Chapter Review 4: Graphs and Transformations | 第4章复习：函数图像与变换

📚 Chapter Review 4: Graphs and Transformations | 第4章复习：函数图像与变换

This chapter review consolidates the key skills for Edexcel A-Level Pure Mathematics Chapter 4: recognising and sketching cubic, quartic and reciprocal graphs, and applying transformations to any given function y = f(x). The focus is on efficient curve sketching and writing new equations after translations, stretches and reflections.

本章复习巩固爱德思 A-Level 纯数学第4章的核心技能：识别并绘制三次、四次和倒数函数图像，以及对任意给定函数 y = f(x) 进行图像变换。重点是高效绘制曲线以及写出平移、伸缩和反射后的新方程。

1. Sketching Cubic Graphs | 绘制三次函数图像

A cubic curve has the general form y = ax³ + bx² + cx + d. The sign of a controls the end behaviour: if a > 0, the curve falls to −∞ on the left and rises to +∞ on the right; if a < 0, this is reversed.

三次曲线的一般式为 y = ax³ + bx² + cx + d。系数 a 的正负决定两端走势：若 a > 0，曲线左侧趋向 −∞、右侧趋向 +∞；若 a < 0 则相反。

To sketch a cubic, find the y-intercept at (0, d) and factorise where possible to locate x-intercepts. A repeated factor gives a stationary point on the x-axis, while a triple factor gives an inflection point that crosses the axis.

绘图时先找出 y 轴截距 (0, d)，并尽可能因式分解以确定 x 轴截距。重复因式表示 x 轴处有静止点；三重因式表示过 x 轴处为拐点。

For example, y = (x − 2)(x + 3)² has a root at x = 2 where the curve crosses the x-axis, and a repeated root at x = −3 where the curve touches the x-axis and turns.

例如，y = (x − 2)(x + 3)² 在 x = 2 处有一个根且曲线穿过 x 轴，在 x = −3 处有重根，曲线在 x 轴处相切并转向。

2. Sketching Quartic Graphs | 绘制四次函数图像

A quartic graph y = ax⁴ + bx³ + cx² + dx + e is shaped like a W or U depending on repeated roots. If a > 0, both ends rise to +∞; if a < 0, both ends fall to −∞.

四次函数图像 y = ax⁴ + bx³ + cx² + dx + e 根据重根不同可呈 W 形或 U 形。若 a > 0，两端都上升至 +∞；若 a < 0，两端都下降至 −∞。

Repeated factors such as (x − p)² create a bounce at the x-axis, while a single factor creates a crossing. A triple factor in a quartic

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