📚 Edexcel A-Level Maths Chapter 4: Graphs and Transformations | 爱德思A-Level数学第四章：函数图像与变换
In Edexcel A-Level Mathematics, Chapter 4 lays the groundwork for understanding how functions behave visually. It covers the shapes of polynomials, reciprocal functions, and the rules for transforming any graph using translations, stretches, and reflections. Mastering these concepts is not only essential for Paper 1 Pure Mathematics but also deepens your intuition for calculus, curve sketching, and solving equations graphically.
在爱德思A-Level数学课程中，第四章为理解函数的图像表现方式打下基础。本章涵盖了多项式函数、反比例函数的形状，以及利用平移、拉伸和反射变换任意图像的方法。掌握这些概念不仅对纯数试卷一至关重要，也能加深你对微积分、曲线描绘以及利用图像解方程的直观理解。
1. Basic Polynomial Shapes and Their Leading Coefficients | 基本多项式形状与首项系数
the polynomial, together with the sign of the leading coefficient, determines the overall 'end behaviour' of its graph. For example, y = x2 is a U-shaped parabola, while y = -x2 opens downwards. Understanding these end behaviours allows you to quickly sketch the rough shape without plotting every point.
多项式函数是若干个 x 的幂次与常系数之和。多项式的次数连同首项系数的正负共同决定了其图像的“末端走向”。例如，y = x² 是开口向上的 U 形抛物线，而 y = -x² 开口向下。理解这些末端走向能让你在不逐点描点的情况下迅速勾勒出大致形状。
When the leading coefficient is positive, the far right of the graph rises for any degree; when negative, it falls. The far-left behaviour depends on whether the degree is even or odd. This simple rule, combined with knowledge of the y-intercept (the constant term), gives you a powerful sketching tool.
当首项系数为正时，不论次数为多少，图像最右端均向上延伸；若为负则向下。最左端的行为取决于次数是偶数还是奇数。这一简单规则，配合对 y 轴截距（常数项）的了解，为你提供了一种强大的描绘工具。
2. Cubic Graphs: y = ax³ + bx² + cx + d | 三次函数图像：y = ax³ + bx² + cx + d
m left, may have up to two turning points, and then goes off to the top right. The general shape resembles an elongated 'S' or a single inflection with no turning points, as seen in y = x³.
三次函数能呈现出若干种典型形状。当首项系数 a > 0 时，图像通常从左下方上升，最多可有两个驻点，然后向右上方延伸。其大致形状类似于拉长的“S”形，或只有一个拐点而无驻点，如 y = x³ 所示。
rised into three linear factors, say y = (x - p)(x - q)(x - r), the graph will cross the x‑ rossing. The middle root often gives a steeper crossing when all three roots are distinct.
若三次函数可分解为三个一次因式，例如 y = (x - p)(x - q)(x - r)，其图像将在 x = p、q、r 处穿过 x 轴。重复根如 (x - p)² 则使曲线在 p 点接触 x 轴而不穿越。当三个根各不相同时，中间的根往往对应更陡峭的穿越行为。
3. Quartic Graphs: y = ax⁴ + bx³ + ... | 四次函数图像：y = ax⁴ + bx³ + ……
Quartic functions, like quadratics, have a W‑shaped (or inverted‑W) appearance for many cases, with up to three turning points. If the leading coefficient a is positive, both ends point upwards, giving a shape similar to a smile. If a is negative, both ends point downwards, like a frown.
四次函数在许多情况下呈现 W 形（或倒 W 形），最多可有三个驻点。若首项系数 a 为正，则两端均指向上方，类似微笑形状；若 a 为负，两端均指向下方，类似皱眉形状。
The presence of squared factors, such as (x - p)²(x - q)², yields a graph that touches the x‑axis at both p and q without crossing. Factorising quartics helps identify x‑intercepts and understand symmetry where it exists. Even‑power polynomials are never one‑to‑one over their whole domain.
若存在平方因式，如 (x - p)²(x - q)²，图像将在 p 和 q 两处接触 x 轴而不穿越。分解四次因式有助于确定 x 轴截距，并在存在对称性的情况下理解其特点。偶次幂多项式在其整个定义域上永远不是一一映射。
4. Reciprocal Graphs and Asymptotes | 反比例函数图像与渐近线
ants, with the axes acting as asymptotes: the graph approaches but never reaches x = 0 and y = 0. For y = k/x with negative k, the branches swap quadrants.
形如 y = k/x 或 y = k/x² 的反比例函数具有鲜明的双曲线形状。y = 1/x 的图像由两支组成，分别位于对角的象限，以坐标轴为渐近线：图像无限接近但永不到达 x = 0 和 y = 0。对于 k 为负的 y = k/x，两分支会交换象限。
The function y = a/x² has both branches lying on the same side of the x‑axis; if a is positive, both branches are above the x‑axis. As x → ±∞, y → 0, and as x → 0, |y| ertical and horizontal asymptotes are crucial landmarks when sketching rational functions.
函数 y = a/x² 的两支位于 x 轴同侧；若 a 为正，两支均在 x 轴上方。当 x → ±∞ 时 y → 0，当 x → 0 时 |y| → ∞。描绘有理函数图像时，垂直渐近线与水平渐近线是不可或缺的关键参照。
5. Recognising Points of Intersection Graphically | 从图像上识别交点
Solving an equation f(x) = g(x) is equivalent to finding the x‑ set of axes and read off approximate solutions. This visual approach is particularly useful for transcendental equations that cannot be solved exactly.
求解方程 f(x) = g(x) 等价于找出 y = f(x) 与 y = g(x) 两图像交点的 x 坐标。无需进行代数移项，你可以将两图像绘制在同一坐标系中并读取近似解。这种可视化的方法对于无法精确求解的超越方程尤其有用。
an see there is exactly one point of intersection in the first quadrant, plus a second intersection you could approximate using a sketched graph. Always label axes clearly when presenting your reasoning.
例如，方程 x² = 1/x 的解即抛物线 y = x² 与双曲线 y = 1/x 的交点。你可以看到在第一象限恰好有一个交点，此外还有第二个交点可通过草图近似取得。展示推理过程时，请务必清晰标注坐标轴。
6. Translating Graphs: y = f(x) + a and y = f(x + a) | 平移变换：y = f(x) + a 与 y = f(x + a)
a shifts the whole curve vertically: up by a units when a > 0, down when a < 0. The shape remains identical; only its vertical position changes.
平移变换将图像上每一点沿给定方向移动相同距离。变换 y = f(x) + a 使整条曲线竖直移动：当 a > 0 时向上平移 a 个单位，a < 0 时向下平移。形状保持不变，仅改变竖直位置。
ph horizontally. If a is positive, the graph moves a units to the left; if negative, it moves to the right. This is a common source of confusion: remember that 'adding inside the bracket moves left'. Replace x with (x + a) and the whole graph is translated along the x‑axis.
与此相反，y = f(x + a) 将图像沿水平方向平移。若 a 为正，图像向左移 a 个单位；若为负则向右移。这是一个常见的混淆点：请牢记“括号内相加向左移”。将 x 替换成 (x + a)，整条图像便沿 x 轴平移。
by 2. These transformations apply identically to any function, including trigonometric, exponential, and log graphs.
例如，以 y = x² 为出发点，曲线 y = x² + 3 是向上平移 3 个单位的抛物线，而 y = (x - 2)² 则向右平移 2 个单位。这些变换同样适用于任何函数，包括三角、指数和对数图像。
7. Stretching Graphs: y = af(x) and y = f(ax) | 拉伸变换：y = af(x) 与 y = f(ax)
A stretch in the y‑ nstant a. The transformation y = af(x) stretches the graph vertically by a scale factor of |a|. If |a| > 1, the graph is pulled away from the x‑axis; if 0 < |a| < 1, it is compressed towards the axis. Negative a also reflects the graph in the x‑axis.
当你用常数 a 乘以整个函数时，就会发生 y 方向的拉伸。变换 y = af(x) 以因子 |a| 沿竖直方向拉伸图像。若 |a| > 1，图像被拉离 x 轴；若 0 < |a| < 1，图像向 x 轴压缩。负的 a 还会令图像关于 x 轴反射。
A stretch in the x‑ion comes from y = f(ax). Here the graph is stretched horizontally by a scale factor of 1/|a|. An 'inside multiple' greater than 1 compresses the graph towards the y‑ n 0 and 1 stretches it outwards. The point (1, f(1)) on the original moves to (1/a, f(1)).
x 方向的拉伸则来自 y = f(ax)。此时图像以因子 1/|a| 沿水平方向拉伸。括号内倍数大于 1 时会将图像向 y 轴压缩，使其变窄；倍数介于 0 与 1 之间则向外拉伸。原图上点 (1, f(1)) 将移到 (1/a, f(1))。
| Transformation | Effect on y = f(x) |
|---|---|
| y = 2f(x) | Vertical stretch by factor 2; all y‑coordinates doubled. |
| y = f(2x) | Horizontal stretch by factor ½; graph becomes narrower. |
| y = -f(x) | Reflect in the x‑axis, equivalent to vertical stretch by -1. |
| y = f(-x) | Reflect in the y‑axis. |
中文翻译：
| 变换 | 对 y = f(x) 的效果 |
|---|---|
| y = 2f(x) | 竖直方向拉伸因子 2；所有 y 坐标加倍。 |
| y = f(2x) | 水平方向拉伸因子 ½；图像变窄。 |
| y = -f(x) | 关于 x 轴反射，相当于竖直拉伸因子 -1。 |
| y = f(-x) | 关于 y 轴反射。 |
8. Reflections: y = -f(x) and y = f(-x) | 反射变换：y = -f(x) 与 y = f(-x)
scale factor is exactly 1 or -1. The transformation y = -f(x) reflects the graph in the x‑axis, flipping all y‑ r opposites. All points that were above the axis move below, and vice versa, leaving the x‑intercepts unchanged.
反射变换是拉伸因子恰好为 1 或 -1 的特例。变换 y = -f(x) 将图像关于 x 轴反射，使所有 y 坐标变为相反数。原在轴上方的点全部移至下方，反之亦然，而 x 轴截距保持不变。
On the other hand, y = f(-x) reflects the graph in the y‑ ch as cos x or x², remain unchanged under y = f(-x) because they are symmetric about the y‑axis.
另一方面，y = f(-x) 将图像关于 y 轴反射。施行该变换时，你将每一个 x 替换为 -x，相当于左右对调。偶函数，如 cos x 或 x²，在 y = f(-x) 下保持不变，因为它们本身就关于 y 轴对称。
When a reflection is combined with a translation, the order in which you perform the steps can alter the final result. For instance, translating first and then reflecting will generally give a different graph than reflecting first and then translating.
当反射与平移结合时，操作的先后顺序会影响最终结果。例如，先平移再反射得到的图像通常与先反射再平移不同。
9. Combining Transformations: Order Matters | 复合变换：顺序至关重要
he innermost bracket outwards. For example, to obtain y = 2f(x + 3) from y = f(x), first apply the horizontal translation x → x + 3 (shift left by 3), then apply the vertical stretch by factor 2.
当一个函数连续经历两次或更多变换时，你必须按正确顺序逐一进行。核心原则为从最内层括号向外操作。例如，要从 y = f(x) 得到 y = 2f(x + 3)，首先进行水平平移 x → x + 3（左移 3），然后再进行竖直方向因子为 2 的拉伸。
horizontal stretch by factor ½ (from the 2 inside) should be performed before the horizontal shift right by 2. This order – stretch, then translate – ensures the translation is not stretched itself.
若表达式写作 y = f(2x - 4)，请先改写为 y = f(2(x - 2))。水平方向因子 ½ 的拉伸（源自括号内 2）应在水平右移 2 个单位之前进行。这样的顺序——先拉伸再平移——可确保平移量本身不被拉伸。
- General rule for y = af(b(x - c)) + d:
- Start with y = f(x).
- Replace x with (x - c) to shift horizontally by c.
- Apply the stretch b in the x‑direction (factor 1/|b|).
- Apply the stretch a in the y‑direction (factor |a|).
- Finally, add d to shift vertically.
中文：
- y = af(b(x - c)) + d 的一般变换顺序：
- 从 y = f(x) 开始。
- 用 (x - c) 替换 x，水平平移 c 个单位。
- 进行 x 方向拉伸 b（因子 1/|b|）。
- 进行 y 方向拉伸 a（因子 |a|）。
- 最后加上 d，竖直平移 d 个单位。
Misordering these steps is one of the most common mistakes in A‑Level exams. Always double‑check by testing a specific point, such as (0, f(0)), to see if the transformed coordinates match your expectation.
在这些步骤中顺序出错是 A-Level 考试中最常见的错误之一。请始终通过检查具体点（例如 (0, f(0))）来复核，看变换后的坐标是否与预期一致。
10. Applying Transformations to Sketch Unfamiliar Curves | 运用变换描绘陌生曲线
Once you have mastered the parent graphs – x², x³, 1/x, √x, eˣ, ln x, sin x, cos x – you can sketch a vast family of related curves without needing a calculator. Simply identify the sequence of transformations that connects the given function to the parent. Label the asymptotes, intercepts, and turning points where possible.
一旦你掌握了父函数的图像——x²、x³、1/x、√x、eˣ、ln x、sin x、cos x——就能在不借助计算器的情况下描绘出大量相关的曲线。只需找出将给定函数与父函数联系起来的变换序列即可。尽可能地标注渐近线、截距和驻点。
1 and the horizontal asymptote to y = 2. The branches keep their hyperbolic shape but are relocated.
例如，要描绘 y = 3/(x - 1) + 2，可先从 y = 1/x 入手。竖直拉伸因子 3，右移 1、上移 2。垂直渐近线移到了 x = 1，水平渐近线移到了 y = 2。两支曲线仍保持双曲线形状，但位置发生了改变。
Similarly, y = -e⁻²ˣ can be viewed as y = eˣ, reflected in both axes, then compressed horizontally and reflected in the x‑ rmation systematically yields an accurate sketch even when the algebra looks intimidating.
类似地，y = -e⁻²ˣ 可视为 y = eˣ 经两轴反射、水平压缩以及 x 轴反射而得。只要系统地追踪每次变换，即使代数形式看起来复杂，也能得出准确的草图。
Remember to always state the equations of any asymptotes in your final answer, especially for rational and exponential graphs. A well‑annotated sketch that clearly shows the effects of each transformation scores full marks in Edexcel examination questions.
请记住，在最终答案中务必标出所有渐近线的方程，尤其对于有理函数和指数函数图像。一幅清晰展示每项变换效果的标注草图，足以在爱德思考试题中获得满分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图