Graphs and Transformations | 函数图像与变换

📚 Graphs and Transformations | 函数图像与变换

Understanding graphs and how they can be transformed is a fundamental skill in A-Level maths. From basic polynomial shapes to trigonometric waves, mastering the effects of translations, stretches, and reflections allows you to quickly sketch complicated functions and solve equations geometrically. This article covers the key graph types and the full set of transformations required for the Edexcel syllabus, showing how each change in an algebraic equation produces a predictable shift, stretch, or flip in the corresponding graph.

理解函数图像及其变换是 A-Level 数学的基本功。从基本多项式形状到三角波形，掌握平移、伸缩和反射的效果能让你快速画出复杂函数的草图，并用几何方法解方程。本文涵盖 Edexcel 考纲所要求的核心图像类型以及全套变换，展示代数式中的每一个改动如何引起图像的平移、拉伸或翻转。

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1. Introduction to Graphs and Transformations | 函数图像与变换概述

A graph of y = f(x) shows the set of points (x, f(x)). Transformations let us modify a known base graph to obtain new graphs without plotting every point. The four main types of transformation are translations, stretches, reflections, and combinations. Each can be applied either to the x‑coordinate (horizontal changes) or the y‑coordinate (vertical changes). Horizontal transformations often appear inside the function argument, while vertical transformations appear outside.

y = f(x) 的图像表示满足关系的点 (x, f(x))。变换使我们能够在已知基本图像的基础上得到新图像，而不需要逐点绘制。四种主要变换类型是平移、伸缩、反射以及它们的组合。每种变换可以作用于 x 坐标（水平变化）或 y 坐标（垂直变化）。水平变换通常出现在函数自变量内部，垂直变换则出现在函数外部。

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2. Basic Graphs: Linear, Quadratic and Cubic | 基本图像：一次、二次与三次函数

Linear functions y = mx + c produce straight lines with gradient m and y‑intercept c. The graph of y = x is a diagonal line through the origin with gradient 1.

一次函数 y = mx + c 产生斜率为 m、y 轴截距为 c 的直线。y = x 的图像是一条经过原点、斜率为 1 的对角线。

Quadratic functions y = x² give a U‑shaped parabola symmetric about the y‑axis, with vertex at (0, 0). Changing the coefficient of x² affects the width and direction: y = ax² opens upward if a > 0 and downward if a < 0.

二次函数 y = x² 是一条关于 y 轴对称的 U 形抛物线，顶点在 (0, 0)。二次项系数改变宽度和开口方向：a > 0 时 y = ax² 开口向上，a < 0 时开口向下。

Cubic functions y = x³ have an S‑shaped curve passing through the origin. The graph is rotationally symmetric about the origin. General cubics y = ax³ + bx² + cx + d can have turning points or inflection points depending on the coefficients.

三次函数 y = x³ 呈 S 形曲线，通过原点并关于原点中心对称。一般三次函数 y = ax³ + bx² + cx + d 根据系数不同，可能具有驻点或拐点。

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3. Reciprocal and Rational Functions | 倒数与有理函数图像

The reciprocal function y = 1/x has two separate branches in the first and third quadrants. The axes are asymptotes: as x → 0, y → ±∞; as x → ±∞, y → 0. It is an odd function with rotational symmetry about the origin.

倒数函数 y = 1/x 在第一和第三象限各有一支曲线。坐标轴是渐近线：x → 0 时 y → ±∞；x → ±∞ 时 y → 0。该函数为奇函数，关于原点旋转对称。

Rational functions like y = 1/(x – a) + b simply translate the basic reciprocal graph: a shifts the vertical asymptote to x = a, and b shifts the horizontal asymptote to y = b. More complex rational functions can have slant asymptotes, but A‑Level work concentrates on vertical and horizontal translations.

形如 y = 1/(x – a) + b 的有理函数只是将基本倒数图像平移：a 使垂直渐近线移动到 x = a，b 使水平渐近线移动到 y = b。更复杂的有理函数可能出现斜渐近线，但在 A‑Level 中主要关注垂直和水平平移。

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4. Exponential and Logarithmic Functions | 指数与对数函数图像

The exponential function y = eˣ and y = aˣ (a > 0) always pass through (0, 1), increase rapidly for x > 0, and approach y = 0 as x → –∞. The x‑axis is a horizontal asymptote. The graph is always above the x‑axis.

指数函数 y = eˣ 和 y = aˣ（a > 0）总经过点 (0, 1)，在 x > 0 时快速增长，x → –∞ 时趋近 y = 0。x 轴为水平渐近线，图像始终位于 x 轴上方。

The natural logarithm y = ln x is the inverse of y = eˣ; its graph is the reflection of y = eˣ in the line y = x. It passes through (1, 0), has the y‑axis as a vertical asymptote, and only exists for x > 0.

自然对数 y = ln x 是 y = eˣ 的反函数，其图像可由 y = eˣ 关于直线 y = x 反射得到。它经过点 (1, 0)，以 y 轴为垂直渐近线，且只定义在 x > 0 上。

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5. Trigonometric Graphs | 三角函数图像

The sine graph y = sin x oscillates between –1 and 1, with period 2π, starting at the origin. Cosine y = cos x has the same amplitude and period but starts at its maximum (0, 1). Tangent y = tan x has period π and vertical asymptotes at x = π/2 + nπ.

正弦函数 y = sin x 在 –1 和 1 之间振荡，周期为 2π，从原点出发。余弦 y = cos x 具有相同的振幅和周期，但从最大值 (0, 1) 开始。正切 y = tan x 的周期为 π，在 x = π/2 + nπ 处有垂直渐近线。

These base graphs are essential for sketching transformed trigonometric functions such as y = a sin(bx + c) + d, where amplitude, frequency, phase shift and vertical shift are all controlled by parameters.

这些基本图像是绘制变换后的三角函数图像的基础，例如 y = a sin(bx + c) + d，其中振幅、频率、相位平移和垂直平移均由各参数控制。

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6. Translations: Shifting Graphs | 平移变换：图像的上下左右移动

A vertical translation replaces f(x) with f(x) + k. When k > 0 the graph moves up by k units; when k < 0 it moves down. For example, y = x² + 3 is the parabola y = x² shifted up by 3.

垂直平移将 f(x) 替换为 f(x) + k。k > 0 时图像向上移动 k 个单位；k < 0 时向下移动。例如 y = x² + 3 就是将抛物线 y = x² 向上平移 3 个单位。

A horizontal translation replaces x with (x – h) inside the function. This shifts the graph to the right by h if h > 0, and to the left if h < 0. Note the counter‑intuitive direction: y = f(x - 2) moves the graph 2 units to the right. The transformation is often written as f(x + a) meaning a translation by vector (–a, 0).

水平平移将函数中的 x 替换为 (x – h)。当 h > 0 时，图像向右平移 h 个单位；h < 0 时向左平移。注意反直觉的方向：y = f(x - 2) 将图像向右移动 2 个单位。该变换常写作 f(x + a)，表示沿向量 (–a, 0) 平移。

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7. Stretches: Vertical and Horizontal Scaling | 伸缩变换：纵向与横向拉伸

A vertical stretch multiplies the whole function by a constant: y = a f(x). If |a| > 1 the graph stretches away from the x‑axis (becomes taller); if 0 < |a| < 1 it compresses towards the x‑axis. Negative a also reflects in the x‑axis, which will be discussed in reflections.

纵向伸缩是将整个函数乘以一个常数：y = a f(x)。当 |a| > 1 时，图像远离 x 轴而拉伸（变高）；当 0 < |a| < 1 时，图像朝 x 轴压缩。a 为负值时还会同时发生关于 x 轴的反射，将在反射一节讨论。

A horizontal stretch operates on the x‑variable: y = f(bx). To maintain the same y‑value, the x‑coordinate is divided by b. Therefore, when |b| > 1 the graph is compressed horizontally towards the y‑axis; when 0 < |b| < 1 it stretches horizontally away from the y‑axis. The transformation is often written as y = f(cx) where c scales the x‑coordinates by factor 1/c.

横向伸缩作用于 x 变量：y = f(bx)。为保持相同的 y 值，x 坐标需除以 b。因此，当 |b| > 1 时，图像水平压缩向 y 轴；当 0 < |b| < 1 时，图像水平拉伸远离 y 轴。该变换常写作 y = f(cx)，其中 x 坐标的比例因子为 1/c。

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8. Reflections: Flipping Graphs | 反射变换：图像翻转

Reflection in the x‑axis is achieved by y = –f(x). Every point (x, y) on the original graph is mapped to (x, –y). For example, y = –x² is an upside‑down parabola.

关于 x 轴的反射通过 y = –f(x) 实现。原图像上的每一点 (x, y) 映射到 (x, –y)。例如 y = –x² 是一个倒置的抛物线。

Reflection in the y‑axis is given by y = f(–x). Points are mapped from (x, y) to (–x, y). This transformation can also be thought of as a horizontal stretch with factor –1. Combining both reflections yields y = –f(–x), which is a rotation of 180° about the origin if applied to odd or even functions appropriately.

关于 y 轴的反射表示为 y = f(–x)。点从 (x, y) 映射到 (–x, y)。该变换也可视为因子为 –1 的横向伸缩。同时应用两种反射得到 y = –f(–x)，对适当的奇函数或偶函数来说，其效果相当于绕原点旋转 180°。

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9. Combining Transformations | 组合变换

When multiple transformations are applied, the order matters. Usually you follow the standard sequence: first handle horizontal shifts and stretches inside the function, then handle vertical stretches and shifts outside. When a transformation is given in the form y = a f(b(x – h)) + k, you would start with the horizontal translation h, then the horizontal stretch/compression by factor 1/b, then the vertical stretch by a, and finally the vertical translation k.

多重变换同时施加时，顺序很关键。通常遵循标准顺序：先处理函数内部的自变量平移和伸缩，再处理外部的垂直伸缩和平移。当变换以 y = a f(b(x – h)) + k 的形式给出时，一般先执行水平平移 h，再进行因子为 1/b 的水平伸缩，然后进行因子为 a 的纵向伸缩，最后是垂直平移 k。

For example, to transform y = √x to y = 2√(3x – 6) + 1, rewrite the inside as 3(x – 2). This reveals a horizontal translation right by 2, then a horizontal compression by factor 1/3, followed by a vertical stretch by factor 2, and finally a vertical shift up by 1. Always express the x‑part in factored form to see the correct sequence.

例如，从 y = √x 变换到 y = 2√(3x – 6) + 1，先将内部改写为 3(x – 2)。由此看出：先向右平移 2，再进行因子为 1/3 的水平压缩，然后进行因子为 2 的纵向拉伸，最后向上平移 1。请务必将 x 部分写成因式分解形式，才能看清正确的变换顺序。

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10. Transformations of Trigonometric Functions | 三角函数图像的变换

For y = a sin(bx + c) + d, the amplitude is |a|, the period is 2π/|b|, the phase shift (horizontal translation) is –c/b, and the vertical shift is d. Similar rules apply to cosine and tangent, with the period of tangent being π/|b|. These parameters allow you to model oscillatory behaviour from tides to sound waves.

对于 y = a sin(bx + c) + d，振幅为 |a|，周期为 2π/|b|，相位平移（水平平移）为 –c/b，垂直平移为 d。类似规则适用于余弦和正切，其中正切周期为 π/|b|。这些参数可让你对从潮汐到声波的各种振动行为进行建模。

Write the function as y = a sin[b(x + c/b)] + d to identify the horizontal translation correctly. For example, y = 3 sin(2x – π) + 1 becomes y = 3 sin[2(x – π/2)] + 1, indicating a right shift of π/2, amplitude 3, period π, and vertical shift of 1.

将函数写成 y = a sin[b(x + c/b)] + d 的形式才能正确识别水平平移。例如 y = 3 sin(2x – π) + 1 可改写为 y = 3 sin[2(x – π/2)] + 1，表明向右平移 π/2，振幅为 3，周期为 π，垂直平移为 1。

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11. Graphs Involving Modulus Functions | 绝对值函数图像变换

The modulus function y = |f(x)| reflects any part of f(x) that lies below the x‑axis in the x‑axis. Points where f(x) ≥ 0 remain unchanged; where f(x) < 0, the graph is flipped upwards. This creates a graph that never goes below the x‑axis.

绝对值函数 y = |f(x)| 将 f(x) 位于 x 轴下方的部分反射到 x 轴上方。f(x) ≥ 0 的部分保持不变；f(x) < 0 的部分向上翻转。这样做出来的图像永远不会低于 x 轴。

The function y = f(|x|) removes the left‑hand side of the graph for x < 0 and replaces it with a reflection of the right‑hand side (for x > 0) in the y‑axis. The result is always symmetric about the y‑axis. For instance, y = |x| is simply the V‑shape, while y = |x² – 4| gives a W‑shape touching the x‑axis at the roots.

函数 y = f(|x|) 则去掉 x < 0 左侧的图像，将其替换为右侧图像（x > 0 部分）关于 y 轴的反射。最终图形总关于 y 轴对称。例如 y = |x| 就是简单的 V 形，而 y = |x² – 4| 则是一个在根处接触 x 轴的 W 形。

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12. Summary and Key Points | 总结与关键点

All graph transformations can be described neatly using function notation. Here is a summary table:

所有图像变换都可以用函数符号清晰地描述。以下为总结表：

Transformation	Notation	Effect on graph
Vertical translation	y = f(x) + k	Shift up by k (down if k < 0)
Horizontal translation	y = f(x – h)	Shift right by h (left if h < 0)
Vertical stretch	y = a f(x)	Multiply y‑coordinates by a (stretch if |a|>1)
Horizontal stretch	y = f(bx)	Divide x‑coordinates by b (compress if |b|>1)
Reflection in x-axis	y = -f(x)	Flip vertically
Reflection in y-axis	y = f(-x)	Flip horizontally

Always check the direction of horizontal transformations carefully, as they seem to operate the reverse of what the sign suggests. In exams, sketch graphs step by step and label coordinates of key points such as intercepts, stationary points and asymptotes after each transformation. Practise with a wide variety of functions: polynomials, reciprocals, exponentials, logarithms and trigonometric curves. Thorough mastery of graphs and transformations will strengthen your problem‑solving across differentiation, integration and equation solving.

务必仔细检查水平变换的方向，因为它们的方向往往与代数符号的直觉相反。在考试中，要逐步画出变换后的图像，并在每一步标出关键点的坐标，例如截距、驻点和渐近线。要针对多项式、倒数函数、指数、对数和三角函数曲线等多种函数进行练习。透彻掌握图像与变换将提升你在微分、积分和方程求解中的解题能力。

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