📚 Mixed Exercise 4: Graphs and Transformations | 混合练习4：图形与变换
d to apply translations, stretches and reflections correctly to a given function y = f(x).
混合练习4是Edexcel A Level纯数学课程第4章的综合性练习，重点巩固图形与变换知识。它考查你绘制三次、四次和反比例曲线的能力，以及能否正确对给定函数 y = f(x) 施加平移、拉伸和反射变换。
1. What Mixed Exercise 4 Covers | 混合练习4涵盖内容
y = x³, y = x⁴, y = 1/x, y = 1/x² and transformed versions, often on the same set of axes.
该混合练习汇集第4章所有核心技能。题目会要求你画出 y = x³、y = x⁴、y = 1/x、y = 1/x² 及其变换后版本，且常在同一坐标系中呈现。
- Sketch standard polynomial and reciprocal graphs / 绘制标准多项式与反比例图形
- Apply translations, stretches and reflections / 应用平移、拉伸和反射变换
- Find coordinates after transformation / 求变换后的坐标
- Solve equations graphically / 利用图形求解方程
2. Recognising Polynomial Graphs | 识别多项式图形
∞, and as x → -∞, y → -∞, so the graph runs from the third quadrant to the first quadrant.
多项式图形的末端走势由其最高次幂决定。对于三次图形如 y = x³，当 x → ∞ 时 y → ∞；当 x → -∞ 时 y → -∞，因此图形从第三象限延伸至第一象限。
y = x³: x → -∞ → y → -∞, x → ∞ → y → ∞
ends point upwards because any negative x becomes positive after an even power. This gives a U-shaped curve that may have up to four real roots.
对于四次图形如 y = x⁴，两端都朝上，因为负数的偶次幂为正。这会产生一条U形曲线，最多可有四个实根。
3. Sketching Cubic Curves | 绘制三次曲线
ch as y = (x - p)²(x - q), the curve touches the x-axis at x = p and does not cross there.
当三次函数分解为 y = (x - p)(x - q)(x - r) 时，图形在 x = p、q、r 处穿过 x 轴。如果某个因式为平方，如 y = (x - p)²(x - q)，曲线在 x = p 处与 x 轴相切，不会穿过。
Cross at single roots: y = (x - a)(x - b)(x - c)
Touch at repeated root: y = (x - a)²(x - b)
| Factor form / 因式形式 | Behaviour at root / 根处行为 |
| (x - a) | Crosses x-axis / 穿过 x 轴 |
| (x - a)² | Touches x-axis / 相切于 x 轴 |
| (x - a)³ | Crosses with a flatter inflection / 以更平缓的拐点穿过 |
4. Sketching Quartic Curves | 绘制四次曲线
x⁴ has a single minimum at the origin and is flatter near x = 0 than the quadratic y = x².
四次图形根据拐点数量，可呈标准U形或W形。最简单的四次函数 y = x⁴ 在原点有一个最小值，并且在 x = 0 附近比二次函数 y = x² 更平缓。
y = x⁴ has minimum at (0, 0)
= a and x = b. This produces a W shape with two minima and one local maximum between them.
如果四次函数写作 y = (x - a)²(x - b)²，图形在 x = a 和 x = b 处都与 x 轴相切。这会形成W形，
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