Transformations of Functions | 函数变换详解

📚 Transformations of Functions | 函数变换详解

Transformations of functions is one of the most frequently tested topics in IGCSE Mathematics. It combines algebraic understanding with visual interpretation, and once you master the patterns, you can solve these questions quickly and confidently. This article covers everything you need to know about translations, reflections, and stretches of graphs, including key rules, worked examples, and common exam traps.

函数变换是 IGCSE 数学中考查频率最高的知识点之一。它把代数理解与图像观察融为一体，一旦掌握了规律，就能快速而自信地解题。本文将全面讲解图像平移、反射和伸缩的核心规则、典型例题以及常见考试陷阱，帮助你完整掌握这一专题。

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1. Function Notation and Graph Basics | 函数记号与图像基础

Before diving into transformations, let us recall the basic notation. A function is written as y = f(x). The graph of f(x) is a set of points (x, y) that satisfy the equation. When we transform a function, we create a new function from the original one, and every transformation changes the graph in a predictable way.

在深入学习变换之前，我们先回顾基础记号。函数通常写作 y = f(x)。函数 f(x) 的图像是满足该方程的所有点 (x, y) 的集合。当我们对函数进行变换时，会基于原函数产生一个新函数，而每种变换都会以可预测的方式改变原图像。

There are three major types of transformations you must know:

你必须掌握三大类变换：

• Translations (shifts) — moving the whole graph without changing its shape
• Translations (平移) — 整体移动图像，不改变其形状
• Reflections — flipping the graph over the x-axis or y-axis
• Reflections (反射) — 将图像关于 x 轴或 y 轴翻转
• Stretches and Compressions — changing the graph’s steepness or width
• Stretches and Compressions (伸缩) — 改变图像的倾斜程度或宽度

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2. Vertical Translation: y = f(x) + a | 垂直平移：y = f(x) + a

When we add a positive constant a to a function, the graph shifts upward by a units. When we subtract a, the graph shifts downward by a units. The shape of the graph never changes — only its vertical position changes.

当我们给函数加上一个正常数 a 时，图像向上平移 a 个单位；减去 a 时，图像向下平移 a 个单位。图像的形状完全不变，只有垂直位置发生变化。

y = f(x) + a shifts the graph of y = f(x) UP by a units
y = f(x) − a shifts the graph of y = f(x) DOWN by a units

Example: If f(x) = x², then y = x² + 3 is a parabola shifted up by 3 units. Every point (x, y) on the original graph becomes (x, y + 3).

例如：若 f(x) = x²，则 y = x² + 3 是将抛物线向上平移 3 个单位后的结果。原图像上的每一个点 (x, y) 都变为 (x, y + 3)。

Key point: vertical translation affects only the y-coordinates. The x-coordinates remain unchanged.

关键点：垂直平移只影响 y 坐标。所有 x 坐标保持不变。

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3. Horizontal Translation: y = f(x − a) | 水平平移：y = f(x − a)

Horizontal translations are trickier. The rule is: y = f(x − a) shifts the graph RIGHT by a units, while y = f(x + a) shifts it LEFT by a units. Notice the opposite direction to the sign inside the bracket.

水平平移更容易出错。规则是：y = f(x − a) 将图像向右平移 a 个单位，而 y = f(x + a) 将图像向左平移 a 个单位。注意括号内符号与移动方向相反。

y = f(x − a) shifts the graph RIGHT by a units
y = f(x + a) shifts the graph LEFT by a units

A common student error is thinking that f(x + 2) shifts the graph right. It does not — it shifts it left. To avoid this mistake, think: to keep the output the same, the input x must become smaller to compensate for the +2 inside the bracket.

一个常见的错误是以为 f(x + 2) 把图像向右平移。事实并非如此——图像是向左移动的。为了避免出错，可以这样理解：为了让函数输出值保持不变，当括号里有 +2 时，输入 x 必须变小来补偿。

Example: If f(x) = √x, then f(x − 4) = √(x − 4) is the square-root graph shifted right by 4 units.

例如：若 f(x) = √x，则 f(x − 4) = √(x − 4) 是将平方根图像向右平移 4 个单位的结果。

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4. Combined Translations | 组合平移

When both horizontal and vertical translations appear together, such as y = f(x − a) + b, the graph shifts right by a units and up by b units. You can apply these in any order — the final position is the same.

当水平和垂直平移同时出现时，例如 y = f(x − a) + b，图像向右平移 a 个单位并向上平移 b 个单位。这两个操作可以按任意顺序进行，最终位置完全相同。

For example, y = (x − 2)² + 1 is the parabola y = x² shifted right by 2 and up by 1. The vertex moves from (0, 0) to (2, 1).

例如，y = (x − 2)² + 1 是将抛物线 y = x² 向右平移 2 个单位、再向上平移 1 个单位的结果。顶点从 (0, 0) 移动到 (2, 1)。

A powerful strategy for combined translations is to track one key point, such as the vertex or the y-intercept. Move that point according to the transformation rules, then draw the rest of the graph accordingly.

处理组合平移的一个高效策略是追踪一个关键点，比如顶点或 y 截距。根据变换规则移动该点，再据此绘制图像的其他部分。

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5. Reflection in the x-axis: y = −f(x) | 关于 x 轴的反射：y = −f(x)

Multiplying the entire function by −1, giving y = −f(x), reflects the graph in the x-axis. Every point (x, y) becomes (x, −y). Points above the x-axis move below it, and vice versa.

将整个函数乘以 −1，即 y = −f(x)，使图像关于 x 轴反射。每个点 (x, y) 变为 (x, −y)。x 轴上方的点移到下方，反之亦然。

y = −f(x) reflects the graph in the x-axis

Example: If f(x) = x² − 1, then y = −(x² − 1) = −x² + 1 reflects the parabola across the x-axis. The vertex (0, −1) becomes (0, 1).

例如：若 f(x) = x² − 1，则 y = −(x² − 1) = −x² + 1 是将抛物线关于 x 轴反射的结果。顶点 (0, −1) 变为 (0, 1)。

Notice that any point lying exactly on the x-axis (y = 0) stays fixed during this reflection.

注意：恰好落在 x 轴上（y = 0）的点在反射过程中保持不动。

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6. Reflection in the y-axis: y = f(−x) | 关于 y 轴的反射：y = f(−x)

Replacing x with −x, giving y = f(−x), reflects the graph in the y-axis. Every point (x, y) becomes (−x, y). The graph is flipped horizontally like a mirror image.

将 x 替换为 −x，即 y = f(−x)，使图像关于 y 轴反射。每个点 (x, y) 变为 (−x, y)。图像像照镜子一样水平翻转。

y = f(−x) reflects the graph in the y-axis

Example: If f(x) = 2x + 3, then f(−x) = −2x + 3. The line x = 0 (the y-axis) acts as the mirror line. The original y-intercept at (0, 3) stays fixed because it lies on the mirror line.

例如：若 f(x) = 2x + 3，则 f(−x) = −2x + 3。直线 x = 0（即 y 轴）作为镜像轴。原来的 y 截距点 (0, 3) 因位于镜像轴上而保持不动。

If the graph of f(x) is symmetric about the y-axis — meaning f(−x) = f(x) for all x — the function is called even, and its reflection will be identical to the original graph.

如果 f(x) 的图像关于 y 轴对称，即对所有 x 有 f(−x) = f(x)，则称该函数为偶函数，其反射图像将与原图像完全重合。

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7. Vertical Stretch: y = a·f(x) | 垂直伸缩：y = a·f(x)

Multiplying the function by a constant a (where a > 0) stretches or compresses the graph vertically. If a > 1, the graph stretches upward (taller); if 0 < a < 1, the graph compresses (flatter).

将函数乘以常数 a（其中 a > 0），会使图像在垂直方向伸缩。若 a > 1，图像向上拉伸（变高）；若 0 < a < 1，图像被压缩（变扁）。

y = a·f(x), a > 1: vertical stretch by scale factor a
y = a·f(x), 0 < a < 1: vertical compression by factor a

Example: If f(x) = x³, then y = 3x³ has the same x-intercepts as the original, but every other y-coordinate is tripled. The point (2, 8) becomes (2, 24).

例如：若 f(x) = x³，则 y = 3x³ 与原函数有相同的 x 截距，但所有其他 y 坐标变为原来的三倍。点 (2, 8) 变为 (2, 24)。

Important: vertical stretches fix the x-axis — points with y = 0 remain exactly where they are. A negative value of a (e.g., y = −2f(x)) combines a vertical stretch with a reflection in the x-axis.

重要：垂直伸缩以 x 轴为固定线——y = 0 的点位置不变。若 a 为负值（例如 y = −2f(x)），则同时包含垂直伸缩和关于 x 轴的反射。

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8. Horizontal Stretch: y = f(kx) | 水平伸缩：y = f(kx)

Replacing x with kx (where k > 0) stretches or compresses the graph horizontally. The critical rule to remember: y = f(kx) compresses the graph by a factor of 1/k. If k > 1, the graph compresses (becomes narrower); if 0 < k < 1, the graph stretches (becomes wider).

将 x 替换为 kx（其中 k > 0）会在水平方向伸缩图像。必须记住的关键规则：y = f(kx) 将图像以 1/k 的比例压缩。若 k > 1，图像变窄（压缩）；若 0 < k < 1，图像变宽（拉伸）。

y = f(kx), k > 1: horizontal compression by scale factor 1/k
y = f(kx), 0 < k < 1: horizontal stretch by factor 1/k

Example: If f(x) = (x − 1)², then f(2x) = (2x − 1)². The distance between any two points along the x-axis is halved, so the graph appears squeezed towards the y-axis.

例如：若 f(x) = (x − 1)²，则 f(2x) = (2x − 1)²。任意两点沿 x 轴的距离减半，因此图像看起来被挤压到 y 轴方向。

As with vertical stretches, horizontal stretches fix the y-axis (x = 0). If k is negative, a horizontal reflection in the y-axis also occurs.

与垂直伸缩类似，水平伸缩以 y 轴（x = 0）为固定线。如果 k 为负值，还会附加关于 y 轴的水平反射。

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9. Transformations of Key Points | 关键点变换法

For any transformation question, tracking key points is the fastest and most reliable method. Choose 2–3 points that are easy to see on the given graph — such as intercepts, vertices, or maxima/minima — apply the transformation to those points, then sketch the full graph through the transformed points.

对于任何变换问题，追踪关键点是最快捷、最可靠的方法。选择在给定图像上容易识别的 2 到 3 个点——如截距、顶点或极大/极小值点——将这些点按变换规则进行处理，然后通过变换后的点画出完整图像。

For example, for a quadratic y = x² with vertex at (0, 0):

例如，对于顶点在 (0, 0) 的二次函数 y = x²：

Transformation | 变换	Rule | 规则	New vertex | 新顶点
y = x² + 2	up 2 | 上移2	(0, 2)
y = (x − 3)²	right 3 | 右移3	(3, 0)
y = −x²	reflect x-axis | x轴反射	(0, 0)
y = 2x²	vertical stretch | 垂直拉伸	(0, 0)
y = (2x)² = 4x²	horizontal compression | 水平压缩	(0, 0)

Notice that vertex stays at (0, 0) for reflections and stretches when it lies on the fixed axis. This consistency makes key-point tracking very useful.

注意，当顶点位于固定轴上时，在反射和伸缩变换下顶点保持在 (0, 0)。这种一致性使关键点追踪法非常实用。

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10. Order of Multiple Transformations | 多重变换的顺序

When more than one transformation is involved, the order matters. The standard approach is: first apply stretches and reflections (multiplications), then apply translations (additions/subtractions). This follows the order of operations.

当涉及多个变换时，顺序很重要。标准处理方式是：先进行伸缩和反射（乘法类操作），然后进行平移（加法/减法类操作）。这遵循运算顺序。

Consider y = −2f(x) + 3. This means:

考虑 y = −2f(x) + 3。这意味着：

• Step 1: reflect in x-axis — multiply by −1 | 第一步：关于 x 轴反射 — 乘以 −1
• Step 2: vertical stretch by factor 2 — multiply by 2 | 第二步：垂直拉伸 2 倍 — 乘以 2
• Step 3: shift up by 3 — add 3 | 第三步：向上平移 3 个单位 — 加 3

It would be an error to shift first and then reflect, because the translation would also get reflected. Always handle the multiplications before the additions.

先平移再反射是错误的，因为平移本身也会被反射。务必先处理乘法运算，再处理加法运算。

For horizontal transformations, the order is similar but the translation is subtracted inside the bracket. For example, y = f(2(x − 1)) means: first compress horizontally by factor 1/2, then shift right by 1.

对于水平变换，顺序类似，但平移在括号内以减法形式出现。例如，y = f(2(x − 1)) 表示：先水平压缩到原来的 1/2，然后向右平移 1 个单位。

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11. Common Exam Questions and Traps | 常见考题与陷阱

Exam questions typically ask you to describe a transformation from one graph to another, or to sketch the transformed graph from a given one. The most common traps include confusing horizontal shift direction, mixing up stretch directions, and forgetting that y = f(kx) compresses rather than stretches when k > 1.

考试通常要求你描述从一个图像到另一个图像的变换，或根据给定图像画出变换后的图像。最常见的陷阱包括搞混水平平移方向、弄反伸缩方向，以及忘记当 k > 1 时 y = f(kx) 是压缩而不是拉伸。

Here are some quick checks to use in the exam:

以下是一些考试中快速自查的小技巧：

• Outside the bracket (e.g., +2, ×3): affects y-values — vertical transformation | 括号外（如 +2、×3）：影响 y 值 — 垂直变换
• Inside the bracket (e.g., x−2, 3x): affects x-values — horizontal transformation | 括号内（如 x−2、3x）：影响 x 值 — 水平变换
• y = f(x − a) → move right; y = f(x + a) → move left | y = f(x − a) → 向右；y = f(x + a) → 向左
• y = f(kx) with k > 1 → squeeze narrower; with 0 < k < 1 → stretch wider | y = f(kx)，k > 1 → 变窄；0 < k < 1 → 变宽
• Negative sign outside → reflect in x-axis; negative inside → reflect in y-axis | 外带负号 → x 轴反射；内带负号 → y 轴反射

Another tip: always sketch the transformed graph on the answer page. Even if the question does not explicitly ask for a sketch, drawing the graph helps you reason correctly and catch errors in your algebra.

另一个技巧：一定要在答题纸上画出变换后的图像。即使题目没有明确要求画图，绘图也能帮助你正确推理并发现代数计算中的错误。

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12. Summary Checklist | 总结清单

Use this summary checklist before your exam to confirm you know all the transformations:

考前使用以下总结清单，确认你掌握了所有变换：

Transformation | 变换	Expression | 表达式	Effect | 效果
Vertical shift | 垂直平移	f(x) + a	Up a units | 上移 a 单位
Horizontal shift | 水平平移	f(x − a)	Right a units | 右移 a 单位
Reflect x-axis | x 轴反射	−f(x)	Flip vertically | 上下翻转
Reflect y-axis | y 轴反射	f(−x)	Flip horizontally | 左右翻转
Vertical stretch | 垂直伸缩	a·f(x), a > 1	Steeper | 变陡峭
Horizontal compression | 水平压缩	f(kx), k > 1	Narrower | 变窄

Mastering function transformations is not about memorising formulas — it is about understanding what each algebraic operation does to the graph. Once you visualise the effect, you can handle any variation, no matter how complicated the expression looks.

掌握函数变换并不在于死记公式，而在于理解每个代数运算对图像做了什么。一旦你能在脑海中想象出变换效果，无论表达式看起来多复杂，你都能从容应对。

Regularly practising with graph-sketching questions will build your intuition far more effectively than reading alone. Combine your algebra skills with visual thinking, and this topic will become one of your strongest areas in the IGCSE Mathematics exam.

经常练习绘图题会比单纯阅读更有效地建立你的直觉。将代数能力与图像思维相结合，这个主题将成为你在 IGCSE 数学考试中最拿分的部分之一。

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