📚 060. Graph Transformations | 图像变换
ak down the key types of transformations, their precise vector effects, and how to apply them to functions such as f(x) = x², sin x, and cos x.
图像变换让你能在不改变函数基本形状的前提下，移动、反射或拉伸函数的图像。掌握这些变换对于IGCSE爱德思数学至关重要，因为它们是许多代数和坐标几何题目的基础。本文将逐一解析主要的变换类型、它们的向量效果，以及如何将其应用到 f(x) = x²、sin x 和 cos x 等函数上。
1. The Big Picture of Graph Transformations | 图像变换总览
ipping the graph across an axis), and stretches (pulling the graph towards or away from an axis). Each transformation can be written as a modification inside the function' , such as y = f(x) + a or y = f(ax). Understanding these forms helps you predict the graph's new position without plotting every single point.
变换 y = f(x) 图像的基本方式有四种：平移（沿水平或垂直方向移动图形）、反射（以坐标轴为镜像翻转图形）和拉伸（朝向或远离坐标轴拉伸图形）。每种变换都可以表示为函数方程内部的修改，例如 y = f(x) + a 或 y = f(ax)。理解这些形式有助于你在不逐点描图的情况下，预测图像的新位置。
At IGCSE level, you are expected to recognise and apply these transformations to both given graphs and algebraic expressions. The key is to link each transformation to its vector or scale factor and to distinguish between transformations that affect the x-coordinate and those that affect the y-coordinate.
在IGCSE阶段，你需要能够识别这些变换并将它们应用到给定的图形和代数表达式上。关键是要把每种变换与其向量或比例因子联系起来，并区分哪些变换影响 x 坐标，哪些影响 y 坐标。
2. Vertical Translations: Moving Up and Down | 垂直平移：上下移动
+ k. If k > 0, the graph shifts upwards by k units; if k < 0, it shifts downwards by |k| units. In vector terms, the translation is represented as (0, k).
垂直平移是将整个图形向上或向下移动，而不改变其形状。这通过在函数外部加上或减去一个常数来实现：y = f(x) + k。如果 k > 0，图像向上移动 k 个单位；如果 k < 0，图像向下移动 |k| 个单位。用向量表示，平移为 (0, k)。
moves the parabola up by 3 units. Every point on the original graph has its y-coordinate increased by 3, while the x-coordinate stays the same. This is one of the simplest yet most frequently tested transformations.
例如，若 f(x) = x²，那么 y = x² + 3 将抛物线向上移动 3 个单位。原图上的每一个点，其 y 坐标都增加 3，而 x 坐标保持不变。这是最简单但也最常考的变换之一。
3. Horizontal Translations: Shifting Left and Right | 水平平移：左右移动
A horizontal translation involves adding or subtracting a constant inside f a > 0, y = f(x + a) shifts the graph to the left by a units. If a < 0, say y = f(x – 2), the graph shifts to the right by 2 units. The translation vector is (–a, 0).
水平平移涉及在函数自变量内部加上或减去一个常数：y = f(x + a)。图像的移动方向与 a 的符号相反。如果 a > 0，y = f(x + a) 将图像向左平移 a 个单位。如果 a < 0，例如 y = f(x – 2)，图像则向右平移 2 个单位。平移向量为 (–a, 0)。
This opposite‑ udents. A useful tip is to think about what value of x makes the bracket zero: for y = f(x – 3), setting x – 3 = 0 gives x = 3, so the graph's 'centre' x = 3, indicating a right shift. Always test with a simple point to confirm the direction.
这种方向相反的效果常常让学生困惑。一个有用的技巧是思考：令括号内为零时 x 取何值。例如 y = f(x – 3)，令 x – 3 = 0 得 x = 3，这意味着图像的“中心”移到了 x = 3 处，表明是向右移动。不妨用一个简单点来验证方向。
4. Reflecting in the x‑axis: Flipping Vertically | 关于 x 轴的反射：垂直翻转
To reflect a graph in the x‑axis, multiply the entire function by –1: y = –f(x). This transformation changes the sign of every y‑coordinate while keeping the x‑coordinates unchanged. Points above the x‑axis move below it, and vice versa, as if the x‑axis acts as a mirror.
要关于 x 轴反射图像，将整个函数乘以 –1：y = –f(x)。这一变换改变了每个 y 坐标的符号，而 x 坐标保持不变。x 轴上方的点翻转到下方，反之亦然，就像 x 轴是一面镜子。
For instance, if f(x) = sin x, then y = –sin x produces a wave that is the upside‑ tays the same, but peaks become troughs. In exam questions, you may be asked to sketch y = –f(x) given the graph of f(x). Simply take key points (x, y) and plot (x, –y).
例如，若 f(x) = sin x，那么 y = –sin x 产生的波形是正弦曲线的倒置版本。振幅保持不变，但波峰变成了波谷。在考试题中，你可能会被要求根据 f(x) 的图像画出 y = –f(x) 的草图。只需将关键点 (x, y) 变为 (x, –y) 即可。
5. Reflecting in the y‑axis: Flipping Horizontally | 关于 y 轴的反射：水平翻转
A reflection in the y‑axis is obtained by replacing x with –x inside the function: y = f(–x). This transformation multiplies every x‑coordinate by –1, flipping the graph left‑to‑right. Points on the right of the y‑axis move to the left, while points on the left move to the right.
关于 y 轴的反射是通过将函数中的 x 替换为 –x 来实现的：y = f(–x)。这一变换将每个 x 坐标乘以 –1，使图像左右翻转。原本在 y 轴右侧的点翻到左侧，左侧的点翻到右侧。
If the original graph is symmetric about the y‑axis (an even function), then y = f(– ctly the same as y = f(x). Otherwise, the shape is mirrored. For example, the graph of y = √–x is the reflection of y = √x in the y‑axis, defined only for x ≤ 0.
如果原图像关于 y 轴对称（偶函数），那么 y = f(–x) 的图像与 y = f(x) 完全相同。否则，形状将被镜像。例如，y = √–x 的图像是 y = √x 关于 y 轴的反射，且仅定义在 x ≤ 0 的区域。
6. Vertical Stretches: Pulling Towards the x‑axis | 垂直拉伸：沿 y 轴方向拉伸
A vertical stretch scales the graph away from or towards the x‑ here a is a positive constant. If a > 1, the graph is stretched vertically by a factor of a, making it taller. If 0 < a < 1, the graph is compressed vertically, appearing squashed towards the x‑axis.
垂直拉伸是将图像沿 y 轴方向拉伸或压缩，形式为 y = a f(x)，其中 a 为正的常数。若 a > 1，图像垂直拉伸 a 倍，变得更高。若 0 < a < 1，图像垂直压缩，看起来被压向 x 轴。
In this transformation, x‑coordinates stay fixed, while every y‑coordinate is multiplied by a. The x‑ (x) = 0. This property is useful when sketching transformed graphs: the zeros do not move.
在这一变换中，x 坐标保持不变，而每个 y 坐标都乘以 a。由于 f(x) = 0 意味着 a f(x) = 0，因此 x 轴截距保持不变。这个性质在画变换后的草图时很有用：零点不会移动。
tric functions, y = 2 sin x doubles the amplitude from 1 to 2. The wave oscillates between –2 and 2 instead of – This is a typical IGCSE question where you must label the new maximum and minimum values.
对于三角函数，y = 2 sin x 将振幅从 1 加倍到 2。波形在 –2 与 2 之间振荡，而不是在 –1 与 1 之间。这是典型的IGCSE题目，要求你标出新的最大值和最小值。
7. Horizontal Stretches: Scaling Along the x‑axis | 水平拉伸：沿 x 轴方向缩放
orizontal stretch is written as y = f(bx), where b > 0. With this transformation, the x‑rdinates are divided by b. If b > 1, the graph is compressed horizontally towards the y‑ y a factor of 1/b. If 0 < b < 1, the graph is stretched horizontally away from the y‑axis. The y‑coordinates remain unchanged.
水平拉伸写作 y = f(bx)，其中 b > 0。通过这一变换，x 坐标除以 b。如果 b > 1，图像水平压缩，向 y 轴靠近，压缩因子为 1/b。如果 0 < b < 1，图像水平拉伸，远离 y 轴。y 坐标保持不变。
, making the period 4π. A common mistake is confusing the effect of b: remember that the x‑values are divided by b, so a larger b means a narrower graph.
例如，y = sin(2x) 压缩了正弦波，使其在 π 内完成一个完整周期，而不是 2π。相反，y = sin(½x) 拉伸了波形，使周期变为 4π。一个常见错误是混淆 b 的作用：记住，x 值除以 b，因此 b 越大，图像越窄。
8. Combined Transformations: Two or More Changes | 组合变换：两种或更多变化
e the bracket) first, then vertical ones (outside). For instance, to sketch y = 2 f(x + 1) – x). Shift left by 1, then stretch vertically by factor 2, and finally translate down by 3.
IGCSE爱德思常考将一个函数进行两次变换的能力。应用的顺序至关重要。当函数同时包含水平和垂直变化时，总是先进行水平变换（括号内的变化），再进行垂直变换（括号外的变化）。例如，要画 y = 2 f(x + 1) – 3 的草图，从 y = f(x) 开始。先向左平移 1 个单位，再垂直拉伸 2 倍，最后向下平移 3 个单位。
If you reverse the order, you might get a different result. For example, translating before a horizontal stretch shifts the stretch centre, changing the final graph. Examiners often award marks for correctly plotting the image of a specific point after a sequence of transformations.
如果你颠倒了顺序，可能会得到不同的结果。例如，在水平拉伸之前进行平移，会改变拉伸的中心，从而改变最终的图形。考官通常会根据连续变换后某个特定点的正确位置来给分。
Combined transformations can be written in vector form when both components are translations, but stretches and reflections must be dealt with separately. Always break the process down step by step and check with key coordinates.
当两个分量均为平移时，组合变换可以写成向量形式，但拉伸和反射必须分开处理。务必逐步分解过程，并用关键坐标进行验证。
9. Applying Transformations to Specific Functions | 将变换应用于具体函数
x. Below is a summary table showing the effect of each transformation on these functions.
要在IGCSE中取得好成绩，你需要能够对 f(x) = x²、f(x) = 1/x、f(x) = sin x 和 f(x) = cos x 等常见函数进行变换。下面是一个总结表，展示了每种变换对这些函数的影响。
| Transformation | f(x) = x² | f(x) = sin x | f(x) = 1/x |
| y = f(x) + 2 | Vertex at (0,2) | Graph oscillates around y = 2 | Asymptotes unchanged, whole curve shifted up |
| y = f(x – 1) | Vertex at (1,0) | Phase shift right by 1 unit | Vertical asymptote moves to x = 1 |
| y = –f(x) | n-shaped parabola (maximum at origin) | Wave reflected vertically | Curve reflected in x‑axis |
| y = f(–x) | No change (even function) | y = sin(–x) = –sin x, reflected in y‑axis | Curve in quadrants 2 and 4 |
| y = 3f(x) | Narrower parabola, passing through (1,3) | Amplitude 3 | y-values tripled, shape distended |
| y = f(2x) | Parabola half as wide | Period π | Graph 'squashed' horizontally |
oblems, identify the base function, then apply transformations stepwise. For instance, y = –2 sin(x – 30°) + 1 involves a vertical stretch by factor 2, a reflection in the x‑axis, a horizontal shift right by 30°, and a vertical shift up by 1.
你可以用这个表格进行快速复习。解题时，先确定基本函数，然后逐步应用变换。例如，y = –2 sin(x – 30°) + 1 包含了垂直拉伸 2 倍、关于 x 轴反射、向右平移 30°，以及向上平移 1。
10. Reading Transformations from Graphs and Equations | 从图像和方程中读出变换
Examiners frequently ask you to describe fully the transformation that maps one graph onto another. The answer must include the type of transformation (translation, reflection, or stretch) and the relevant details (vector, axis, scale factor, and direction).
考官常常要求你完整描述一个图形到另一个图形的变换。答案必须包括变换类型（平移、反射或拉伸）和相关细节（向量、轴、比例因子和方向）。
For a translation, give the column vector. For a reflection, specify the mirror line (x‑axis, y‑ stretch, state whether it is vertical or horizontal and give the scale factor. Vague answers like 'it moves up' lose marks; you must write 'translation by vector (0, 3)'.
对于平移，给出列向量。对于反射，指定镜面线（x 轴、y 轴或其他直线，如 y = x）。对于拉伸，说明是垂直还是水平拉伸，并给出比例因子。模糊的答案如“它向上移动”会丢分；你必须写成“平移，向量为 (0, 3)”。
If the question gives two algebraic forms, such as y = x² and y = (x + 4)² – 1, the answer is 'translation by vector (–4, –1)' retch, identify the factor by comparing key coordinates, e.g., the distance from the axis.
如果题目给出两个代数形式，例如 y = x² 和 y = (x + 4)² – 1，答案是“平移，向量为 (–4, –1)”。如果展示的是拉伸，可通过比较关键坐标（如到轴的距离）来确定因子。
11. Common Pitfalls and How to Avoid Them | 常见陷阱及避免方法
transformations to the wrong coordinate. Remember that y = f(x + 2) subtracts 2 from the x‑coordinate of every point, yet many students add 2. Use the 'inside opposite' rule: if the sign inside the bracket is plus, shift left; if minus, shift right.
一个常见错误是将水平变换应用到错误的坐标上。记住 y = f(x + 2) 是将每个点的 x 坐标减去 2，但许多学生却加上 2。牢记“内部相反”规则：若括号内为正号，则向左平移；若为负号，则向右平移。
ntal stretch, and always do horizontal changes before vertical ones. For example, y = f(2x – 4) + 1 must first be written as y = f(2(x – followed by a horizontal compression by factor 1/2, and finally a vertical shift up by 1.
另一个陷阱是弄错组合变换的顺序。先进行水平平移再进行水平拉伸，并且始终在垂直变换之前完成水平变换。例如，y = f(2x – 4) + 1 必须先写成 y = f(2(x – 2)) + 1；水平方向先向右平移 2，然后水平压缩 1/2 倍，最后垂直向上平移 1。
forget that stretches and reflections keep some points fixed. In a vertical stretch, the x‑intercepts stay put; in a horizontal stretch, the y‑intercept may remain if b does not change the function's value at x=0. Double‑check these invariants to speed up sketching.
学生们还常忘记拉伸和反射会固定某些点。垂直拉伸时，x 轴截距不动；水平拉伸时，若 b 不改变 x=0 处的函数值，则 y 轴截距可能保持不变。复核这些不变量可以加快画草图的速度。
12. Exam‑Style Practice and Final Tips | 考试风格练习与最终提示
To master graph transformations, practise with past paper questions that mix translations, stretches, and reflections. Start by identifying the basic function, then list the transformations in correct order. Always sketch the intermediate steps lightly to avoid errors.
要掌握图像变换，请练习真题中混合了平移、拉伸和反射的题目。首先识别基本函数，然后按正确顺序列出变换。始终轻轻画出中间步骤，以避免错误。
In the exam, if you are given the graph of y = f(x) and asked to draw y = – l stretch by factor 2 (replace x with ½x makes graph twice as wide), then reflect in the x‑axis (multiply y by – d finally shift up by 2. Label clearly the images of labelled points to secure full marks.
在考试中，如果给出 y = f(x) 的图像，并要求画出 y = –f(½x) + 2，请系统性地操作：首先水平拉伸 2 倍（用 ½x 替换 x 使图像变为两倍宽），然后关于 x 轴反射（y 乘以 –1），最后向上平移 2。清楚标出标记点的像，以确保满分。
Finally, memorise the standard vector forms: vertical translation (0, k), horizontal translation (–a, 0). For stretches, use 'stretch parallel to y‑axis scale factor a' or 'stretch parallel to x‑axis scale factor 1/b'. Precise terminology gains you marks and shows examiners you understand the underlying concepts.
最后，记住标准向量形式：垂直平移 (0, k)，水平平移 (–a, 0)。对于拉伸，使用“平行于 y 轴拉伸，比例因子为 a”或“平行于 x 轴拉伸，比例因子为 1/b”。准确的术语能让你得分，并向考官表明你理解了基本概念。
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