A-Level Pure Maths Unit 4: Graphs and Transformations – Higher Homework | A-Level 纯数学第4单元：函数图像与变换——进阶作业

📚 A-Level Pure Maths Unit 4: Graphs and Transformations – Higher Homework | A-Level 纯数学第4单元：函数图像与变换——进阶作业

This revision article covers the core skills in Edexcel A-Level Pure Mathematics Unit 4, pages 64-99. The focus is graph sketching and the algebra of transformations for polynomial functions. Use this as a higher homework guide to build fluency with translations, stretches, reflections, and graphical intersections.

本篇复习文章覆盖 Edexcel A-Level 纯数学第4单元（64-99页）的核心技能。重点在于多项式函数的图像绘制与变换代数。将本文作为进阶作业指南，帮助你熟练掌握平移、伸缩、对称以及图像交点等技能。

1. What Do We Mean by Transformation? | 什么是图像变换？

A transformation maps every point (x, y) on a graph to a new point (x’, y’). We can describe this using a mapping formula, for example (x, y) → (x + a, y + b). In Unit 4, we mainly study transformations that keep the shape of the graph unchanged but move or stretch it.

变换将图像上的每一点 (x, y) 映射到一个新点 (x’, y’)。我们可以用映射公式来描述，例如 (x, y) → (x + a, y + b)。在第4单元中，我们主要研究保持图形形状不变、但移动或拉伸位置的变换。

For a function y = f(x), vertical changes affect the y-coordinate while horizontal changes affect the x-coordinate. Remember that horizontal transformations often look ‘backwards’ because we replace x with x + a to shift left by a units.

对于函数 y = f(x)，垂直方向的变化影响 y 坐标，而水平方向的变化影响 x 坐标。请记住，水平变换常常看起来是“反向”的，因为我们将 x 替换为 x + a 后图像向左平移 a 个单位。

2. Translations: y = f(x) + a and y = f(x + a) | 平移变换：y = f(x) + a 与 y = f(x + a)

A vertical translation is given by y = f(x) + a. If a > 0, the graph moves up by a units; if a < 0, it moves down. For example, y = x² + 3 is the parabola y = x² shifted up by 3 units.

垂直平移由 y = f(x) + a 给出。若 a > 0，图像向上移动 a 个单位；若 a < 0，则向下移动。例如 y = x² + 3 就是抛物线 y = x² 向上平移 3 个单位。

A horizontal translation is written y = f(x + a). Here y = f(x + a) shifts the graph left by a units when a > 0, and right by a units when a < 0. So y = (x – 2)² moves the base parabola 2 units to the right.

水平平移写作 y = f(x + a)。当 a > 0 时，y = f(x + a) 将图像向左平移 a 个单位；当 a < 0 时，则向右平移。因此 y = (x – 2)² 将基础抛物线向右移动 2 个单位。

Vertical translation: y = f(x) + a   |   Horizontal translation: y = f(x + a)

3. Stretches: y = a f(x) and y = f(ax) | 伸缩变换：y = a f(x) 与 y = f(ax)

Vertical stretches are written y = a f(x). If a > 1, the graph is stretched vertically by a factor of a, moving points away from the x-axis. If 0 < a < 1, the graph is compressed vertically. Negative a also reflects the graph in the x-axis.

垂直伸缩写作 y = a f(x)。若 a > 1，图像在垂直方向拉伸为原来的 a 倍，使各点远离 x 轴。若

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