📚 058: Graph Transformations | 图像变换
covers all the key transformations of the graph of y = f(x) required for Edexcel IGCSE, with clear explanations and examples.
图像变换使我们能够通过对基本图形进行简单的平移、伸缩和反射来快速绘制复杂函数的草图。在IGCSE数学中，掌握这些变换对于理解函数行为和图像求解方程至关重要。本文涵盖Edexcel IGCSE考试中y = f(x)图形所有关键变换，并提供清晰解释和示例。
1. Vertical Translation: y = f(x) + a | 垂直平移：y = f(x) + a
ng its shape. If a > 0, the graph shifts upwards by a units; if a < 0, it shifts downwards by |a| units. This is equivalent to adding a constant to the y‑coordinate of every point on the original graph.
垂直平移将y = f(x)的图像向上或向下移动，而不改变其形状。若a > 0，图像向上平移a个单位；若a < 0，则向下平移|a|个单位。这相当于给原图像上每个点的y坐标加上一个常数。
s vertex from (0, 0) to (0, 3). Similarly, y = sin x + 2 raises the entire sine wave by 2.
对于函数f(x) = x²，y = x² + 3的图像是抛物线向上平移3个单位，顶点从(0, 0)移动到(0, 3)。类似地，y = sin x + 2将整个正弦波向上提升2个单位。
Transformation: (x, y) → (x, y + a)
2. Horizontal Translation: y = f(x – a) | 水平平移：y = f(x – a)
A horizontal translation shifts the graph left or right. Replacing x with (x – a) moves the graph a units to the right if a > 0, and |a| units to the left if a < 0. This can feel counter‑intuitive because the sign inside the bracket is opposite to the direction of movement.
水平平移将图像向左或向右移动。用(x – a)替换x，若a > 0，图像向右平移a个单位；若a < 0，则向左平移|a|个单位。这可能与直觉相反，因为括号内的符号与移动方向相反。
For y = (x – the basic parabola y = x² is shifted 3 units to the right, with vertex at (3, 0). For y = |x + 2|, the V‑shaped graph moves 2 units left, placing the vertex at (–2, 0).
对于y = (x – 3)²，基本抛物线y = x²向右平移3个单位，顶点位于(3, 0)。对于y = |x + 2|，V形图向左平移2个单位，顶点位于(–2, 0)。
Transformation: (x, y) → (x + a, y)
3. Vertical Stretch: y = k f(x) | 垂直伸缩：y = k f(x)
onstant k stretches the graph vertically. If k > 1, the graph is stretched away from the x‑axis; if 0 < k < 1, it is compressed towards the x‑axis. The x‑intercepts remain unchanged because f(x) = 0 implies k f(x) = 0.
将整个函数乘以常数k会使图像垂直伸缩。若k > 1，图像沿y轴方向拉伸；若0 < k < 1，则向x轴压缩。x轴截距保持不变，因为f(x) = 0时，k f(x) = 0。
For y = 2 sin x, the amplitude doubles from 1 to 2, so the peaks reach 2 and troughs reach –2. For y = ½ x², the parabola becomes wider because every y‑coordinate is halved compared to y = x².
对于y = 2 sin x，振幅从1增加到2，因此波峰达到2，波谷达到–2。对于y = ½ x²，抛物线变宽，因为与y = x²相比，每个y坐标都减半了。
Transformation: (x, y) → (x, k y)
4. Horizontal Stretch: y = f(b x) | 水平伸缩：y = f(b x)
Replacing x with b x causes a horizontal stretch or compression. If b > 1, the graph is compressed towards the y‑axis by a factor of 1/b; if 0 < b < 1, it is stretched away from the y‑axis. The y‑intercept stays the same because f(0) is unchanged.
用b x替换x会导致水平伸缩。若b > 1，图像向y轴方向压缩，压缩比例为1/b；若0 < b < 1，则图像沿x轴方向拉伸。y轴截距不变，因为f(0)不变。
les into 360°. For y = (0.5x)², the parabola widens, as the squaring now happens on slower‑changing x‑values.
对于y = sin(2x)，周期从360°减半到180°，在360°内挤压出两个完整周期。对于y = (0.5x)²，抛物线变宽，因为平方运算现在作用于变化更慢的x值。
Transformation: (x, y) → (x / b, y)
5. Reflection in the x‑axis: y = –f(x) | 关于x轴的反射：y = –f(x)
Taking the negative of the whole function reflects the graph in the x‑axis. Points above the x‑axis are flipped below, and vice versa. The x‑intercepts remain fixed because changing sign of zero still leaves zero.
对整个函数取负会将图像关于x轴反射。x轴上方的点翻转到下方，反之亦然。x轴截距保持不变，因为零的符号变化后仍为零。
If f(x) = x², then y = –x² is an upside‑down parabola, opening downwards with vertex at (0, 0). For y = –|x|, the V‑shape points downwards.
若f(x) = x²，则y = –x²是倒置的抛物线，开口向下，顶点在(0, 0)。对于y = –|x|，V形图指向下方。
Transformation: (x, y) → (x, –y)
6. Reflection in the y‑axis: y = f(–x) | 关于y轴的反射：y = f(–x)
Replacing x with –x reflects the graph in the y‑axis. This transformation swaps left and right sides. Even functions, where f(–x) = f(x), are symmetric about the y‑axis and remain unchanged under this reflection.
用–x替换x会将图像关于y轴反射。这一变换交换左右两侧。偶函数满足f(–x) = f(x)，关于y轴对称，在此反射下保持不变。
For y = (–x)³ = –x³, the cubic graph is reflected across the y‑axis, which also turns it upside down compared to the original y = x³. For y = cos(–x) = cos x, the graph is unchanged because cosine is an even function.
对于y = (–x)³ = –x³，三次函数图像关于y轴反射，同时也使其与原函数y = x³相比上下颠倒。对于y = cos(–x) = cos x，图像不变，因为余弦是偶函数。
Transformation: (x, y) → (–x, y)
7. Combined Transformations: Order Matters | 组合变换：顺序很重要
as y = a f(b x + c) + d, but the safest method is to track the point (x, y) step by step.
当一个函数中应用两个或多个变换时，顺序可能会影响最终图像。通常，当函数写为y = a f(b x + c) + d时，先应用水平变换，但最稳妥的方法是逐步跟踪点(x, y)的变化。
l shift gives (p + 3, q), then the vertical stretch gives (p + 3, 2q). Reversing the order (stretch first) would stretch the unshifted point, leading to an incorrect horizontal position.
考虑y = 2 f(x + 3)。从y = f(x)上的点(p, q)开始，水平平移得到(p + 3, q)，然后垂直伸缩得到(p + 3, 2q)。颠倒顺序（先伸缩）会拉伸未平移的点，导致水平位置错误。
A useful tip: rewrite complicated functions using a sequence of changes: inside the bracket (horizontal shift, horizontal stretch/reflection), then outside (vertical stretch/reflection, vertical shift).
一个有用技巧：用一系列变化重写复杂函数：括号内（水平平移、水平伸缩/反射），然后括号外（垂直伸缩/反射、垂直平移）。
General form: y = a f(b(x – h)) + k
Here, h is the horizontal shift, 1/b the horizontal stretch factor, a the vertical stretch factor, and k the vertical shift.
这里，h是水平平移量，1/b是水平伸缩因子，a是垂直伸缩因子，k是垂直平移量。
8. Impact on Key Points and Asymptotes | 对关键点和渐近线的影响
Graph transformations move every point, including intercepts, turning points, and asymptotes. When sketching a transformed graph, it is often easiest to locate the new positions of a few well‑chosen original points. For rational functions, vertical and horizontal asymptotes shift according to the same rules.
图像变换移动每一个点，包括截距、转折点和渐近线。在绘制变换后的图像时，通常最容易的方法是定位几个精心选择的原始点的新位置。对于有理函数，垂直和水平渐近线也按照相同规则移动。
For the reciprocal function y = 1/x, applying y = 1/(x – horizontal asymptote from y = 0 to y = 1. The graph is then a hyperbola centred at (2, 1).
对于反比例函数y = 1/x，应用变换y = 1/(x – 2) + 1将垂直渐近线从x = 0移到x = 2，水平渐近线从y = 0移到y = 1。此时图像是以(2, 1)为中心的双曲线。
Always check intercepts: set x = 0 to find the new y‑intercept, and set y = 0 to find new x‑intercepts after transformation. These confirm the sketch.
始终检查截距：设x = 0求变换后的y轴截距，设y = 0求变换后的x轴截距。这些可以验证草图。
9. Recognising Transformations from Equations | 从方程识别变换
often ask you to describe the transformation that maps one graph to another. For instance, "Describe the transformation that maps y = x² onto y = (x – 4)² + 3". You must specify the type and the amount: "translation by vector (4, 3)".
考试题目经常要求描述将一个图形映射到另一个图形的变换。例如，“描述将y = x²映射到y = (x – 4)² + 3的变换”。你必须明确类型和数量：“平移向量为(4, 3)”。
If a graph is stretched, state the stretch factor and direction: "stretch parallel to the y‑axis with scale factor 2". For reflections, name the mirror line: "reflection in the x‑axis".
如果图形被拉伸，说明拉伸因子和方向：“平行于y轴、比例因子为2的拉伸”。对于反射，要指明镜面线：“关于x轴的反射”。
Use the exact wording from the specification to avoid losing marks. A vector notation for translations is expected: "translation by vector (a, b)" where a is horizontal shift and b is vertical shift.
使用考纲中的精确措辞以避免失分。平移要求使用向量表示：“平移向量为(a, b)”，其中a是水平移动，b是垂直移动。
10. Sketching Transformed Graphs | 绘制变换后的图形
Sketching a transformed graph does not require plotting numerous points. Instead, start with the basic shape of f(x), then apply transformations one by one, marking the new positions of key features such as maxima, minima, and intercepts.
绘制变换后的图形不需要绘制大量点。相反，从f(x)的基本形状开始，然后逐个应用变换，标记最值点、截距等关键特征的新位置。
For y = –2 f(x – nown graph of y = f(x), shift right by 1, stretch vertically by factor 2, reflect in the x‑axis, and finally shift up by 3. Drawing the intermediate steps helps to avoid mistakes.
对于y = –2 f(x – 1) + 3，取已知的y = f(x)图形，向右平移1个单位，垂直拉伸到原来的2倍，关于x轴反射，最后向上平移3个单位。绘制中间步骤有助于避免错误。
Label the axes clearly, and indicate the scale if the graph has been stretched. Show the coordinates of at least three clearly transformed points to demonstrate accuracy.
清晰地标记坐标轴，如果图形被拉伸，要标明比例。显示至少三个清晰转换后的点坐标以证明准确性。
11. Transformations and Invariant Points | 变换与不变点
An invariant point is a point that does not move under a transformation. For example, the origin (0,0) is invariant under stretches if it lies on the axis of stretch, and points on the mirror line are invariant under reflection. Recognising these points helps to anchor the transformed graph.
不变点是指在变换下不移动的点。例如，若原点(0,0)位于伸缩轴上，则在拉伸下它保持不变；镜面线上的点在反射下保持不变。识别这些点有助于固定变换后的图像。
In y = f(x) + 5, no point on the original graph is invariant because every y‑coordinate changes. In y = f(–x), points on the y‑axis (x = 0) are invariant because their x‑coordinate stays zero.
在y = f(x) + 5中，原图上没有不变的点，因为每个y坐标都变了。在y = f(–x)中，y轴上的点（x = 0）是不变的，因为它们的x坐标仍为零。
Questions may ask you to identify invariant points after a combination of transformations. Set up equations using the transformation rules to solve for coordinates that map to themselves.
题目可能要求你在组合变换后识别不变点。利用变换规则建立方程，求解映射到自身的坐标。
12. Applications in Graph Solving | 在图像求解中的应用
k and find intersection points. A horizontal line y = k is a vertical translation of the x‑axis, so solutions to f(x) = 0 are just the x‑intercepts.
变换是图像求解方程的有力工具。例如，要解f(x) = k，可以绘制y = f(x)和y = k并找到交点。水平线y = k是x轴的垂直平移，因此f(x) = 0的解就是x轴截距。
To solve f(x) = g(x), you can rearrange to f(x) – g(x) = 0 and sketch the graph of y = f(x) – g(x), finding its x‑ Familiarity with how transformations affect graphs makes these approaches straightforward.
要解f(x) = g(x)，可以重排为f(x) – g(x) = 0，然后绘制y = f(x) – g(x)的图像，找到其x轴截距。熟悉变换如何影响图形会使这些方法变得简单明了。
In IGCSE, typical tasks include finding the number of solutions of an equation by comparing a transformed graph with a given line or curve, making graph transformations indispensable.
在IGCSE中，典型任务包括通过比较变换后的图形与给定直线或曲线来确定方程解的个数，这使得图像变换不可或缺。
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