📚 Graph Transformations: Animation Practice Type G-1-3 Question Analysis | 函数图像变换：动画练习 G-1-3 题型解析
This article provides a detailed breakdown of the typical question types encountered in the ‘Graph Animation Practice G-1-3’ set, focusing on function transformations such as translations, stretches, and reflections. By mastering these core techniques, students can dynamically visualise changes in graph positions and shapes, which is essential for A-level mathematics and beyond.
本文详细解析了“图像动画练习 G-1-3”中的典型题型，重点讨论函数图像的平移、伸缩和反射等变换。掌握这些核心技巧后，学生能够动态地想象图像位置和形状的变化，这对 A-level 数学及后续学习至关重要。
1. Understanding the Base Function y = f(x) | 理解原函数 y = f(x)
Before applying any transformation, it is vital to recognise the shape and key features of the base function. Typical base functions include y = x², y = √x, y = 1/x, y = sin x, and y = eˣ. Knowing intercepts, asymptotes, and turning points allows you to predict the effect of transformations accurately.
在进行任何变换之前，识别原函数的形状和关键特征至关重要。常见的原函数包括 y = x²、y = √x、y = 1/x、y = sin x 和 y = eˣ。了解截距、渐近线和极值点有助于准确预判变换的效果。
For example, the graph of y = x² is a parabola opening upwards with vertex at (0,0). When we later apply a vertical translation, the vertex moves accordingly.
例如，y = x² 的图像是一条开口向上的抛物线，顶点在 (0,0)。当我们后续对它进行垂直平移时，顶点也会随之移动。
2. Vertical Translations: y = f(x) + a | 垂直平移：y = f(x) + a
A vertical translation moves the graph up or down without changing its shape. The transformation y = f(x) + a shifts the graph by a units upwards if a > 0, and downwards if a < 0.
垂直平移会上下移动图像而不改变其形状。变换 y = f(x) + a 会将图像向上移动 a 个单位（若 a > 0），或向下移动（若 a < 0）。
In animation-type questions, you often see a slider for ‘a’ that dynamically moves the entire curve. Key points such as maxima, minima, and intercepts all shift by exactly a in the y-direction.
在动画类题目中，你经常会看到一个控制“a”的滑块，它能动态地移动整条曲线。极大值点、极小值点和截距等关键点都会在 y 方向上精确移动 a 个单位。
3. Horizontal Translations: y = f(x + b) | 水平平移：y = f(x + b)
A horizontal translation is often counter-intuitive: y = f(x + b) shifts the graph to the left by b units if b > 0, and to the right if b < 0. The plus sign inside the bracket corresponds to a leftward movement.
水平平移常常与直觉相反：y = f(x + b) 会将图像向左移动 b 个单位（若 b > 0），向右移动（若 b < 0）。括号内的加号对应着向左移动。
Animations help to illustrate this by moving the graph horizontally while preserving its shape. For a quadratic, the x-coordinate of the vertex becomes -b. For trigonometric functions, this represents a phase shift.
动画能够帮助展示这一点，让图像在水平方向移动而形状不变。对于二次函数，顶点的 x 坐标变为 -b。对于三角函数，这就代表着相位的移动。
4. Combining Vertical and Horizontal Shifts | 组合垂直与水平平移
When a function is given in the form y = f(x + b) + a, both translations are applied simultaneously. The graph of y = f(x) moves left/right by b and up/down by a. The order of these shifts does not affect the final position.
当函数以 y = f(x + b) + a 的形式给出时，两种平移会同时生效。y = f(x) 的图像会左右移动 b，上下移动 a。这两次移动的先后顺序并不影响最终的位置。
In a typical G-1-3 animated exercise, you might be asked to match a transformed graph to its equation. Checking the new position of a known key point (like the vertex) quickly reveals the values of a and b.
在典型的 G-1-3 动画练习中，你可能会被要求将变换后的图像与它的方程配对。检查某个已知关键点（如顶点）的新位置，就能迅速判断 a 和 b 的值。
5. Vertical Stretches: y = c f(x) | 垂直伸缩：y = c f(x)
A vertical stretch multiplies all y-coordinates by a factor c. If c > 1, the graph stretches away from the x-axis; if 0 < c < 1, it compresses towards the x-axis. Negative values of c also reflect the graph in the x-axis.
垂直伸缩会将所有 y 坐标乘以因子 c。如果 c > 1，图像会沿 y 轴方向拉伸；如果 0 < c < 1，图像会向 x 轴方向压缩。c 为负值时还会将图像关于 x 轴反射。
For y = cf(x), the x-intercepts remain unchanged because f(x)=0 implies c*0=0. The overall steepness and turning point heights are scaled, which can be vividly shown via animated stretching.
对于 y = cf(x)，x 截距保持不变，因为 f(x)=0 时 c*0=0。整体的陡峭程度和极值点的高度都会被缩放，动画拉伸可以生动地展现这一过程。
6. Horizontal Stretches: y = f(dx) | 水平伸缩：y = f(dx)
A horizontal stretch transforms the x-coordinates by dividing by d. The function y = f(dx) compresses the graph towards the y-axis if d > 1, and stretches it away if 0 < d < 1. This is the opposite of what many students initially expect.
水平伸缩会将 x 坐标除以 d。函数 y = f(dx) 在 d > 1 时会将图像向 y 轴方向压缩，在 0 < d < 1 时则向两侧拉伸。这与许多学生最初的预期相反。
Animations clearly demonstrate that the period of a trigonometric function like sin(dx) becomes 2π/d, so larger d means shorter period. The y-intercept remains fixed since x=0 gives f(0).
动画可以清晰地展示，像 sin(dx) 这样的三角函数周期会变为 2π/d，因此 d 越大周期越短。y 截距保持不动，因为 x=0 时函数值为 f(0)。
7. Reflections in the Axes | 坐标轴反射
Reflections produce a mirror image of the graph. y = -f(x) reflects the graph in the x-axis, while y = f(-x) reflects it in the y-axis. Both can be combined with stretches and translations.
反射会产生图像的镜像。y = -f(x) 将图像关于 x 轴反射，而 y = f(-x) 则将图像关于 y 轴反射。这两种变换都可以与伸缩和平移组合使用。
In animated practice, a toggle flips the graph across an axis, helping students link algebraic signs to geometric symmetry. Even functions satisfy f(-x) = f(x) and are symmetric about the y-axis; odd functions satisfy f(-x) = -f(x).
在动画练习中，一个开关就能让图像关于某个轴翻转，帮助学生建立代数符号与几何对称之间的联系。偶函数满足 f(-x) = f(x)，关于 y 轴对称；奇函数满足 f(-x) = -f(x)。
8. Order of Multiple Transformations | 多种变换的顺序
When a function involves several transformations, the order matters if horizontal changes are combined with horizontal translations. A safe approach is to rewrite the function in the form y = a f(b(x + h)) + k, then apply transformations from the inside out: horizontal shift (h), then horizontal stretch (1/b), then vertical stretch (a), then vertical shift (k).
当函数包含多种变换时，如果水平方向的伸缩与平移相结合，顺序就很重要。一个稳妥的方法是先将函数写成 y = a f(b(x + h)) + k 的形式，然后由内向外依次施加变换：水平平移 (h)，接着水平伸缩 (1/b)，然后垂直伸缩 (a)，最后垂直平移 (k)。
Many G-1-3 questions test this sequence explicitly. For example, transforming y = f(x) to y = 2f(3x + 6) + 1 requires rewriting to y = 2f(3(x + 2)) + 1, showing a shift left by 2, horizontal compression by factor 3, vertical stretch by 2, and shift up by 1.
许多 G-1-3 题目专门考察这一顺序。例如，将 y = f(x) 变换为 y = 2f(3x + 6) + 1，需要先改写为 y = 2f(3(x + 2)) + 1，这意味着向左平移 2，水平压缩为原来的 1/3，垂直拉伸 2 倍，再向上平移 1。
9. Interpreting the Transformed Graph: Finding Original Coordinates | 解读变换后的图像：找出原始坐标
Given a transformed graph and its equation, you may be asked to find the original coordinates of a specific point before transformation, or vice versa. This involves reversing each step methodically.
给定一个变换后的图像及其方程，你可能会被要求找出某个特定点在变换之前的原坐标，或者反过来。这就需要有条理地逆向执行每一步变换。
If a point (p, q) on y = f(x) maps to (r, s) on y = a f(b(x + h)) + k, then r = p/b – h and s = a q + k. Working backwards, p = b(r + h) and q = (s – k)/a. This algebraic link is fundamental in coordinate geometry.
如果 y = f(x) 上的点 (p, q) 映射为 y = a f(b(x + h)) + k 上的点 (r, s)，那么 r = p/b – h 且 s = a q + k。逆推可得 p = b(r + h) 且 q = (s – k)/a。这种代数联系是解析几何的基础。
10. Solving Equations Using Graph Transformations | 利用图像变换解方程
Transformations can be used to solve equations graphically. The number of solutions to f(x) = g(x) is the number of intersections between y = f(x) and y = g(x). If g(x) is derived from f(x) by transformations, the intersection points can often be deduced by shifting or stretching the axes mentally.
变换可用于通过图像解方程。方程 f(x) = g(x) 的解的个数就是 y = f(x) 与 y = g(x) 图像的交点个数。如果 g(x) 是 f(x) 经过变换得到的，通常可以通过在头脑中平移或伸缩坐标轴来推断交点。
For instance, to solve f(x + 2) = 4, you can think of finding where the graph of y = f(x) after a left shift of 2 meets the horizontal line y = 4. Alternatively, let u = x + 2, solve f(u) = 4, then back-substitute.
例如，要解 f(x + 2) = 4，可以想象将 y = f(x) 的图像向左移动 2 个单位后，它与水平线 y = 4 的交点。也可以令 u = x + 2，解 f(u) = 4，再回代。
11. Asymptote Behaviour Under Transformations | 渐近线在变换下的表现
Rational and exponential functions have asymptotes that shift along with the graph. If y = f(x) has a horizontal asymptote y = L, then y = f(x) + a has asymptote y = L + a. A vertical asymptote x = H moves to x = H – b under y = f(x + b).
有理函数和指数函数具有渐近线，这些渐近线会随着图像的移动而改变。如果 y = f(x) 有一条水平渐近线 y = L，那么 y = f(x) + a 的水平渐近线就是 y = L + a。垂直渐近线 x = H 在变换 y = f(x + b) 下会移动到 x = H – b。
In an animation, you can observe how the dashed asymptote lines glide together with the curve, reinforcing the concept that these boundaries are a fixed part of the graph’s structure.
在动画中，你可以观察到虚线渐近线如何与曲线一同滑动，这强化了一个概念：这些边界是图像结构的固定组成部分。
12. Practice Strategy and Common Pitfalls | 练习策略与常见误区
To master G-1-3 type questions, always start by identifying the base function, then note the sequence of operations. Draw a quick sketch for each intermediate step. Common mistakes include forgetting that f(x + 2) shifts left, or applying horizontal stretches before factoring out the coefficient of x.
要掌握 G-1-3 题型，始终从识别原函数开始，然后记录运算顺序。为每个中间步骤快速画一个草图。常见的错误包括忘记 f(x + 2) 是向左平移，或者在提取 x 的系数之前就进行水平伸缩。
Using interactive graphs where you can drag sliders to see instant changes is extremely effective. It builds an intuitive link between algebraic changes and geometric effects, turning abstract rules into concrete visual memory.
使用可交互的图像，通过拖动滑块即时观察变化，是非常有效的方法。它能在代数变化与几何效果之间建立直觉联系，将抽象的规则转化为具体的视觉记忆。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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