Combined Transformations of Graphs | 图像组合变换

📚 Combined Transformations of Graphs | 图像组合变换

In A-Level mathematics, understanding how functions behave when their equations are altered is essential. Combined transformations occur when more than one transformation is applied to a graph, such as a translation followed by a stretch, or a reflection combined with a shift. Mastering this concept requires a clear grasp of transformation order and the algebraic effect on the function.

在A-Level数学中，理解函数方程变化时的行为至关重要。组合变换是指对图像施加不止一种变换，例如先平移再伸缩，或反射与平移结合。掌握这一概念需要对变换顺序及其代数影响有清晰的认识。

1. Basic Transformations Review | 基本变换回顾

The four fundamental transformations are translations, reflections, stretches, and compressions. A translation moves the graph horizontally or vertically without changing its shape. A reflection flips the graph across an axis. A stretch or compression changes the graph’s scale along the x- or y-axis.

四种基本变换是平移、反射、伸缩和压缩。平移沿水平或垂直方向移动图像而不改变其形状。反射使图像关于某条轴翻转。伸缩或压缩改变图像沿x轴或y轴的比例尺度。

For a function y = f(x), the following standard forms apply:

对于函数 y = f(x)，以下标准形式适用：

y = f(x) + a (vertical translation up by a)
y = f(x – a) (horizontal translation right by a)
y = -f(x) (reflection in x-axis)
y = f(-x) (reflection in y-axis)
y = a·f(x) (vertical stretch factor a)
y = f(x/a) (horizontal stretch factor a)

These basic rules form the building blocks for combined transformations.

这些基本规则构成组合变换的基础。

2. Order of Transformations | 变换顺序

When multiple transformations are applied, the order matters, especially when both horizontal and vertical transformations involve translation and scaling. The correct order follows the same logic as function composition: innermost operations affect x and are applied first.

当施加多个变换时，顺序很重要，尤其是当水平和垂直变换同时涉及平移和缩放时。正确的顺序遵循函数复合的逻辑：最内层的运算影响x，需先进行。

A common rule is: for horizontal transformations, the order is reflections/stretches first, then translations. For vertical transformations, translations are usually applied after scaling.

常用规则是：对于水平变换，先进行反射/伸缩，再进行平移；对于垂直变换，通常先缩放后平移。

For example, to obtain y = 3f(2x – 4) + 1, you must first apply the horizontal stretch factor 1/2, then the horizontal translation right by 2 (not 4), because the transformation is f(2(x – 2)).

例如，要得到 y = 3f(2x – 4) + 1，必须先进行水平伸缩因子1/2，然后水平向右平移2个单位（不是4），因为变换是 f(2(x – 2))。

3. Combining Horizontal Transformations | 水平变换组合

Horizontal transformations act on the input variable x. When a function involves multiple horizontal changes, they must be simplified into a single form. For instance, y = f(2x + 3) can be rewritten as f(2(x + 3/2)), meaning a stretch by factor 1/2 followed by a translation left by 3/2.

水平变换作用于输入变量x。当一个函数包含多个水平变化时，必须简化成单一形式。例如，y = f(2x + 3) 可重写为 f(2(x + 3/2))，意味着先进行1/2的伸缩，然后向左平移3/2。

The order for horizontal transformations is: first stretch/compress, then translate. This order is essential because translations are inside the function and are affected by the stretch factor.

水平变换的顺序是：先伸缩，后平移。这个顺序至关重要，因为平移位于函数内部，会受伸缩因子影响。

An important example: y = f(3 – 2x). Rewrite as f(-2(x – 3/2)). So the process is: reflect in y-axis, stretch horizontally by 1/2, then translate right by 3/2.

一个重要例子：y = f(3 – 2x)。重写为 f(-2(x – 3/2))。过程是：关于y轴反射，水平伸缩1/2，然后向右平移3/2。

4. Combining Vertical Transformations | 垂直变换组合

Vertical transformations act on the output value. For example, y = 2f(x) + 3 means first stretch vertically by factor 2, then shift up by 3. The order here is simpler: scaling before translation.

垂直变换作用于输出值。例如，y = 2f(x) + 3 表示先垂直伸缩因子2，然后上移3。这里顺序更简单：缩放在前，平移在后。

This is because vertical translations are added outside the function, so they are not affected by the vertical stretch. If you translate first, the stretch would also scale the translation, giving a different result.

这是因为垂直平移在函数外部添加，不受垂直伸缩影响。如果你先平移，伸缩也会缩放平移量，得到不同结果。

For y = a·f(x) + b, the order is: first scale by a, then translate by b. This rule is universal for vertical combinations.

对于 y = a·f(x) + b，顺序是：先按a缩放，再按b平移。这一规则对垂直组合普遍适用。

5. Combined Horizontal and Vertical Transformations | 水平与垂直变换结合

When both horizontal and vertical transformations occur in the same equation, they are independent of each other. Horizontal changes only affect x, while vertical changes affect y. Therefore, the combined effect can be analysed separately and then applied in a logical sequence.

当同一方程中同时出现水平和垂直变换时，它们彼此独立。水平变化只影响x，垂直变化只影响y。因此，可以分别分析组合效应，然后按逻辑顺序应用。

Consider y = 2f(3x – 6) + 4. First, handle the horizontal part: f(3x – 6) = f(3(x – 2)), so stretch horizontally by 1/3, then translate right by 2. Next, apply the vertical part: stretch vertically by 2, then translate up by 4.

考虑 y = 2f(3x – 6) + 4。首先处理水平部分：f(3x – 6) = f(3(x – 2))，所以先水平伸缩1/3，然后向右平移2。接下来应用垂直部分：垂直伸缩2，然后上移4。

The overall sequence can be: horizontal stretch → horizontal translation → vertical stretch → vertical translation. This order works consistently for all combined transformations.

整体顺序可以是：水平伸缩 → 水平平移 → 垂直伸缩 → 垂直平移。这一顺序对所有组合变换都一致适用。

6. Effect of Transformations on Key Points | 关键点的变换

When sketching a transformed graph, start from known key points of the original function, such as intercepts, turning points, and asymptotes. Apply the transformations to each coordinate separately. The x-coordinate is affected only by horizontal transformations, and the y-coordinate only by vertical ones.

绘制变换后的图像时，应从原函数的已知关键点出发，如交点、极值点和渐近线。对每个坐标分别施加变换。x坐标只受水平变换影响，y坐标只受垂直变换影响。

For a point (a, b) on y = f(x), the point on y = p·f(q(x – h)) + k becomes (h + a/q, p·b + k). Notice the order for x: divide by q first, then add h; for y: multiply by p first, then add k.

对于 y = f(x) 上的点 (a, b)，在 y = p·f(q(x – h)) + k 上的对应点为 (h + a/q, p·b + k)。注意x的顺序：先除以q，再加h；y的顺序：先乘p，再加k。

This method is reliable because it avoids confusion about which transformation happens first algebraically.

这个方法可靠，因为它避免了代数上先进行哪个变换的混淆。

7. Using Algebraic Representation | 代数表示法

To apply combined transformations correctly, it is helpful to rewrite the function in a standard factored form. For example, y = 2f(2x + 4) – 6 can be rewritten as y = 2f(2(x + 2)) – 6. This clearly shows the horizontal stretch factor 1/2, the horizontal shift left by 2, the vertical stretch factor 2, and the vertical shift down by 6.

要正确应用组合变换，最好将函数重写为标准因式分解形式。例如，y = 2f(2x + 4) – 6 可重写为 y = 2f(2(x + 2)) – 6。这清晰显示水平伸缩因子1/2、水平左移2、垂直伸缩因子2和垂直下移6。

The general form is y = a·f(b(x + c)) + d, where |a| is the vertical stretch factor (negative reflects in x-axis), |1/b| is the horizontal stretch factor (negative reflects in y-axis), c is the horizontal shift (sign reversed), and d is the vertical shift.

一般形式为 y = a·f(b(x + c)) + d，其中 |a| 是垂直伸缩因子（负值表示x轴反射），|1/b| 是水平伸缩因子（负值表示y轴反射），c 是水平平移（符号相反），d 是垂直平移。

Always factor out the coefficient of x before identifying horizontal shifts.

务必先提出x的系数，再识别水平平移。

8. Inverse Transformations | 逆变换

Sometimes you are given the transformed graph and asked to find the original function. In such cases, apply inverse transformations in reverse order. For example, if a point has been moved by a translation, the inverse translation reverses the shift.

有时你已知变换后的图像，需要求原函数。在这种情况下，按相反顺序应用逆变换。例如，如果一个点经过平移，逆平移就是反向移动。

Suppose the transformation from f to g is g(x) = 3f(2x – 4) + 1. To recover f, first subtract 1, then divide by 3, then replace x with (x + 4)/2, which is the algebraic inverse of the horizontal part.

假设从f到g的变换是 g(x) = 3f(2x – 4) + 1。要恢复f，先减1，再除以3，然后用 (x + 4)/2 替换x，这是水平部分的代数逆。

Inverse transformations are useful for solving problems involving unknown original points.

逆变换在解决包含未知原始点的问题时很有用。

9. Sketching Combined Transformations | 绘制组合变换图像

To sketch a combined transformation, first mark the key features of the original graph. Then apply horizontal transformations to x coordinates and vertical transformations to y coordinates one by one. Use tracing or mental visualisation to check that the shape is preserved.

绘制组合变换图像时，先标出原图的关键特征。然后对x坐标应用水平变换，对y坐标应用垂直变换，逐项进行。通过追踪或心理可视化检查形状是否保持。

Remember that translations do not change shape, but stretches and reflections may do so. A vertical stretch increases the amplitude, while a horizontal stretch changes the period (for periodic functions).

记住平移不改变形状，但伸缩和反射可能改变。垂直伸缩增加振幅，水平伸缩改变周期（对于周期函数）。

A good practice is to sketch the graph in stages: first the stretch, then the translation. For example, for y = 2f(x) + 1, first sketch 2f(x), then shift up by 1.

好的做法是分阶段绘图：先画伸缩，再画平移。例如，对于 y = 2f(x) + 1，先画 2f(x)，然后上移1。

10. Common Mistakes | 常见错误

One common mistake is confusing the direction of horizontal translations. For y = f(x + 3), the graph moves left by 3, not right. Another mistake is applying a vertical translation inside a horizontal context, or forgetting the stretch factor inside the function when finding the new x-coordinate.

常见错误是混淆水平平移的方向。对于 y = f(x + 3)，图像左移3，不是右移。另一个错误是在水平上下文中应用垂直平移，或求解新x坐标时忘记函数内部的伸缩因子。

Using transformations incorrectly: for example, treating y = f(2x – 4) as a shift right by 4. The correct form is f(2(x – 2)), so a stretch factor 1/2 followed by a shift right by 2.

变换使用不当：例如，将 y = f(2x – 4) 看作右移4。正确形式是 f(2(x – 2))，因此先伸缩1/2，然后右移2。

Another frequent error is applying the vertical stretch after a vertical translation. Always perform the stretch before the translation for vertical changes.

另一个常见错误是在垂直平移后应用垂直伸缩。对于垂直变化，务必先伸缩后平移。

11. Exam Tips | 考试技巧

In Edexcel A-Level exams, combined transformation questions often appear as part of curve sketching or finding the equation of a new function. Always write the transformation steps explicitly and show the intermediate graph if required.

在Edexcel A-Level考试中，组合变换问题常出现在曲线绘制或求新函数方程中。务必明确写出变换步骤，并在需要时画出中间图像。

Use the “inside-out” method: for the argument of the function, factor out any coefficient before identifying shifts. For the output, apply vertical scaling before vertical shifting. This reduces error.

使用“由内而外”的方法：对于函数参数，先提出系数再识别平移；对于输出，先垂直缩放再垂直平移。这能减少错误。

Practice with simple functions like y = x² and y = |x|. Sketch the original, then apply combinations step by step. This builds intuition for more complex cases.

使用简单函数如 y = x² 和 y = |x| 进行练习。先画原图，然后逐步应用组合。这能帮助建立对更复杂情况的直觉。

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Combined Transformations of Graphs | 图像组合变换