📚 Combined 143: Transformations and Their Combinations | 组合143: 函数变换及其组合
In Edexcel A‑ tretches, and reflections applied to a function y = f(x), and the more challenging skill of combining two or more of these transformations in the correct order. Understanding how to describe and interpret combined transformations using function notation will help you tackle difficult exam questions with confidence.
在Edexcel A‑Level数学中，掌握图像变换对于函数、三角学和微积分至关重要。本专题涵盖对函数 y = f(x) 施加的平移、拉伸和反射，以及更具挑战性的技能——以正确顺序组合两个或多个变换。理解如何用函数符号描述和解释组合变换，将帮助你自信地应对高难度考题。
1. Overview of Graph Transformations | 图像变换概览
. When these are applied to y = f(x), they can be expressed as changes inside the function argument (horizontal) or outside the function (vertical). For example, y = f(x + a) is a horizontal translation, while y = f(x) + a is a vertical one.
变换会改变函数图像的位置、形状或方向。基本类型有平移、拉伸和反射。当这些变换应用于 y = f(x) 时，可以表示为对函数自变量（水平方向）或函数外部（垂直方向）的改变。例如，y = f(x + a) 是水平平移，而 y = f(x) + a 是垂直平移。
2. Translations: Shifting the Graph | 平移：图像的移动
ts the graph to the right by p. A vertical translation of +q units is y = f(x) + q, moving the graph up by q. Negative values shift in the opposite direction. In combined transformations, translations are often applied at the final stage to avoid confusion.
平移会整体移动图像而不改变其形状。水平平移 +p 个单位表示为 y = f(x − p)：将 x 替换为 x − p 使图像向右移动 p。垂直平移 +q 个单位表示为 y = f(x) + q，将图像向上移动 q。负值则向相反方向移动。在组合变换中，平移通常在最后阶段应用以避免混淆。
3. Stretches: Scaling the Graph | 拉伸：图像的缩放
1/k is written as y = f(kx). If k > 1, the graph is compressed horizontally towards the y‑axis; if 0 < k < 1, it is stretched away from the y‑s. A vertical stretch by factor a is y = a f(x): a > 1 stretches the graph away from the x‑axis, and 0 < a < 1 compresses it towards the x‑axis. Remember that horizontal stretches affect the x‑coordinates and are represented by the reciprocal factor inside the function.
拉伸会改变点到坐标轴的距离。以比例因子 1/k 的水平拉伸写为 y = f(kx)。若 k > 1，图像水平向 y 轴压缩；若 0 < k < 1，则远离 y 轴拉伸。垂直拉伸比例因子 a 表示为 y = a f(x)：a > 1 使图像远离 x 轴拉伸，0 < a < 1 则向 x 轴压缩。请记住，水平拉伸影响 x 坐标，并在函数内部用倒数因子表示。
4. Reflections: Flipping the Graph | 反射：图像的翻转
Reflections create a mirror image across an axis. y = −f(x) reflects the graph in the x‑axis, changing the sign of the y‑coordinates. y = f(−x) reflects in the y‑axis, changing the sign of the x‑ reflections are involved, the order can affect the final result if not treated carefully.
反射会生成关于坐标轴的镜像。y = −f(x) 表示图像关于 x 轴反射，改变 y 坐标的符号。y = f(−x) 关于 y 轴反射，改变 x 坐标的符号。在组合变换中，反射可视为比例因子为 −1 的拉伸特例。当涉及多次反射时，如果不仔细处理，顺序可能影响最终结果。
5. Using Function Notation to Describe Transformations | 用函数符号描述变换
m f(k(x − p)). For instance, 2x − 4 = 2(x − 2). Then the steps are: horizontal stretch by factor 1/2, translation right by 2, vertical stretch by factor 3, and vertical translation up by 1. Always deal with the horizontal stretch before the horizontal translation, unless the brackets indicate otherwise.
Edexcel 考题常要求你将 f(x) 映射到一个新函数，如 3f(2x − 4) + 1。处理这类问题的最佳方式是先将内部改写为 f(k(x − p)) 的形式。例如，2x − 4 = 2(x − 2)。于是步骤为：水平拉伸比例因子 ½，向右平移 2，垂直拉伸因子 3，再向上平移 1。除非括号另有指示，否则总是先处理水平拉伸，再进行水平平移。
6. Order of Combined Transformations | 组合变换的顺序
When a function is written as a f(b(x + c)) + d, the correct order for horizontal changes is from the inside out: stretch/reflection first, then translation. For vertical changes, the order is from the outside in: stretch/reflection first, then translation. However, if the expression is given as f(px + q) without factorisation, you must factorise first to see the true horizontal transformation sequence.
当函数写成 a f(b(x + c)) + d 的形式时，水平变化的正确顺序是从内向外：先拉伸/反射，然后平移。垂直变化的顺序是从外向内：先拉伸/反射，然后平移。然而，如果给出的表达式为 f(px + q) 且未因式分解，你必须先因式分解才能看出真正的水平变换顺序。
7. Horizontal vs Vertical Combinations: Which Comes First? | 水平与垂直的组合：谁先谁后？
transformations are independent and can be applied in either order relative to each other – y = 2f(3x + 6) − 1, first rewrite as 2f(3(x + 2)) − 1. The sensible sequence is: horizontal stretch by 1/3, left translation by 2, then vertical stretch by 2, and finally vertical translation down by 1. You could swap the two stretches without affecting the outcome, but keep each category’s internal order.
水平变换和垂直变换相互独立，可以任意顺序进行——但关键在于水平变换链条内部顺序重要，垂直变换链条内部顺序也重要。例如，要将 y = f(x) 变换为 y = 2f(3x + 6) − 1，首先重写为 2f(3(x + 2)) − 1。合理顺序是：水平拉伸 1/3，左移 2，然后垂直拉伸 2，最后垂直下移 1。你可以交换两个拉伸的顺序而不影响结果，但需保持每类变换的内部顺序。
8. Combined Transformations on Trigonometric Functions | 三角函数上的组合变换
equently tested with combined transformations. For y = a sin(bx + c) + d, the amplitude is |a| tal translation clear. A common mistake is forgetting that the horizontal stretch/compression must be applied before the phase shift.
三角函数图像如 y = sin x 经常在组合变换中考查。对于 y = a sin(bx + c) + d，振幅为 |a|，周期为 360°/b（弧度制为 2π/b），相位平移为 −c/b，垂直平移为 d。重写为 a sin(b(x − (−c/b))) + d 可清晰显示水平平移。一个常见错误是忘记在相位平移之前先进行水平拉伸或压缩。
9. Transformations Involving Modulus and Inverse Functions | 涉及绝对值和反函数的变换
When the modulus function appears, e.g., y = |f(x)|, any part of the graph below the x‑axis is reflected upwards. For y = f(|x| e positive x portion. Combining these with stretches or translations requires careful step‑by‑ y = f⁻¹(x) by remembering that its graph is the reflection of y = f(x) in the line y = x.
当出现绝对值函数时，例如 y = |f(x)|，位于 x 轴下方的图像部分向上反射。对于 y = f(|x|)，负 x 区域的图像被舍弃，并以正 x 部分关于 y 轴的反射代替。将这些变换与拉伸或平移结合时，需要逐步仔细绘图。对于反函数，可对 y = f⁻¹(x) 施加变换，同时记住其图像是 y = f(x) 关于直线 y = x 的反射。
10. Finding the Image of a Point After Multiple Transformations | 求点经过多重变换后的像
e. For y = 2f(3x − 6) + 1, let the original point be (p, q). Inside the function: x → (x + 6)/3, so new x is (p + 6)/3. Outside: multiply y by 2 then add 1, giving 2q + 1. Always check by substituting back into the transformed equation if possible.
已知点 (p, q) 在 y = f(x) 上，可通过按正确顺序对坐标施加变化来追踪其在组合变换下的像。对于 y = 2f(3x − 6) + 1，设原像点为 (p, q)。内部变换：x → (x + 6)/3，因此新 x 坐标为 (p + 6)/3。外部变换：y 乘 2 再加 1，得到 2q + 1。如有可能，务必代回变换后的方程进行验证。
11. Sketching Graphs with Combined Transformations | 绘制组合变换的图像
basic graph of y = f(x). Apply horizontal transformations first: shift or stretch in the x‑ , and label asymptotes if the function has them. For combined transformations including a reflection, treat the reflection as a negative stretch and apply it at the appropriate stage.
从 y = f(x) 的基本图像开始。首先施加水平变换：在 x 方向平移或拉伸，得到 y = f(k(x − p))。然后施加垂直变换：拉伸并平移，得到 y = a f(k(x − p)) + d。在每一阶段标出关键点，若函数有渐近线也需标注。对于包含反射的组合变换，可将反射视为负的拉伸并在适当阶段应用。
12. Common Pitfalls and Exam Strategy | 常见误区与应试策略
Many students mistakenly apply translation before stretch when dealing with expressions like f(2x + 4), forgetting to factorise to f(2(x + 2)). Others confuse the direction of horizontal translations: f(x + 3) moves left, not right. In the exam, always rewrite the transformed function in the standard form a f(k(x − p)) + d before identifying the sequence. Check your answer by testing a point and by ensuring the graph looks consistent with transformations. Finally, practice with past Edexcel questions that involve multiple transformations on quadratic, cubic, trigonometric, and logarithmic functions.
许多学生在处理类似 f(2x + 4) 的表达式时，会错误地先平移再拉伸，忘记因式分解为 f(2(x + 2))。还有人混淆水平平移方向：f(x + 3) 向左移动，而非向右。在考试中，务必先将变换后的函数改写为标准形式 a f(k(x − p)) + d，再确定变换顺序。通过测试一个点并确保图像与变换相符来检查答案。最后，多练习涉及二次、三次、三角和对数函数的多重变换的Edexcel真题。
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