📚 Graphs 7: Transformations of Graphs | 图形7：图形的变换
In IGCSE Mathematics, understanding how graphs can be transformed is essential for sketching functions quickly and solving equations visually. This topic covers translations, reflections, stretches, and combinations of these transformations, as well as using graphs to find solutions.
在IGCSE数学中，理解图像如何被变换对于快速绘制函数草图以及直观解方程至关重要。本主题涵盖平移、反射、拉伸以及这些变换的组合，还包括利用图像求解方程。
1. Introduction to Function Transformations | 函数变换简介
A function transformation changes the position or size of a graph. The basic function f(x) can be altered by applying operations such as addition, multiplication, or sign changes. These changes follow specific rules that affect the graph's coordinates.
函数变换改变图像的位置或大小。基本函数 f(x) 可以通过加法、乘法或符号变化等操作来改变。这些变化遵循特定的规则，会影响图像的坐标。
Common transformations include vertical and horizontal shifts, reflections across axes, and vertical or horizontal stretches. Knowing these rules allows you to predict the new graph without plotting many points.
常见的变换包括垂直和水平移动、关于坐标轴的反射以及垂直或水平拉伸。了解这些规则可以让你在不绘制许多点的情况下预测新图形。
2. Vertical Translations (y = f(x) + a) | 垂直平移（y = f(x) + a）
f(x) moves the entire graph up if a > 0, and down if a < 0. The shift is vertical, and the x-coordinates of points remain unchanged.
在函数 y = f(x) 上加一个常数 a，如果 a > 0，整个图像向上移动；如果 a < 0，则向下移动。这种移动是垂直的，各点的 x 坐标保持不变。
y = f(x) + a → translation by vector (0, a)
y = f(x) + a → 通过向量 (0, a) 平移
3 = x² + 3 shifts the parabola up by 3 units. The minimum point moves from (0,0) to (0,3).
例如，如果 f(x) = x²，那么 f(x) + 3 = x² + 3 将抛物线向上平移 3 个单位。最低点从 (0,0) 移动到 (0,3)。
3. Horizontal Translations (y = f(x + a)) | 水平平移（y = f(x + a)）
ment. The graph of y = f(x + a) moves left if a > 0 and right if a < 0. This seems counter‑ntuitive initially, but it helps to think of the x-coordinate compensating for the change.
水平移动是通过在函数自变量内部加一个常数产生的。y = f(x + a) 的图像当 a > 0 时向左移动，当 a < 0 时向右移动。这起初似乎违反直觉，但可以理解为 x 坐标在补偿这种变化。
y = f(x + a) → translation by vector (-a, 0)
y = f(x + a) → 通过向量 (-a, 0) 平移
- 2)² is the parabola shifted 2 units to the right. The vertex moves from (0,0) to (2,0).
对于 f(x) = x²，y = (x - 2)² 的图像是向右平移 2 个单位的抛物线。顶点从 (0,0) 移动到 (2,0)。
4. Reflections in the x-axis (y = -f(x)) | 关于x轴的反射（y = -f(x)）
Multiplying the whole function by -1 reflects the graph in the x-axis. Positive y-values become negative and vice versa. Points on the x-axis stay fixed.
将整个函数乘以 -1 会使图像关于 x 轴反射。正的 y 值变为负，负的变为正。x 轴上的点保持不变。
y = -f(x) → reflect in the x-axis
y = -f(x) → 关于 x 轴反射
For example, the graph of y = -√x is the reflection of y = √x across the x-axis. All y-coordinates are multiplied by -1.
例如，y = -√x 的图像是 y = √x 关于 x 轴的反射。所有的 y 坐标都乘以 -1。
5. Reflections in the y-axis (y = f(-x)) | 关于y轴的反射（y = f(-x)）
Replacing x with -x inside the function reflects the graph in the y-axis. The shape is mirrored left-to-right. Even functions remain unchanged; odd functions may also change sign.
在函数内部用 -x 替换 x，会使图像关于 y 轴反射。形状左右镜像。偶函数保持不变；奇函数也可能改变符号。
y = f(-x) → reflect in the y-axis
y = f(-x) → 关于 y 轴反射
For an exponential function like y = eˣ, the reflection y = e⁻ˣ produces a decreasing curve instead of increasing, mirroring across the y-axis.
对于像 y = eˣ 这样的指数函数，反射 y = e⁻ˣ 会产生一条递减的曲线，而不是递增，是关于 y 轴的镜像。
6. Vertical Stretches (y = a f(x)) | 垂直拉伸（y = a f(x)）
Multiplying the function by a constant a > 1 stretches the graph vertically away from the x-axis. If 0 < a < 1, it compresses the graph towards the x-axis. Negative a also reflects the graph in the x-axis.
将函数乘以常数 a > 1 会使图像在垂直方向上远离 x 轴拉伸。如果 0 < a < 1，则会使图像向 x 轴压缩。负的 a 还会同时关于 x 轴反射。
y = a f(x) → vertical stretch by factor a
y = a f(x) → 垂直拉伸，缩放因子为 a
ubles the amplitude, stretching the sine curve vertically. Points on the x-axis stay stationary.
考虑 f(x) = sin x。那么 y = 2 sin x 将振幅加倍，在垂直方向上拉伸了正弦曲线。x 轴上的点保持不动。
7. Horizontal Stretches (y = f(ax)) | 水平拉伸（y = f(ax)）
Multiplying x by a constant a inside the function compresses the graph horizontally towards the y-axis if a > 1, and stretches it away if 0 < a < 1. This transformation affects x-coordinates by factor 1/a.
在函数内部将 x 乘以常数 a，如果 a > 1，会使图像在水平方向上向 y 轴压缩；如果 0 < a < 1，则会进行水平拉伸。这种变换将 x 坐标按因子 1/a 缩放。
y = f(ax) → horizontal stretch by factor 1/a
y = f(ax) → 水平拉伸，缩放因子为 1/a
es the parabola horizontally, making it narrower. For y = (x/2)² = x²/4, the graph widens.
对于 f(x) = x²，y = (2x)² = 4x² 会将抛物线在水平方向上压缩，使其更窄。对于 y = (x/2)² = x²/4，则图形变宽。
8. Combining Transformations | 组合变换
Graphs can undergo several transformations in sequence. The order matters: applying transformations from the 'outside' (vertical) and 'inside' (horizontal) correctly ensures the intended result. Typically, perform stretches and reflections before translations.
图像可以依次进行多个变换。顺序很重要：正确地应用“外部”（垂直）和“内部”（水平）变换可以确保得到预期结果。通常，先进行拉伸和反射，再进行平移。
(x + 3) - 1, first shift left by 3, then stretch vertically by factor 2, and finally shift down by 1. The order inside the argument: f(x+3) first, then the 2 and -1 outside.
示例：要将 f(x) 变换为 y = 2 f(x + 3) - 1，首先向左平移 3 个单位，然后垂直拉伸 2 倍，最后向下平移 1 个单位。自变量内部的顺序：先 f(x+3)，再外部乘以2并减1。
Use a table of coordinates of key points from the original function to apply each transformation step-by-step. This method reduces mistakes in sketching.
使用原函数关键点的坐标表，逐步应用每个变换。这种方法可以减少草图绘制中的错误。
9. Using Graphs to Solve Equations (f(x) = 0, etc.) | 利用图像解方程
(x) = k, draw the horizontal line y = k and find the x-coordinates of intersection points.
利用 y = f(x) 的图像，可以通过找到 x 轴截距来解 f(x) = 0。如果需要解 f(x) = k，绘制水平线 y = k，并找到交点的 x 坐标。
identify the x-values where the graphs cross. The intersection points give the solutions.
对于像 f(x) = g(x) 这样的方程，同时绘制 y = f(x) 和 y = g(x)，并确定图像相交处的 x 值。交点提供解。
This graphical method is especially useful for solving equations that are difficult to solve algebraically, such as cubic or reciprocal equations.
这种图像法对于解难以用代数方法求解的方程特别有用，例如三次方程或倒数方程。
10. Using Graphs to Solve Inequalities | 利用
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