📚 Transforming Functions | 函数图像的变换
Function transformations are fundamental tools in A‑Level Mathematics, allowing you to build new graphs from simple parent functions by shifting, stretching, or reflecting them. Mastering these ideas is essential for curve sketching, equation solving, and understanding the behaviour of functions in various contexts. This article walks through every type of transformation specified in the Edexcel A‑Level syllabus, with clear rules, examples, and common pitfalls to avoid.
函数图像的变换是 A‑Level 数学中极其重要的工具，通过平移、伸缩或反射简单的基本函数，我们可以构造出新的图像。掌握这些思想对于绘制曲线、求解方程以及理解函数在不同情境下的行为至关重要。本文将全面梳理 Edexcel 考试大纲中涉及的每一种变换，提供清晰的规则、示例以及需要避免的常见错误。
1. The Core Idea | 核心思想
re applying a rule that changes every point (x, y) on the original graph to a new point (x', y' ransformation are translation (shift), stretch, and reflection. Understanding whether a transformation affects the x‑coordinate or the y‑coordinate – and in what way – is the key to getting the correct new equation.
当我们谈论函数 y = f(x) 的变换时，指的是对原图像上的每一点 (x, y) 施加一个规则，将其移动到新的点 (x', y')，得到的图像就代表一个新的函数。四种基本变换类型是平移、伸缩和反射。理解某种变换究竟影响的是横坐标 x 还是纵坐标 y——以及以何种方式影响——是写出正确新方程式的关键。
2. Vertical Translations | 垂直平移
n without changing its shape. The transformation y = f(x) + a adds a constant a to every y‑coordinate. If a > 0, the graph shifts up by a units; if a < 0, it shifts down by |a| e, starting with f(x) = x², the graph of y = x² + 3 is the basic parabola shifted up by 3.
垂直平移将图像沿竖直方向上下移动，而不改变其形状。变换 y = f(x) + a 会在每个纵坐标 y 上加上常数 a。若 a > 0，图像向上移动 a 个单位；若 a < 0，则向下移动 |a| 个单位。这可能是最直观的一种变换：直接在函数的输出上增加一个量。例如，从 f(x) = x² 出发，y = x² + 3 的图像就是将基本抛物线向上平移 3 个单位。
3. Horizontal Translations | 水平平移
te direction to the sign. For y = f(x + 3), the graph shifts 3 units to the left. For y = f(x − 3), it shifts 3 units to the right. The point (p, q) on y = f(x) moves to (p − a, q) on the transformed graph. Think of it as setting the input x + a = 0 to find where the previously zero input now occurs.
水平平移将图像向左或向右移动，其表达式为 y = f(x + a)。务必注意：括号内加上一个正数，图像反而朝反方向移动。例如 y = f(x + 3) 将图像向左平移 3 个单位；y = f(x − 3) 则向右平移 3 个单位。原图像上的点 (p, q) 在变换后的图像上移动到 (p − a, q)。可以这样思考：把新的输入 x + a 设为零，就能看出原先输入为零的点现在移动到了何处。
4. Vertical Stretches | 垂直伸缩
A vertical stretch multiplies all y‑ a constant factor a, giving y = a f(x). When a > 1, the graph is stretched away from the x‑axis; when 0 < a < 1, it is compressed towards the x‑axis. If a is negative, the stretch is combined with a reflection in the x‑axis. The x‑ f(x) = sin x into y = 2 sin x doubles the amplitude, making peaks reach ±2 instead of ±1.
垂直伸缩将所有纵坐标乘以一个常数因子 a，得到 y = a f(x)。当 a > 1 时，图像沿竖直方向背离 x 轴拉伸；当 0 < a < 1 时，则向 x 轴压缩。如果 a 为负，则该伸缩同时伴有关于 x 轴的反射。x 轴上的截距保持不变，因为 f(x)=0 时 a·0=0。例如，将 f(x) = sin x 变为 y = 2 sin x，振幅将变成原来的两倍，波峰达到 ±2 而不是 ±1。
5. Horizontal Stretches | 水平伸缩
Horizontal stretching changes the x‑ x). Here the factor 1/a dictates how the graph is stretched or compressed horizontally. If |a| > 1, the graph is compressed towards the y‑axis by factor 1/a; if 0 < |a| < 1, it is stretched away from the y‑ or example, y = sin(2x) completes two full cycles in an interval where sin x completes one – that is a compression by factor ½. The y‑intercept (where x=0) stays the same because f(0) does not change.
水平伸缩改变的是横坐标，写作 y = f(ax)。这里因子 1/a 决定了图像在水平方向上是拉伸还是压缩。若 |a| > 1，图像朝 y 轴方向压缩为原来的 1/|a|；若 0 < |a| < 1，则背离 y 轴拉伸。同样要注意这种倒数关系。例如 y = sin(2x) 在一个原本 sin x 完成一个周期的区间内完成了两个完整周期——这就是以 ½ 的倍数压缩。y 轴截距（x=0 处的点）保持不变，因为 f(0) 不会改变。
6. Reflections | 反射
are two fundamental reflections. The transformation y = −f(x) reflects the graph in the x‑axis: every y‑coordinate changes sign. The transformation y = f(−x) reflects the graph in the y‑axis: every x‑coordinate changes sign. For example, starting from f(x) = √x, the graph y = −√x is the reflection of y = √x in the x‑axis; the graph y = √−x is only defined for x ≤ 0 and is the reflection of y = √x in the y‑axis. Combining −f(−x) gives a rotation of 180° about the origin.
有两种基本的反射。变换 y = −f(x) 将图像关于 x 轴反射：所有纵坐标改变符号。变换 y = f(−x) 将图像关于 y 轴反射：所有横坐标改变符号。例如，从 f(x) = √x 出发，y = −√x 是 y = √x 关于 x 轴的反射；y = √−x 只有当 x ≤ 0 时才有定义，它是 y = √x 关于 y 轴的反射。组合 −f(−x) 则得到关于原点旋转 180° 的图像。
7. Combining Transformations | 组合变换
b(x + c)) + d. The standard convention is to follow the order of operations relative to x: first perform transformations inside the bracket (horizontal shifts and stretches), then those outside (vertical stretches and shifts). More precisely, from f(x) to a f(bx + c) + d, you can rewrite as a f(b(x + c/b)) + d. Then the horizontal transformation is: replace x by (x + c/b), which is a shift left by c/b, followed by the stretch factor 1/b in the x‑ ore the stretch without adjusting for the factor b, so always factorise the bracket first.
当一个函数经过多次连续变换时，顺序是很关键的。Edexcel 大纲要求你能处理 y = a f(b(x + c)) + d 这类形式。标准的做法是遵循与 x 相关的运算顺序：先处理括号内部的变换（水平移动和伸缩），再处理外部的变换（垂直伸缩和平移）。更准确地说，从 f(x) 到 a f(bx + c) + d，通常将内部改写为 a f(b(x + c/b)) + d。此时水平变换为：把 x 替换为 (x + c/b)，即先向左平移 c/b，然后在 x 方向以 1/b 的因子伸缩。外部则乘以 a 并加上 d。许多同学错误地先平移再伸缩却忽略了因子 b 的影响，因此务必先将括号内的式子提出 b 进行因式分解。
8. Applying to Trigonometric Functions | 应用于三角函数
hift y = sin(x − 30°) moves the graph 30° to the right. Vertical stretch y = 3 sin x changes the amplitude to 3, while horizontal compression y = sin 2x halves the period. When transformations are combined, e.g. y = 2 sin(3x − 60°), rewrite as 2 sin[3(x − 20°)] to see the shift correctly: horizontal shift right by 20°, then period compression by ⅓, then vertical stretch by 2. Edexcel questions frequently ask for the sequence of transformations that maps one trig graph to another, and mark schemes reward the correct order and clear reasoning.
三角函数的图像常常能优雅地展示各种变换。对于 y = sin x，垂直平移 y = sin x + 1 将整个波形上移。水平平移 y = sin(x − 30°) 将图像向右移动 30°。垂直伸缩 y = 3 sin x 将振幅变为 3，水平压缩 y = sin 2x 则将周期减半。当变换组合在一起时，例如 y = 2 sin(3x − 60°)，应改写为 2 sin[3(x − 20°)] 才能正确看出平移：先向右平移 20°，然后周期压缩为原来的 ⅓，最后垂直拉伸为原来的 2 倍。Edexcel 考题经常要求描述将一个三角函数图像变为另一个的变换序列，阅卷标准也格外重视正确的顺序和清晰的推理。
9. Sketching Graphs Using Transformations | 利用变换绘制图像草图
A powerful exam technique is to start with a well‑ ansformation at a time, plotting key points after each step. For example, to sketch y = −2√x − 1 + 4, begin with y = √x. Shift right by 1: y = √x − 1. Vertically stretch by factor 2: y = 2√x − 1. Reflect in the x‑axis: y = −2√x − 1. Finally, shift up by 4: y = −2√x − 1 + 4. At each stage, identify where the start point (for the root function, originally (0,0)) moves to. This structured approach reduces errors and ensures the final graph is accurate.
一种非常有效的考试技巧是：从一个熟悉的母函数出发，每次只施加一种变换，并在每一步后标出关键点。例如，要绘制 y = −2√x − 1 + 4 的草图，可以从 y = √x 开始。先向右平移 1 个单位：y = √x − 1。然后沿竖直方向以因子 2 拉伸：y = 2√x − 1。接着关于 x 轴反射：y = −2√x − 1。最后向上平移 4：y = −2√x − 1 + 4。在每一个阶段都跟踪起点（对于根函数，原点是 (0,0)）的位置变化。这种有条理的做法可以减少错误，保证最终图像准确无误。
10. Transformations and Asymptotes | 变换与渐近线
f(x), a vertical translation f(x) + d moves the asymptote to y = c + d. A vertical stretch a f(x) keeps the asymptote at y = a·c (unless c = 0, then it stays y = 0). Horizontal asymptotes remain unaffected by horizontal transformations. For vertical asymptotes x = p, a horizontal shift f(x + a) moves the asymptote to x = p − a; a horizontal stretch f(ax) changes it to x = p/a. Always identify asymptotes explicitly when sketching transformed curves.
含有渐近线的有理函数及指数/对数函数在变换下，渐近线也会随之移动。若 f(x) 有水平渐近线 y = c，垂直平移 f(x) + d 会将渐近线移至 y = c + d。垂直伸缩 a f(x) 则使渐近线变为 y = a·c（除非 c=0，此时渐近线仍为 y=0）。水平渐近线不受水平变换的影响。对于垂直渐近线 x = p，水平平移 f(x + a) 会将渐近线移到 x = p − a；水平伸缩 f(ax) 则将其变为 x = p/a。在绘制变换后的曲线时，一定要明确标出渐近线的位置。
11. Recognising Transformations from Equations | 从方程识别变换
nt to isolate horizontal shifts from stretches. For example, if f(x) = ln x and the new graph has equation y = 3 ln(4 − 2x), rewrite as y = 3 ln[−2(x − 2)]. Start with ln x: reflect in y‑axis to get ln(−x), then shift left 2? − 2)] means we want the transformations step by step: from ln x to ln(−x) (reflection in y‑ tically: write argument as −2x + 4, factor −2: −2(x − 2). So transformations: reflect in y‑ ln(−x)), stretch horizontally by factor ½ (ln(−2x)), then shift right by 2 (ln(−2(x − 2)))? oute: translate horizontal stretch and shift in terms of ln[−2(x − 2)]: first reflect in y‑ actual shift is 2 units right, as derived. This illustrates why factorising is essential.
给定一个变换后的方程，你可以通过将新函数与基本函数 f(x) 进行对比，来描述一系列变换。先对自变量部分进行因式分解，把水平平移和伸缩分开。例如，设 f(x) = ln x，新图像方程为 y = 3 ln(4 − 2x)，先改写为 y = 3 ln[−2(x − 2)]。从 ln x 开始：关于 y 轴反射得到 ln(−x)，然后向右平移 2？要小心：ln[−2(x − 2)] 意味着变换步骤为：先 ln x → ln(−x)（y 轴反射），再将 x 替换为 (x − 2) 得到 ln(−(x − 2)) = ln(−x + 2)，这还不是最终形式。更好的方法是系统化操作：自变量为 −2x + 4，提取 −2 得 −2(x − 2)。因此变换为：先 y 轴反射 (ln(−x))，然后水平方向以 ½ 的因子伸缩 (ln(−2x))，最后向右平移 2 得到 ln(−2(x − 2))。但要注意，反射之后，平移必须与因子配合。最稳妥的方式是基于 ln[−2(x − 2)] 理解：先反射、再伸缩、最后平移。最终平移确实是向右 2 个单位。这充分说明了提取公因子的重要性。
12. Summary and Common Pitfalls | 总结与常见错误
To summarise the basic mappings: y = f(x) + a – vertical translation by a; y = f(x + a) – horizontal translation by −a; y = a f(x) – vertical stretch by factor a; y = f(ax) – horizontal stretch by factor 1/a; y = −f(x) – reflection in x‑axis; y = f(−x) – reflection in y‑ etting the opposite direction for horizontal shifts, confusing the stretch factor in the x‑ the wrong order when both horizontal and vertical changes are present. Always check the effect on a specific point, like the turning point or intercept, to verify your transformed graph. With practice and methodical steps, you can confidently master function transformations for the Edexcel A‑Level exam.
总结基本映射关系：y = f(x) + a —— 垂直平移 a；y = f(x + a) —— 水平平移 −a；y = a f(x) —— 垂直伸缩因子 a；y = f(ax) —— 水平伸缩因子 1/a；y = −f(x) —— 关于 x 轴反射；y = f(−x) —— 关于 y 轴反射。常见错误包括：忘记水平平移的方向是相反的，混淆 x 方向的伸缩因子（它是系数的倒数），在描述变换序列前没有对自变量进行因式分解，以及当水平和垂直变换同时存在时搞错了应用顺序。始终通过检验特定位点（如转折点或截距）的效果来验证变换后的图像。经过充分练习并有条不紊地分步操作，你一定能在 Edexcel A‑Level 考试中自信地掌握函数的各种变换。
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