x 轴反射 |
| 4 | 垂直平移 |
Following this order ensures the correct final graph. Always check the position of key points, such as turning points or intercepts, after each intermediate step.
按此顺序可保证最终图像正确。在每一个中间步骤之后，务必检查转折点、截距等关键点的位置。
10. Solving Equations Graphically | 利用图像解方程
ically, plot both curves on the same axes; the x-coordinates of their intersections are the solutions. This is especially useful for equations that are difficult to handle algebraically, such as cubic equations or intersections of a line and a curve.
图像提供了求解方程的直观方法。要在图像上解 f(x) = g(x)，将两条曲线画在同一坐标系中；它们交点的 x 坐标即为解。这对于难以代数求解的方程（如三次方程或直线与曲线的交点）尤其有用。
grid, and hence find approximate solutions. The solutions are read from the intersection points, so accuracy in plotting and reading scales is essential.
题目通常会引导你：先画出 y = x³ - 3x，然后在同一格网上画出 y = x + 1，从而找出近似解。解由交点读出，因此绘图和读取刻度的准确性非常关键。
11. Using Graphs to Find Gradients and Turning Points | 利用图像求斜率和转折点
Although you will later study differentiation, graphs can approximate gradients of curves. To estimate the gradient at a point, draw a tangent to the curve, then calculate its slope using rise/run. This skill is tested in IGCSE Edexcel when exact algebraic methods are not available.
虽然稍后会学习微分，但图像可以估算曲线的斜率。要估算某点的斜率，在该点处作出曲线的切线，然后用纵差除以横差计算其斜率。在没有精确代数方法可用时，IGCSE Edexcel 会考查这项技能。
Turning points (maximum or minimum points) are where the gradient is zero — the tangent is horizontal. On a sketched graph, you can locate the coordinates of turning points visually and then verify using symmetry or substitution.
转折点（最大值或最小值点）是斜率为零的地方——切线水平。在草图上，可以用肉眼确定转折点的坐标，然后用对称性或代入法加以验证。
12. Common Exam Mistakes and How to Avoid Them | 常见考试错误与避免方法
hift to the right. Remember: inside the bracket, addition moves left. Another common error is drawing the reciprocal graph with the wrong asymptotic behaviour — it must approach, but never touch, the axes.
学生经常混淆水平平移的方向，误以为 y = f(x + 2) 是向右移动。记住：括号内是加号时向左移动。另一个常见错误是绘制倒数函数图像时渐近行为不正确——它必须趋近坐标轴但永不接触。
When combining transformations, wrongly sequencing a stretch before a translation can wreck the position of the graph. Use the order: horizontal shift, horizontal stretch, vertical stretch, vertical shift. Also, check your sketch by testing one simple coordinate to ensure it lies on the transformed curve.
在组合变换时，错误地先拉伸再平移会破坏图像的位置。请使用顺序：水平移动、水平拉伸、垂直拉伸、垂直移动。同时，通过测试一个简单的坐标来检查你的草图，确保它在变换后的曲线上。
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