📚 3 Graphs | 图形的三种基本变换
Graph transformations allow us to sketch new functions quickly by shifting, stretching, or reflecting a known parent function. In IB Mathematics, mastering these three fundamental moves — translation, stretch, and reflection — is essential for analysis and modelling. Whether you are working with quadratics, exponentials, or trigonometric curves, understanding how the equation controls the graph’s position and shape gives you a powerful visual toolkit.
图形变换使我们能够通过对已知基本函数进行平移、伸缩或反射，快速绘制出新函数的草图。在 IB 数学中，掌握平移、伸缩和反射这三种基本动作，对于函数分析和建模至关重要。无论你处理的是二次函数、指数函数还是三角函数曲线，理解方程如何控制图形的位置与形状，都会为你提供一套强大的视觉工具箱。
1. The Language of Transformations | 变换的语言
A transformation maps each point (x, y) on an original graph to a new point (x', y' h)) + k. The parameters a, b, h, k correspond to vertical stretch, horizontal stretch, horizontal shift and vertical shift respectively. Learning to read these letters is the first step.
变换将原始图形上的每一点 (x, y) 映射为新的点 (x', y')。在函数符号中，我们通常记母函数为 y = f(x)，然后应用诸如 y = a f(b(x - h)) + k 的变化。参数 a、b、h、k 分别对应垂直伸缩、水平伸缩、水平平移和垂直平移。学会解读这些字母是第一步。
2. Vertical Translations: y = f(x) + k | 垂直平移
ward, while y = sin x - 2 shifts the sine wave two units down. There is no change in the x‑coordinates; only the y‑coordinates are affected.
在输出值上加上常数 k 会把每一个点向上 (k > 0) 或向下 (k < 0) 移动。例如，y = x² + 3 将抛物线向上平移三个单位，而 y = sin x - 2 将正弦波向下平移两个单位。x 坐标保持不变，只有 y 坐标受到影响。
3. Horizontal Translations: y = f(x - h) | 水平平移
the graph h units to the right when h > 0 and to the left when h < 0. This often feels counter‑ x from (0, 0) to (4, 0), not leftwards. Pay close attention to the sign inside the bracket.
变换 y = f(x - h) 将图形向右平移 h 个单位（当 h > 0 时），向左平移（当 h < 0 时）。这一点常常与直觉相反：y = (x - 4)² 将顶点从 (0, 0) 移动到 (4, 0)，而不是向左。务必注意括号内的符号。
4. Vertical Stretches: y = a f(x) | 垂直伸缩
Multiplying the whole function by a factor a causes a vertical stretch when |a| > 1 or a vertical compression when 0 < |a| < 1. If a is negative, the graph is also reflected in the x‑ section). For instance, y = 2 cos x doubles the amplitude of the cosine wave, while y = ½√x compresses the square‑root graph vertically by half.
将整个函数乘以系数 a，当 |a| > 1 时产生垂直拉伸，当 0 < |a| < 1 时产生垂直压缩。如果 a 为负，图形同时还会关于 x 轴反射（参见后续小节）。例如，y = 2 cos x 将余弦波的振幅加倍，而 y = ½√x 将平方根图形垂直压缩为原来的一半。
5. Horizontal Stretches: y = f(bx) | 水平伸缩
ing x with bx in the function argument horizontally compresses the graph by a factor of 1/|b| when |b| > 1, and stretches it when 0 < |b| < 1. For example, y = sin(2x) completes a full period in π units instead of 2π — the wave is compressed. Similarly, y = (⅓x)³ stretches the cubic graph horizontally.
将函数自变量 x 替换为 bx，当 |b| > 1 时会将图形水平压缩为原来的 1/|b|，当 0 < |b| < 1 时则水平拉伸。例如，y = sin(2x) 在 π 个单位内完成一个完整周期，而不是 2π — 波形被压缩了。类似地，y = (⅓x)³ 则将三次函数图形水平拉伸。
6. Reflections in the x‑axis: y = -f(x) | 关于 x 轴的反射
Placing a minus sign in front of f(x) flips the graph across the x‑axis. Every y‑coordinate changes sign while the x‑ uadratic parent, y = -x² opens downward instead of upward; its vertex stays at the origin.
在 f(x) 前面加上负号会将图形关于 x 轴翻转。每一个 y 坐标改变符号，而 x 坐标保持不变。这是垂直伸缩中 a = -1 的特殊情况。对于二次母函数，y = -x² 开口向下而不是向上，其顶点仍位于原点。
7. Reflections in the y‑axis: y = f(-x) | 关于 y 轴的反射
Replacing x with -x inside the argument reflects the graph in the y‑axis. All points (x, y) become (-x, y). This is equivalent to a horizontal stretch by a factor of -1. Even functions like cos x and x² are unchanged by this reflection because they already possess y‑axis symmetry.
将自变量 x 替换为 -x 会使图形关于 y 轴反射。所有点 (x, y) 变为 (-x, y)。这等价于以系数 -1 进行水平伸缩。像 cos x 和 x² 这样的偶函数由于已经具有 y 轴对称性，因此经过这种反射后不会改变。
8. Combining Transformations: Order Matters | 组合变换：顺序很重要
e argument is factored. For example, y = -2 f(3x + 6) + 1 must be written as y = -2 f(3(x + 2)) + 1 to identify h = -2 correctly.
当需要应用多个变换时，标准顺序是：先进行水平伸缩/反射（参数 b），然后是水平平移 (h)，再进行垂直伸缩/反射 (a)，最后是垂直平移 (k)。务必先将函数写成 y = a f(b(x - h)) + k 的形式，确保自变量被因式分解。例如，y = -2 f(3x + 6) + 1 必须写成 y = -2 f(3(x + 2)) + 1，才能正确识别出 h = -2。
9. Worked Example: Transforming a Quadratic | 示例：变换二次函数
rtex is at (2, -1), and the parabola is narrower than the parent. If we reversed the order — shifting first then stretching — we would get a different (and incorrect) equation.
考虑母函数 y = x²。先进行垂直伸缩，系数为 3，然后向右平移 2 个单位，向下平移 1 个单位。新函数为 y = 3(x - 2)² - 1。其顶点位于 (2, -1)，且抛物线比母函数更窄。如果我们颠倒顺序——先平移再伸缩——就会得到不同且错误的表达式。
10. Worked Example: Transforming an Exponential | 示例：变换指数函数
Start with y = 2ˣ. Apply a horizontal translation 4 units left, a reflection in the x‑axis, and a vertical shift of +5. Stepwise: left shift gives y = 2x+4; reflection yields y = -2x+4; finally shift up gives y = -2x+4 from y = 0 to y = 5. The graph passes through (-4, 4) because when x = -4, y = -1 + 5 = 4.
从 y = 2ˣ 开始。先向左平移 4 个单位，再做关于 x 轴的反射，最后向上平移 5 个单位。逐步来看：左移得到 y = 2x+4；反射得到 y = -2x+4；最后上移得到 y = -2x+4 + 5。水平渐近线从 y = 0 移动到 y = 5。图形经过点 (-4, 4)，因为当 x = -4 时，y = -1 + 5 = 4。
11. Special Cases: Absolute Value and Reciprocal Graphs | 特殊情况：绝对值与倒数图形
The absolute value transformation y = |f(x)| reflects any part of the graph below the x‑axis upward. The function y = f(|x|) copies the right‑ an be transformed: y = a/(x - h) + k produces a hyperbola with asymptotes x = h and y = k.
绝对值变换 y = |f(x)| 会将 x 轴下方的部分向上反射。函数 y = f(|x|) 则将右侧图形复制到左侧，形成一个偶函数。与此同时，像 y = 1/x 这样的倒数图形也可以被变换：y = a/(x - h) + k 生成一条以 x = h 和 y = k 为渐近线的双曲线。
12. Summary and IB Exam Tips | 小结与 IB 考试技巧
Remember the four key parameters: a (vertical stretch/reflection), b (horizontal stretch/reflection), h (horizontal shift), k (vertical shift). Always write the function in the standard form before listing transformations. When sketching, apply transformations in the correct order and label the new positions of key points. Check your answer by substituting a simple x‑value into both the parent and the transformed function. With practice, graph transformations become one of the most reliable marks in the IB exam.
记住四个关键参数：a（垂直伸缩/反射）、b（水平伸缩/反射）、h（水平平移）、k（垂直平移）。在列出变换步骤之前，始终将函数写成标准形式。绘图时，按正确顺序应用变换，并标注关键点的新位置。将简单的 x 值代入母函数和变换后的函数，以检验你的答案。经过练习，图形变换将成为 IB 考试中最有把握的得分点之一。
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