📚 2 Changing places: relationships and connections | 位置变化：数学中的关系与联系
In Edexcel A-Level Mathematics, the phrase ‘changing places’ can be interpreted through graph transformations, coordinate mappings, and function relationships. When a curve is translated, reflected, or stretched, every point changes place in a controlled way, and these new positions preserve connections such as roots, asymptotes, turning points, domain and range. Understanding these relationships is essential for Pure Mathematics papers and for modelling real motion.
在 Edexcel A-Level 数学中，“位置变化”可以通过图像变换、坐标映射和函数关系来理解。当曲线被平移、反射或拉伸时，每一个点都以可控方式改变位置，而这些新位置仍然保留与零点、渐近线、极值点、定义域和值域的联系。掌握这些关系对于纯数试卷和实际运动建模都至关重要。
1. Understanding ‘Changing Places’ in A-Level Maths | 理解 A-Level 数学中的“位置变化”
In the Edexcel specification, transformations are not just a set of rules to memorise; they describe connections between an original graph y = f(x) and a new graph after a change of place. We call these basic changes translations, reflections, and stretches. Each transformation maps every point (x, y) on the original curve to a new point (x′, y′).
在 Edexcel 考纲中，变换不仅是一组需要记忆的规则；它们描述了原图像 y = f(x) 与位置变化后新图像之间的联系。这些基本变化称为平移、反射和拉伸。每一种变换都会把原曲线上的每个点 (x, y) 映射到新点 (x′, y′)。
A key exam skill is to identify the parent function and then describe exactly how the graph has moved. This skill appears in Pure Year 1 and Year 2 papers, often asking you to sketch a transformed curve, find new coordinates of turning points, or state new domain and range.
一项关键的考试技能是识别母函数，然后准确描述图像如何移动。这一技能出现在纯数学 Year 1 和 Year 2 试卷中，题目通常要求你画出变换后的曲线、求出极值点的新坐标，或者写出新的定义域和值域。
2. Key Transformations: Translations | 关键变换：平移
A vertical translation changes the y-coordinate of every point by the same amount. The graph of y = f(x) + a is the graph of y = f(x) moved upwards by a units if a > 0, and downwards by |a| units if a < 0. This preserves shape, but roots are changed and horizontal asymptotes move.
垂直平移使每个点的 y 坐标改变相同的量。y = f(x) + a 的图像是 y = f(x) 的图像沿 y 轴方向平移 a 个单位：a > 0 时向上，a < 0 时向下。形状不变，但零点会改变，水平渐近线也会移动。
A horizontal translation changes the input: y = f(x + a) moves the graph left by a units when a > 0, because the input reaches a given value earlier. Many students expect ‘plus’ to move right, but the graph goes the opposite way. For example, y = f(x − 3) is a translation 3 units to the right.
水平平移改变输入：y = f(x + a) 将图像向左平移 a 个单位（a > 0），因为输入更早达到给定值。许多学生以为“加”会向右移动，但图像移动方向正好相反。例如，y = f(x − 3) 表示向右平移 3 个单位。
3. Reflections in the Coordinate Axes | 坐标轴反射
Reflections create mirror-image connections. The transformation y = −f(x) reflects the graph in the x-axis; x-intercepts stay fixed, while maxima become minima and minima become maxima. The transformation y = f(−x) reflects the graph in the y-axis; y-intercepts stay fixed, and even functions remain unchanged.
反射产生镜像联系。变换 y = −f(x) 将图像关于 x 轴反射；x 轴截距保持不变，而极大值变为极小值，极小值变为极大值。变换 y = f(−x) 将图像关于 y 轴反射；y 轴截距保持不变，偶函数完全不变。
Edexcel also tests the modulus transformation y = |f(x)|. This keeps all parts of the graph above the x-axis the same, and reflects any part below the x-axis upwards. The connection is that every negative y-value changes sign, while positive y-values do not change place.
Edexcel 还会考察模函数变换 y = |f(x)|。这一变换保持 x 轴上方的图像不变，并将 x 轴下方的任何部分向上反射。其联系在于每个负的 y 值改变符号，而正的 y 值不发生位置变化。
4. Stretches and Enlargements | 拉伸与缩放
A vertical stretch y = a f(x) multiplies all y-coordinates by a. If a > 1 the graph moves away from the x-axis; if 0 < a < 1 it is compressed towards the x-axis. A horizontal stretch y = f(a x) replaces x by a x, so x-coordinates are divided by a: the graph is compressed horizontally if a > 1 and stretched if 0 < a < 1.
垂直拉伸 y = a f(x) 将所有 y 坐标乘以 a。若 a > 1，图像远离 x 轴；若 0 < a < 1，则向 x 轴压缩。水平拉伸 y = f(a x) 用 a x 替换 x，因此 x 坐标要除以 a：a > 1 时水平压缩，0 < a < 1 时水平拉伸。
To avoid errors, remember that the stretch factor outside the function affects y-values directly, while the factor inside the function affects x-values inversely. This inverse connection is one of the most common sources of mistakes in exam answers.
为避免错误，请记住：函数外部的拉伸系数直接影响 y 值，而函数内部的系数以倒数方式影响 x 值。这种逆向联系是考试答案中最常见的错误来源之一。
| Transformation | New graph | Coordinate effect |
|---|---|---|
| Vertical translation | y = f(x) + a | (x, y) → (x, y + a) |
| Horizontal translation | y = f(x + a) | (x, y) → (x − a, y) |
| Reflection in x-axis | y = −f(x) | (x, y) → (x, −y) |
| Reflection in y-axis | y = f(−x) | (x, y) → (−x, y) |
| Vertical stretch | y = a f(x) | (x, y) → (x, a y) |
| Horizontal stretch | y = f(a x) | (x, y) → (x/a, y) |
5. Combined Transformations and Order | 复合变换及其顺序
When two transformations are combined, order matters. For example, transforming f(x) to 2 f(x + 3) involves a horizontal translation left by 3, then a vertical stretch by factor 2. If we stretch first and translate after, the result may differ. A safe method is to track a general point (x, y) and write the new coordinates step by step.
当两种变换复合时，顺序很重要。例如，从 f(x) 变换到 2 f(x + 3) 需要先水平向左平移 3 个单位，再垂直拉伸为原来的 2 倍。如果先拉伸后平移，结果可能不同。一个稳妥的方法是跟踪一般点 (x, y)，逐步写出新坐标。
Consider transforming y = x² to y = 2(x − 3)² + 1. Start with y = x², then translate right by 3 to get y = (x − 3)², then apply a vertical stretch factor 2 to get y = 2(x − 3)², and finally translate up by 1. The connections between each stage show why the final vertex is at (3, 1).
以 y = x² 变换为 y = 2(x − 3)² + 1 为例。先由 y = x² 开始，向右平移 3 个单位得到 y = (x − 3)²，再施加垂直拉伸系数 2 得到 y = 2(x − 3)²，最后向上平移 1 个单位。各阶段之间的联系说明了为什么最终顶点位于 (3, 1)。
In exam questions, describe a sequence as ‘horizontal translation, then reflection, then stretch’ only if the order stated matches the algebra. Checking with a specific point, such as the origin or a known intercept, can quickly reveal whether the order is correct.
在考试题中，只有当所述顺序与代数表达式匹配时，才可以把变换顺序描述为“先水平平移，再反射，再拉伸”。用一个具体点（例如原点或已知截距）进行检验，可以快速判断顺序是否正确。
6. Mapping Coordinates: Connecting Old and New Places | 坐标映射：连接新旧位置
A very reliable way to connect old and new places is to express each transformation as a mapping. For y = f(x) + a, the mapping is (x, y) → (x, y + a); for y = f(x + a), if we set the new variable X = x + a, then x = X − a, so the old point (x, y) maps to (X − a, y). Writing the old point in terms of the new point makes the direction clear.
将新旧位置联系起来的一个可靠方法，是把每种变换表示为坐标映射。对于 y = f(x) + a，映射为 (x, y) → (x, y + a)；对于 y = f(x + a)，若设新变量 X = x + a，则 x = X − a，因此旧点 (x, y) 映射为 (X − a, y)。用新点表示旧点能让方向变得清晰。
For example, if the graph of y = f(x) has a turning point at (2, −5), then y = f(x + 4) has the same turning point moved to (2 − 4, −5) = (−2, −5). The connection is purely horizontal because the change is inside the function.
例如，若 y = f(x) 的图像在 (2, −5) 处有一个极值点，那么 y = f(x + 4) 的对应极值点移动到 (2 − 4, −5) = (−2, −5)。这一联系纯粹是水平方向的，因为变化发生在函数内部。
7. Functions as Relationships: Domain and Range | 作为关系的函数：定义域与值域
Transformations also change the domain and range. For y = f(x + a), the domain shifts horizontally by a; for y = f(x) + a, the range shifts vertically by a. Reflections swap positive and negative parts, while vertical stretches multiply the range and horizontal stretches divide the domain.
变换也会改变定义域和值域。对于 y = f(x + a)，定义域水平平移 a；对于 y = f(x) + a，值域垂直平移 a。反射会交换正负部分，而垂直拉伸会乘以值域，水平拉伸会除以定义域。
Suppose f(x) has domain 0 ≤ x ≤ 4 and range 1 ≤ f(x) ≤ 5. Then y = 2 f(x − 1) has domain 1 ≤ x ≤ 5, because x − 1 must be between 0 and 4, and range 2 ≤ y ≤ 10,
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