Mixed Exercise 4: Graphs and Transformations (Edexcel A-Level Maths) | 混合练习4：图形与变换（爱德思A-Level数学）

📚 Mixed Exercise 4: Graphs and Transformations (Edexcel A-Level Maths) | 混合练习4：图形与变换（爱德思A-Level数学）

Welcome to this in-depth revision guide for Mixed Exercise 4 from the Edexcel A-Level Pure Mathematics textbook. This mixed exercise consolidates the entire chapter on graphs and transformations, challenging you to combine cubic and quartic curve sketching with translations, stretches, and reflections. We will break down the most common question types, reinforce the crucial distinction between transformations of the form y = f(x + a) and y = f(x) + a, and sharpen your ability to read transformation descriptions directly from equations. Whether you are preparing for an end‑of‑topic test or the final A‑Level exam, this article will help you master the key skills tested in Mixed Exercise 4.

欢迎来到这篇针对爱德思A-Level纯数学教材混合练习4的深入复习指南。该混合练习将图形与变换这一整章的知识融会贯通，要求你把三次和四次函数图像的绘制与平移、拉伸和反射结合起来灵活运用。我们将解析最常见的考题类型，强化对形如 y = f(x + a) 和 y = f(x) + a 的变换之间关键区别的理解，并让你能从方程中直接读出变换描述。无论你是在准备单元测验还是最终的A-Level考试，这篇文章都将帮助你掌握混合练习4所考查的核心技能。

1. Overview of Mixed Exercise 4: What to Expect | 混合练习4概览：考查要点

Mixed Exercise 4 appears at the end of Chapter 4 (Graphs and Transformations) in the Pearson Edexcel Pure Mathematics Year 1/AS textbook. It contains a carefully curated set of exam‑style questions that require you to move fluently between the algebraic form of a function and its geometric representation. Expect to see a mix of sketching tasks, ‘describe the transformation’ questions, and problems that ask you to find the equation of a transformed curve given its original equation and a description or sequence of changes. Many questions also test your ability to identify the coordinates of key points—such as turning points and intercepts—after a transformation has been applied.

混合练习4出现在培生爱德思纯数学第一册/AS教材第4章（图形与变换）的末尾。它包含了一套精心设计的考试风格题目，要求你能在函数的代数形式和它的几何表示之间自如转换。你将看到绘图任务、“描述变换”问题，以及给出原方程和一段变化描述后求变换后曲线方程的题目。许多题目还会考查你应用变换后确定关键点（如驻点和截距）坐标的能力。

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2. Sketching Cubic and Quartic Graphs: The Foundation | 绘制三次与四次函数图像：基础功

Before tackling transformations, Mixed Exercise 4 assumes you can sketch basic cubic (degree 3) and quartic (degree 4) graphs. For y = ax³ + bx² + cx + d, sign of a determines whether the curve rises to the right (a > 0) or falls to the right (a < 0). Quartic graphs of the form y = ax⁴ + bx³ + ... have a general ‘W’ or ‘M’ shape depending on the leading coefficient a. You must also recall how repeated roots affect the graph: a double root (x − p)² will touch the x‑axis at p without crossing it, while a triple root (x − p)³ produces a point of inflection with crossing. Always begin by factorising where possible and note the y‑intercept (x = 0).

在应对变换之前，混合练习4要求你能够画出基本的三次（三次方）和四次（四次方）函数图像。对于 y = ax³ + bx² + cx + d，a 的正负决定了曲线是向右上升（a > 0）还是向右下降（a < 0）。形如 y = ax⁴ + bx³ + … 的四次图像会根据首项系数 a 呈现大致的“W”或“M”形状。你还需要牢记重根如何影响图形：二重根 (x − p)² 会在 p 处触碰 x 轴而不穿过，三重根 (x − p)³ 则会产生一个带有拐点并穿过的图形。务必在可能的时候进行因式分解，并标注 y 轴截距（x = 0）。

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3. Horizontal and Vertical Translations: The Most Common Steps | 水平和垂直平移：最常见的步骤

A horizontal translation moves the whole graph left or right. The transformation y = f(x + a) shifts the graph of y = f(x) by −a units in the x‑direction: a positive a moves the graph a units to the left, a negative a moves it to the right. For a vertical translation, y = f(x) + a moves the graph a units upward when a > 0 and downward when a < 0. Mixed Exercise 4 often disguises translations inside cubics, for example, asking you to sketch y = (x − 2)³ + 1 by recognising it as a translation of y = x³ by the vector (2, 1).

水平平移将整条曲线向左或向右移动。变换 y = f(x + a) 将 y = f(x) 的图像沿 x 轴方向平移 −a 个单位：a 为正时图像向左移动 a 个单位，a 为负时向右移动。对于垂直平移，y = f(x) + a 在 a > 0 时将图像向上移动 a 个单位，a < 0 时向下移动。混合练习4常将平移隐藏在三次函数中，例如要求你通过识别 y = (x − 2)³ + 1 是 y = x³ 经向量 (2, 1) 平移得到来进行绘图。

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4. Stretches in the x and y Directions: Scaling Shapes | 沿 x 和 y 方向的拉伸：缩放图形

A stretch in the y‑direction multiplies all y‑coordinates by a scale factor k. This is written as y = k f(x). When k > 1, the graph is stretched away from the x‑axis; when 0 < k < 1, it is compressed towards the x‑axis. A stretch in the x‑direction, written as y = f(ax), multiplies the x‑coordinates by the reciprocal 1/a. Because this operates inside the function bracket, many students get the direction wrong: a > 1 squashes the graph horizontally, while 0 < a < 1 stretches it horizontally. Mixing these up is a classic exam pitfall, so always test with a single point, such as the turning point, to verify your thinking.

沿 y 方向的拉伸会将所有 y 坐标乘以一个比例因子 k，记作 y = k f(x)。当 k > 1 时，图像向远离 x 轴的方向拉伸；当 0 < k < 1 时，图像朝 x 轴压缩。沿 x 方向的拉伸写作 y = f(ax)，它将 x 坐标乘以倒数 1/a。由于这一操作发生在函数括号内部，许多学生会搞错方向：a > 1 会让图像水平压缩，而 0 < a < 1 则会让图像水平拉伸。混淆这两种情况是考试中的经典陷阱，因此务必通过一个点（例如驻点）进行检验，以确认你的判断。

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5. Reflections in the Coordinate Axes: Flipping Graphs | 在坐标轴上的反射变换：翻转图形

Two fundamental reflections are tested: y = −f(x) reflects the graph in the x‑axis, flipping all positive y‑values to negative and vice versa. The reflection y = f(−x) flips the graph in the y‑axis: the shape is mirrored left‑to‑right. Mixed Exercise 4 sometimes combines these, such as asking for the effect of y = −f(−x), which is a rotation by 180° about the origin. When working with cubics and quartics, also notice that if a function is odd, f(−x) = −f(x), and if it is even, f(−x) = f(x); these symmetry properties can simplify your work enormously.

考查两种基本的反射变换：y = −f(x) 将图像关于 x 轴反射，把所有正的 y 值翻转为负值，反之亦然。反射变换 y = f(−x) 则是关于 y 轴翻转图像：图形左右镜像。混合练习4有时会将这些组合在一起，比如要求你说明 y = −f(−x) 的效果，即图像绕原点旋转 180°。在处理三次和四次函数时，还要注意：如果函数是奇函数，则 f(−x) = −f(x)；如果是偶函数，则 f(−x) = f(x)。这些对称性质可以大大简化你的工作。

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6. Composite Transformations: The Crucial Order | 复合变换：顺序至关重要

When a function undergoes two or more transformations in one equation, careful sequencing is essential. For example, consider transforming y = f(x) into y = 2 f(x + 3). There are two possible routes: (i) translate left by 3, then stretch vertically by factor 2; (ii) stretch vertically by factor 2, then translate left by 3. Both produce the same final graph. However, if you transform y = f(x) into y = f(2x + 3), rewriting as y = f(2(x + 1.5)) reveals that the horizontal translation by −1.5 occurs after the horizontal stretch by factor 1/2. Mixed Exercise 4 will ask you to describe a sequence in the correct order, and many marks depend on getting this right. Use the ‘bracket method’ for x‑transformations to avoid losing track of the inside changes.

当一个函数在一个方程中经历两次或更多变换时，仔细的先后顺序至关重要。例如，考虑将 y = f(x) 变换为 y = 2 f(x + 3)。有两种可能的路径：(i) 先向左平移 3，再垂直拉伸因子 2；(ii) 先垂直拉伸因子 2，再向左平移 3。两者得到的最终图像是一样的。但如果你要将 y = f(x) 变换为 y = f(2x + 3)，将其改写为 y = f(2(x + 1.5)) 就可看出，水平平移 −1.5 发生在水平拉伸因子 1/2 之后。混合练习4会要求你按正确顺序描述变换序列，很多分数都取决于这一点。处理 x 变换时使用“括号法”，避免迷失括号内部的变化。

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7. Using Transformations to Find Equation Roots | 利用变换求方程的根

Transformations can map the known roots of one equation to the roots of another. If y = f(x) crosses the x‑axis at x = p, q, r, then y = f(x + a) will cross at p − a, q − a, r − a, because all x‑coordinates are shifted. Similarly, y = f(ax) produces roots p/a, q/a, r/a. Mixed Exercise 4 may ask: ‘Given the curve y = (x + 1)(x − 2)(x − 4), find the roots of the equation y = (2x + 1)(2x − 2)(2x − 4).’ Recognising that the second equation is a horizontal stretch of the first enables you to write down the roots without fully expanding. This approach saves time and reduces algebraic errors.

变换可以将一个方程的已知根映射到另一个方程的根。如果 y = f(x) 与 x 轴交于 x = p, q, r，那么 y = f(x + a) 将与轴交于 p − a, q − a, r − a，因为所有的 x 坐标都平移了。类似地，y = f(ax) 产生的根为 p/a, q/a, r/a。混合练习4可能会这样问：“已知曲线 y = (x + 1)(x − 2)(x − 4)，求方程 y = (2x + 1)(2x − 2)(2x − 4) 的根。”识别出第二个方程是第一个的水平拉伸，你就可以直接写出根而无须完全展开。这种方法能节省时间并减少代数错误。

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8. Applying Transformations to Asymptotes | 变换应用于渐近线

Although Mixed Exercise 4 is centred on polynomials, some questions touch on rational functions and their asymptotes, linking to later topics. If y = 1/x has a horizontal asymptote y = 0 and a vertical asymptote x = 0, then y = 1/(x − 2) + 3 has a vertical asymptote at x = 2 (right shift) and a horizontal asymptote at y = 3 (up shift). Transformations apply to asymptotes in exactly the same way as they apply to the curve itself. When you describe a transformation, always state how each asymptote moves; a common examination requirement is: “Write down the equations of the asymptotes of the transformed curve.”

尽管混合练习4以多项式为中心，但有些题目会涉及有理函数及其渐近线，并连接后续的课题。如果 y = 1/x 有一条水平渐近线 y = 0 和一条垂直渐近线 x = 0，那么 y = 1/(x − 2) + 3 的垂直渐近线为 x = 2（向右平移），水平渐近线为 y = 3（向上平移）。变换作用于渐近线的方式与作用于曲线本身的方式完全相同。当你在描述一个变换时，务必说明每条渐近线是如何移动的；一个常见的考试要求是：“写出变换后曲线渐近线的方程。”

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9. Matching Equations to Graphs: Reading Visual Patterns | 匹配方程与图像：阅读视觉模式

A favourite task in Mixed Exercise 4 presents four or five graphs and an equal number of equations, asking you to pair them up correctly. To succeed, scan key features: the end behaviour (what happens as x → ±∞), the position of intercepts and turning points, and the effect of any negative signs. For example, y = (2 − x)(x + 1)² and y = (x − 2)(x + 1)² differ by a factor of −1 in the linear term, which produces a reflection in the y‑axis combined with a horizontal translation—or simply a different set of roots. Rather than plotting points, use logic: identify the sign of the leading coefficient first; then locate the y‑intercept; finally check the x‑intercept behaviour (crossing vs. touching).

混合练习4中有一个常见的任务，即给出四到五幅图像和同样数量的方程，要求你正确配对。要取得成功，你需要扫描关键特征：末端走势（当 x → ±∞ 时的变化趋势）、截距和驻点的位置，以及任何负号的影响。例如，y = (2 − x)(x + 1)² 和 y = (x − 2)(x + 1)² 的一次项相差一个因子 −1，这会产生一个关于 y 轴的反射伴以水平平移——或仅仅是一组不同的根。与其描点绘图，不如运用逻辑：首先确定首项系数的正负；然后找出 y 轴截距；最后检查 x 截距的性质（穿过还是触碰）。

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10. Solving Transformation Problems Step by Step | 逐步解决变换问题

A structured approach transforms a messy transformation description into a clean solution. Follow these steps: (1) Identify the parent function y = f(x). (2) List all transformations in the exact order they will be executed, using function notation. For instance, “stretch vertically by factor 3, then reflect in the x‑axis” becomes step A: y = 3 f(x), step B: y = −(3 f(x)) = −3 f(x). (3) Work out the new coordinates of any given key points. (4) Apply transformations to the whole equation, simplifying where necessary. Mixed Exercise 4 often asks for the final equation of a transformed cubic, so once you have constructed the new equation, expand it to standard form if the question demands it.

用结构化的方法可以把一段杂乱的变换描述转化为清晰的解答。请遵循以下步骤：(1) 确定母函数 y = f(x)。(2) 按执行顺序列出所有变换，并使用函数符号。例如，“先垂直拉伸因子 3，再关于 x 轴反射”就变成步骤 A：y = 3 f(x)，步骤 B：y = −(3 f(x)) = −3 f(x)。(3) 计算所有给定关键点的新坐标。(4) 把变换应用到整个方程，必要时进行化简。混合练习4经常要求写出变换后三次函数的最终方程，因此一旦你构建出了新方程，若题目要求则将它展开为标准形式。

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11. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Even confident students lose marks on Mixed Exercise 4 due to predictable slips. The most frequent error is misapplying horizontal transformations: writing y = f(x + 2) and thinking the graph moves right. Remember the ‘+2’ sits inside the bracket, so the shift is left by 2. Another mistake is failing to factorise the coefficient of x before identifying horizontal stretches and translations—always write f(ax + b) as f(a(x + b/a)). Also, do not forget that a stretch must be described with its factor and direction, e.g., “stretch by factor 2 in the y‑direction” is precise; “multiply y‑values by 2” is also acceptable. Vague language such as “it gets bigger” earns no marks. Finally, when a graph is reflected in the y‑axis, check that the x‑intercepts change sign, but the y‑intercept stays the same only for even functions; for odd functions, the y‑intercept may change sign as well.

即使是自信的学生，也会因一些可预见的失误在混合练习4中丢分。最常见的错误是错误应用水平变换：写成 y = f(x + 2) 却认为图像会向右移动。请记住，“+2”位于括号内部，因此是向左平移 2 个单位。另一个错误是在识别水平拉伸和平移之前未能提取 x 的系数——务必把 f(ax + b) 写成 f(a(x + b/a))。此外，不要忘记拉伸必须描述其因子和方向，例如，“沿 y 方向拉伸因子 2”就是准确的；“将 y 值乘以 2”也是可以接受的。诸如“图形变大了”这样含糊的语言是得不到分数的。最后，当图像关于 y 轴反射时，注意 x 截距会变号，而 y 截距只有在偶函数时才不变；对于奇函数，y 截距也可能变号。

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12. Final Tips for Acing Mixed Exercises | 高手技巧

To excel in Mixed Exercise 4 and similar revision sets, practise reading transformation information both from the equation and from a sketch. Draw a quick set of coordinate axes and label key points before and after transformations—this visual check catches sign errors instantly. When you get stuck on a multi‑step problem, break it down and tackle one change at a time, writing intermediate equations. Use the transformation rules as a checklist:

要在混合练习4和类似的复习题集中取得优异成绩，你需要练习既能从方程中、也能从草图里读出变换信息。画一组简易的坐标轴，并标注变换前后的关键点——这种视觉检查能立刻发现符号错误。当你在多步骤问题中卡住时，不妨将其分解开来，一次只处理一处变化，并写下中间方程。将变换规则当作一份清单来使用：

• y = f(x) + a → vertical translation by a
• y = f(x + a) → horizontal translation by −a
• y = k f(x) → vertical stretch by k
• y = f(ax) → horizontal stretch by 1/a
• y = −f(x) → reflection in x‑axis
• y = f(−x) → reflection in y‑axis
• y = f(x) + a → 垂直平移 a
• y = f(x + a) → 水平平移 −a
• y = k f(x) → 垂直拉伸 k
• y = f(ax) → 水平拉伸 1/a
• y = −f(x) → 关于 x 轴反射
• y = f(−x) → 关于 y 轴反射

By internalising these facts and practising Mixed Exercise 4 repeatedly, you will build the speed and confidence needed for the exam.

通过内化这些知识点并反复练习混合练习4，你将在考试中培养出所需的速度和信心。

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