📚 A-Level Maths: Single Transformation of Curves | A-Level数学：单一函数图像变换精讲
Graph transformations form one of the core building blocks in A-Level mathematics, enabling you to manipulate the position, shape, and orientation of curves with precision. This article focuses on single transformations—translation, stretch, and reflection—applied to the graph of a function y = f(x). Mastering these concepts empowers you to sketch unfamiliar graphs quickly, interpret function notation flexibly, and solve equations graphically.
图像变换是A-Level数学中的核心模块，让你能够精准地操控曲线的位置、形状和朝向。本文聚焦于应用于函数 y = f(x) 图像的单一变换——平移、伸缩和反射。掌握这些概念后，你将能够快速绘制陌生函数的草图、灵活解读函数记号并利用图像求解方程。
1. What Are Graph Transformations? | 什么是图像变换？
A graph transformation changes the appearance or position of a curve according to a set of algebraic rules. When we apply a transformation to y = f(x), we obtain a new function whose graph is a mapped version of the original. Single transformations alter only one aspect at a time, making them the essential first step before tackling combinations.
图像变换是指按照一组代数规则改变曲线的外观或位置。当我们对 y = f(x) 施加一个变换时，会得到一个新函数，其图像是原图像的映射版本。单一变换每次只改变一个方面，因此是解决组合变换前必不可少的第一步。
Every point (x, y) on the original graph moves to a new coordinate according to the specific transformation. Recognising these point-wise mappings is key: translations shift the entire graph, stretches pull or compress it along axes, and reflections flip it across a line.
原图像上的每个点 (x, y) 都会根据特定的变换移动到新的坐标。识别这些逐点映射至关重要：平移会整体移动图像，伸缩会沿坐标轴拉伸或压缩图像，反射则会以一条直线为轴翻转图像。
2. Vertical Translation: f(x) + a | 垂直平移：f(x) + a
The transformation y = f(x) + a moves the graph vertically. If a > 0, the graph shifts upwards by a units; if a < 0, it shifts downwards by |a| units. The x-coordinates of all points remain unchanged, while every y-coordinate becomes y + a.
变换 y = f(x) + a 可将图像垂直移动。若 a > 0，则图像向上平移 a 个单位；若 a < 0，则向下平移 |a| 个单位。所有点的 x 坐标保持不变，而每个 y 坐标变为 y + a。
For example, adding 3 to f(x) = x² gives y = x² + 3, which lifts the entire parabola three units upward without altering its shape. The vertex moves from (0, 0) to (0, 3).
例如，将 f(x) = x² 加上 3 得到 y = x² + 3，整个抛物线向上提升三个单位而形状不变。顶点从 (0, 0) 移动到了 (0, 3)。
In mapping notation: (x, y) → (x, y + a). This is often described by a translation vector (0
a).
用映射记号表示为：(x, y) → (x, y + a)。通常可用平移向量 (0
a) 描述。
3. Horizontal Translation: f(x + a) | 水平平移：f(x + a)
Horizontal translation is governed by y = f(x + a). Here, the graph moves in the opposite direction to the sign of a: if a > 0, the graph shifts left by a units; if a < 0, it shifts right by |a| units. This counter-intuitive behaviour is a common source of errors.
水平平移由 y = f(x + a) 控制。这里的移动方向与 a 的符号相反：若 a > 0，图像向左平移 a 个单位；若 a < 0，则向右平移 |a| 个单位。这种与直觉相反的行为是常见的错误来源。
Why does f(x + 2) move left? To produce the same output as f(x) at a given position, the input must occur two units earlier. Thus every point shifts left to maintain the relationship.
为什么 f(x + 2) 会向左移动？为了在某个位置得到与 f(x) 相同的输出，输入必须提前两个单位。因此每个点都向左移动以维持该关系。
The mapping is (x, y) → (x – a, y). The plus sign inside the bracket corresponds to a negative horizontal shift in terms of the coordinate.
映射为 (x, y) → (x – a, y)。括号中的加号对应于坐标的负向水平移动。
4. Vertical Stretch: a f(x) | 垂直伸缩：a f(x)
Multiplying the whole function by a constant a yields y = a f(x), which stretches the graph vertically by scale factor a. If |a| > 1, the graph is stretched away from the x-axis; if 0 < |a| < 1, it is compressed towards the x-axis. A negative a also introduces a reflection (covered later).
将整个函数乘以常数 a 得到 y = a f(x)，这会将图像在垂直方向上拉伸，伸缩因子为 a。若 |a| > 1，图像远离 x 轴拉伸；若 0 < |a| < 1，则向 x 轴压缩。负的 a 还会引入反射（稍后讲解）。
Each point’s y-coordinate is multiplied by a, while the x-coordinate stays the same: (x, y) → (x, a y). For y = 2 sin(x), the amplitude doubles, but the period remains 2π.
每个点的 y 坐标乘以 a，而 x 坐标保持不变：(x, y) → (x, a y)。对于 y = 2 sin(x)，振幅翻倍，但周期仍为 2π。
Critical points where the graph crosses the x-axis (y = 0) remain unchanged because a × 0 = 0.
图像与 x 轴的交点（y = 0）保持不变，因为 a × 0 = 0。
5. Horizontal Stretch: f(ax) | 水平伸缩：f(ax)
The transformation y = f(ax) stretches the graph horizontally with scale factor 1/a. When a > 1, the graph is compressed horizontally towards the y-axis; when 0 < a < 1, it is stretched away from the y-axis. Again, the effect is reciprocal to the coefficient.
变换 y = f(ax) 会在水平方向上以因子 1/a 伸缩图像。当 a > 1 时，图像向 y 轴压缩；当 0 < a < 1 时，图像远离 y 轴拉伸。同样，效果与系数的关系是互为倒数的。
The mapping becomes (x, y) → (x / a, y). For y = sin(2x), the graph completes a full cycle in π instead of 2π, so the period halves.
映射变为 (x, y) → (x / a, y)。对于 y = sin(2x)，图像在 π 内完成一个完整周期，而不是 2π，因此周期减半。
Points on the y-axis (x = 0) are unaffected because 0/a = 0. This trick helps identify invariant points.
位于 y 轴上的点 (x = 0) 不受影响，因为 0/a = 0。利用这一特性可以识别不动点。
6. Reflection in the x-axis: –f(x) | 关于 x 轴的反射：–f(x)
Replacing f(x) with –f(x) reflects the graph across the x-axis. Every y-coordinate changes sign, while x-coordinates remain unchanged. The mapping is (x, y) → (x, –y).
将 f(x) 替换为 –f(x) 会将图像关于 x 轴反射。每个 y 坐标改变符号，而 x 坐标保持不变。映射为 (x, y) → (x, –y)。
For y = –√x, the graph is the reflection of the standard square root curve below the x-axis. This transformation is also equivalent to a vertical stretch by a factor of –1.
对于 y = –√x，其图像是标准平方根曲线在 x 轴下方的反射。该变换也等价于因子为 –1 的垂直拉伸。
Points on the x-axis stay fixed because –0 = 0.
x 轴上的点保持不动，因为 –0 = 0。
7. Reflection in the y-axis: f(–x) | 关于 y 轴的反射：f(–x)
To reflect the graph in the y-axis, we use y = f(–x). Each x-coordinate is multiplied by –1, giving the mapping (x, y) → (–x, y). The graph appears as a mirror image across the y-axis.
要将图像关于 y 轴反射，我们使用 y = f(–x)。每个 x 坐标乘以 –1，映射为 (x, y) → (–x, y)。图像表现为关于 y 轴的镜像。
This transformation is particularly useful for testing symmetries: if f(–x) = f(x) the graph is even, and if f(–x) = –f(x) it is odd.
该变换对检验对称性尤为有用：若 f(–x) = f(x)，图像是偶函数；若 f(–x) = –f(x)，则是奇函数。
For y = e⁻ˣ, the graph is the reflection of y = eˣ across the y-axis, showing exponential decay instead of growth.
对于 y = e⁻ˣ，其图像是 y = eˣ 关于 y 轴的反射，呈现指数衰减而非增长。
8. Combining Transformations: Sequences of Single Steps | 组合变换：单一变换的顺序
Although this article focuses on single transformations, exam questions often require applying two or more in sequence. The order matters. Changing the sequence of horizontal and vertical transformations can lead to different graphs.
尽管本文聚焦于单一变换，考试题往往要求按顺序应用两个或更多变换。顺序至关重要。改变水平变换和垂直变换的顺序可能导致不同的图像。
A safe strategy is to apply any horizontal transformations (including horizontal translation and stretch) first, then handle vertical ones. For instance, to sketch y = 2f(x + 3) – 1, you would: start with f(x), translate left by 3, stretch vertically by factor 2, and finally translate down by 1.
一个稳妥的策略是：先进行所有水平变换（含水平平移和伸缩），然后再处理垂直变换。例如，要绘制 y = 2f(x + 3) – 1 的图像，步骤如下：从 f(x) 出发，向左平移 3 个单位，垂直拉伸为 2 倍，最后向下平移 1 个单位。
Always consider the order inside the brackets first. When you have f(ax + b), rewrite it as f(a(x + b/a)) and apply the translation before the stretch inside, but when applying to x, the stretch comes first if you use mapping correctly.
务必优先考虑括号内的操作顺序。当遇到 f(ax + b) 时，改写为 f(a(x + b/a))，在括号内部先平移后伸缩；但若从映射的角度正确操作，对 x 而言实际是先伸缩后平移。
9. Applying Transformations to Common Functions | 常见函数的变换应用
Recognising how single transformations affect well-known curves—such as quadratics, cubics, reciprocals, and trigonometric functions—helps build intuition. A translated quadratic y = (x – 2)² is the parabola y = x² shifted right by 2.
识别单一变换如何影响已知曲线（如二次函数、三次函数、反比例函数和三角函数）有助于建立直观感受。平移后的二次函数 y = (x – 2)² 就是将抛物线 y = x² 向右移动 2 个单位。
For trigonometric graphs, y = sin(x) + 1 lifts the entire wave up by 1, making it oscillate around y = 1 instead of y = 0. y = cos(2x) doubles the frequency.
对于三角函数图像，y = sin(x) + 1 将整条波形向上提升 1，使其围绕 y = 1 振荡而非围绕 y = 0。y = cos(2x) 则将频率加倍。
With reciprocal graphs y = 1/x, a vertical translation y = 1/x + 3 moves the horizontal asymptote from y = 0 to y = 3. A horizontal translation y = 1/(x – 1) moves the vertical asymptote from x = 0 to x = 1.
对于反比例图像 y = 1/x，垂直平移 y = 1/x + 3 会将水平渐近线从 y = 0 移到 y = 3。水平平移 y = 1/(x – 1) 则会将垂直渐近线从 x = 0 移到 x = 1。
| Transformation | Effect on y = f(x) | Example |
|---|---|---|
| f(x) + a | Vertical shift by a | y = x² + 5 |
| f(x + a) | Horizontal shift by –a | y = (x – 3)² |
| a f(x) | Vertical stretch by factor a | y = 3 sin(x) |
| f(ax) | Horizontal stretch by factor 1/a | y = cos(0.5x) |
| –f(x) | Reflection in x-axis | y = –|x| |
| f(–x) | Reflection in y-axis | y = e⁻ˣ |
10. Tips for Sketching and Exams | 绘图技巧与考试建议
When sketching transformed curves, always begin with the basic shape of f(x) and mark key points. Then apply the single transformation step-by-step, adjusting those points accordingly. Labelling coordinates of at least three key points often earns method marks.
绘制变换曲线时，务必从 f(x) 的基本形状入手，并标注关键点。然后逐步应用单一变换，相应地调整这些点。标注至少三个关键点的坐标常常能获得方法分。
Check for invariant points: where does the transformation leave points unchanged? For vertical translations, there are none; for x-axis reflection, points with y = 0 stay; for y-axis reflection, points with x = 0 stay. These anchors can guide your sketch.
检查不动点：变换在哪些地方使点保持不变？对于垂直平移，没有不动点；对于 x 轴反射，y = 0 的点不动；对于 y 轴反射，x = 0 的点不动。这些锚点能指导你的草图。
Double-check the direction of horizontal translations: f(x – 1) moves right, f(x + 2) moves left. A quick numerical test—plug in x = 0 and see what input gives the same y—can prevent sign errors.
务必核对水平平移的方向：f(x – 1) 向右移动，f(x + 2) 向左移动。一个快速的数值测试——代入 x = 0 并观察哪个输入能得到相同的 y——可以避免符号错误。
Finally, familiarise yourself with the effect of each transformation on asymptotes, intercepts, and domain/range. Single transformations often preserve the overall shape but shift its position or scale.
最后，要熟悉每种变换对渐近线、截距以及定义域/值域的影响。单一变换通常保持整体形状，但会改变其位置或尺度。
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