📚 Function Graph Transformations | 函数图像变换
Graph transformations are one of the most frequently tested topics in IB Mathematics (Analysis & Approaches and Applications & Interpretation). Mastering the rules for shifting, stretching, and reflecting graphs allows you to sketch complex functions quickly and understand how algebraic changes affect visual representations.
函数图像变换是 IB 数学（分析与方法、应用与解释）中最高频的考点之一。掌握平移、伸缩和反射的规则，可以帮助你快速绘制复杂函数的草图，并理解代数变化如何影响图形的视觉表现。
1. Vertical Translation | 垂直平移
Adding a constant k to a function shifts its graph vertically. For y = f(x) + k, every point (x, y) on the original graph moves to (x, y + k).
在函数中加上常数 k，图像会沿垂直方向平移。对于 y = f(x) + k，原图像上每一点 (x, y) 都会移动到 (x, y + k)。
y = f(x) + k, k > 0 upward shift; k < 0 downward shift
y = f(x) + k，k > 0 向上平移；k < 0 向下平移
The domain of the function remains unchanged, while the range shifts by k units. For example, if f(x) = x², then f(x) + 3 = x² + 3 moves the parabola up by 3 units, changing its vertex from (0, 0) to (0, 3).
函数的定义域保持不变，而值域整体平移 k 个单位。例如，若 f(x) = x²，则 f(x) + 3 = x² + 3 将抛物线向上移动 3 个单位，顶点从 (0, 0) 变为 (0, 3)。
2. Horizontal Translation | 水平平移
For y = f(x - h), the graph shifts horizontally by h units. The key rule to remember is the "opposite direction" effect: x - h with h > 0 shifts the graph to the right, and x + h shifts it to the left.
对于 y = f(x - h)，图像沿水平方向平移 h 个单位。需要牢记的关键规则是"反向效果"：x - h 在 h > 0 时图像向右平移，而 x + h 时图像向左平移。
y = f(x - h), h > 0 shift right; h < 0 shift left
y = f(x - h)，h > 0 向右平移；h < 0 向左平移
This counterintuitive rule confuses many students. Think of it this way: to achieve the same output value y, the input x must now be larger by h, so the whole graph moves right. For instance, y = (x - 2)² is the parabola y = x² shifted 2 units to the right, with vertex at (2, 0).
这一反直觉规则让许多学生感到困惑。你可以这样理解：为了得到相同的输出值 y，输入 x 必须比原来大 h，因此整个图像向右移动。例如，y = (x - 2)² 是抛物线 y = x² 向右平移 2 个单位后的结果，其顶点位于 (2, 0)。
3. Vertical Stretch and Compression | 垂直伸缩
Multiplying a function by a positive constant a produces a vertical stretch or compression: y = a · f(x). If |a| > 1, the graph is vertically stretched by factor a; if 0 < |a| < 1, it is vertically compressed. The x-intercepts remain fixed.
将函数乘以正数 a 会产生垂直拉伸或压缩：y = a · f(x)。若 |a| > 1，图像在垂直方向拉伸 a 倍；若 0 < |a| < 1，则在垂直方向压缩。x 轴截距保持不变。
y = a·f(x), |a| > 1 stretch; 0 < |a| < 1 compression
y = a·f(x)，|a| > 1 拉伸；0 < |a| < 1 压缩
Each point (x, y) maps to (x, a y). A negative value of a would also include a reflection in the x-axis, which we will discuss shortly. In IB questions, you are often asked to identify the amplitude change of trigonometric functions — for example, y = 3sin x stretches the sine wave vertically by a factor of 3, changing its amplitude from 1 to 3.
每个点 (x, y) 映射为 (x, a y)。若 a 为负数，则还包含关于 x 轴的反射，我们稍后将讨论。IB 题目常要求你识别三角函数的振幅变化——例如 y = 3sin x 将正弦波垂直拉伸 3 倍，振幅从 1 变为 3。
4. Horizontal Stretch and Compression | 水平伸缩
For y = f(bx), the graph is horizontally stretched or compressed by a factor of 1b (positive b). If |b| > 1, the graph is compressed horizontally; if 0 < |b| < 1, it is stretched. The y-intercept remains fixed at (0, f(0)).
对于 y = f(bx)，图像在水平方向以 1b 的因子（b 为正数）被拉伸或压缩。若 |b| > 1，图像被水平压缩；若 0 < |b| < 1，则被水平拉伸。y 轴截距保持在 (0, f(0))。
y = f(bx), stretch factor = 1/b
y = f(bx)，伸缩因子 = 1/b
Again, the direction seems reversed. If you double the input (b = 2), the function reaches the same output value twice as quickly, so the graph appears narrower — a horizontal compression. If b = 12, the graph appears wider. This concept is critical for understanding the period of sine and cosine functions: y = sin(2x) has period π, not 2π.
同样，方向看起来是反的。如果将输入加倍（b = 2），函数达到相同输出值的速度快了一倍，所以图像显得更窄——即水平压缩。若 b = 12，图像则显得更宽。这一概念对理解正弦和余弦函数的周期至关重要：y = sin(2x) 的周期是 π，而非 2π。
5. Reflection in the x-axis | 关于 x 轴的反射
Multiplying the entire function by -1 creates a reflection across the x-axis: y = -f(x). Every point (x, y) becomes (x, -y), flipping the graph upside down.
将整个函数乘以 -1 会产生关于 x 轴的反射：y = -f(x)。每一点 (x, y) 变为 (x, -y)，使图像上下翻转。
y = -f(x) ⇔ reflection in the x-axis
y = -f(x) ⇔ 关于 x 轴的反射
For example, y = -ex is the mirror image of the exponential growth curve y = ex about the horizontal axis. The x-intercepts stay the same, but the range is negated: if f(x) ∈ [0, ∞), then -f(x) ∈ (-∞, 0].
例如，y = -ex 是指数增长曲线 y = ex 关于水平轴的镜像。x 轴截距保持不变，但值域取相反数：若 f(x) ∈ [0, ∞)，则 -f(x) ∈ (-∞, 0]。
6. Reflection in the y-axis | 关于 y 轴的反射
Replacing x with -x creates a reflection across the y-axis: y = f(-x). Every point (x, y) maps to (-x, y).
将 x 替换为 -x 会产生关于 y 轴的反射：y = f(-x)。每一点 (x, y) 映射为 (-x, y)。
y = f(-x) ⇔ reflection in the y-axis
y = f(-x) ⇔ 关于 y 轴的反射
A function whose graph is symmetric about the y-axis satisfies f(-x) = f(x) and is called even. An odd function satisfies f(-x) = -f(x) and is symmetric about the origin. Reflecting an exponential function like y = 2x yields y = 2-x = (12)x, a decaying exponential.
图像关于 y 轴对称的函数满足 f(-x) = f(x)，称为偶函数。奇函数满足 f(-x) = -f(x)，关于原点对称。例如，将指数函数 y = 2x 反射得到 y = 2-x = (12)x，即衰减型指数函数。
7. Absolute Value Transformations | 绝对值变换
There are two distinct absolute value transformations, and students often mix them up.
绝对值变换有两种不同类型，学生经常混淆。
The first is y = |f(x)|: the part of the graph below the x-axis is reflected upward, while the part above remains unchanged. The whole output becomes non-negative. For example, y = |x - 2| - 1 creates a V-shaped graph where the section of the line y = x - 3 below the axis is flipped up.
第一种是 y = |f(x)|：图像位于 x 轴下方的部分被向上反射，位于上方的部分保持不变。所有输出值变为非负数。例如，y = |x - 2| - 1 的 V 形图中，直线 y = x - 3 位于轴下方的部分被翻转到上方。
The second is y = f(|x|): the part of the graph for x ≥ 0 is kept, and the graph for x < 0 is replaced by the mirror image of the right-hand branch. The result is always an even function — symmetric about the y-axis.
第二种是 y = f(|x|)：保留 x ≥ 0 部分的图像，而 x < 0 部分被替换为右侧分支的镜像。结果始终是偶函数——关于 y 轴对称。
y = |f(x)| reflects negative parts upward; y = f(|x|) mirrors the right side to the left
y = |f(x)| 将负值部分向上反射；y = f(|x|) 将右侧镜像到左侧
8. Combined Transformations | 组合变换
Real-world IB problems rarely test a single transformation in isolation. The general transformed function takes the form:
实际 IB 题目很少单独考一种变换。一般形式的变换函数为：
y = a·f(b(x - h)) + k
y = a·f(b(x - h)) + k
Here, k controls vertical translation, h controls horizontal translation, a controls vertical stretch/reflection, and b controls horizontal stretch/reflection. The order of operations matters: horizontal transformations apply to x before the function is evaluated, and vertical transformations apply after.
其中 k 控制垂直平移，h 控制水平平移，a 控制垂直伸缩/反射，b 控制水平伸缩/反射。变换的先后顺序很重要：水平变换作用于 x 之后再进行函数求值，垂直变换则作用于函数值之后。
A recommended strategy is to apply transformations in this order: horizontal stretch/reflection, then horizontal translation; vertical stretch/reflection, then vertical translation. This aligns with the algebraic structure of the expression.
推荐的变换顺序为：先进行水平伸缩/反射，再进行水平平移；先进行垂直伸缩/反射，再进行垂直平移。这与表达式的代数结构相一致。
9. Transformations of Trigonometric Functions | 三角函数变换
In IB Mathematics, trigonometric transformations are among the most heavily examined applications of graph transformations. The general sine function is written as:
在 IB 数学中，三角函数变换是图像变换应用中最常考查的内容之一。一般正弦函数可写作：
y = A·sin(B(x - C)) + D
y = A·sin(B(x - C)) + D
Here, |A| is the amplitude, 2π|B| is the period, C is the horizontal phase shift, and D is the vertical shift. For example, y = 2sin(3x - π2) + 1 can be rewritten as 2sin(3(x - π6)) + 1, revealing: amplitude 2, period 2π3, phase shift right by π6, and vertical shift up by 1.
其中 |A| 为振幅，2π|B| 为周期，C 为水平相位移，D 为垂直位移。例如，y = 2sin(3x - π2) + 1 可改写为 2sin(3(x - π6)) + 1，由此可看出：振幅为 2，周期为 2π3，相位向右平移 π6，垂直向上平移 1。
10. Common Pitfalls in IB Exams | IB 考试常见陷阱
Let us examine the most frequent mistakes students make in IB examinations.
让我们来看看 IB 考试中学生最常犯的错误。
Confusing y = f(x - 2) with a left shift — the horizontal direction is always opposite to the sign. | 将 y = f(x - 2) 误认为是左移——水平方向始终与符号相反。
Forgetting that y = f(2x) compresses, not stretches, the graph by a factor of 12. | 忘记 y = f(2x) 是将图像以 12 的因子压缩而非拉伸。
Applying horizontal transformations before factoring out coefficients: y = f(2x - 4) is a shift of y = f(2x), equivalent to y = f(2(x - 2)), i.e., shift right by 2, not by 4. | 在提出系数之前就进行水平平移：y = f(2x - 4) 是 y = f(2x) 的平移，等价于 y = f(2(x - 2))，即右移 2 个单位而非 4 个单位。
Treating |f(x)| and f(|x|) as identical transformations. | 将 |f(x)| 与 f(|x|) 视为相同变换。
When finding the image of a specific point, applying vertical transformations before horizontal ones inconsistently. | 求特定点像点时，不一致地先应用垂直变换再应用水平变换。
11. Worked Example | 实战例题
Let us apply all the rules in a typical IB-style question. Given f(x) = x², describe the transformations that map f to g(x) = 2(x - 1)² + 3.
让我们用一道典型的 IB 风格题目来运用所有规则。已知 f(x) = x²，描述将 f 映射到 g(x) = 2(x - 1)² + 3 的变换。
The transformation is y = 2(x - 1)² + 3, which fits the general form y = a f(b(x - h)) + k with a = 2, b = 1, h = 1, k = 3.
变换式为 y = 2(x - 1)² + 3，符合一般形式 y = a f(b(x - h)) + k，其中 a = 2，b = 1，h = 1，k = 3。
Step 1: Vertically stretch y = x² by a factor of 2 to obtain y = 2x². Step 2: Shift the graph right by 1 unit to obtain y = 2(x - 1)². Step 3: Shift upward by 3 units to obtain g(x) = 2(x - 1)² + 3.
第一步：将 y = x² 垂直拉伸 2 倍，得到 y = 2x²。第二步：将图像向右平移 1 个单位，得到 y = 2(x - 1)²。第三步：向上平移 3 个单位，得到 g(x) = 2(x - 1)² + 3。
Check the vertex: the original parabola has vertex (0, 0). After a vertical stretch, it remains (0, 0). After shifting right 1 and up 3, the new vertex is (1, 3), which matches g(x) directly. This kind of point-checking is an excellent verification method in exam conditions.
验证顶点：原抛物线顶点为 (0, 0)。垂直拉伸后仍为 (0, 0)。右移 1 个单位再上移 3 个单位后，新顶点为 (1, 3)，与 g(x) 直接吻合。在考试条件下，这种点位验证法是一种极好的检查手段。
12. Summary Table | 总结表
The table below summarises all key transformations for quick revision before your IB exam.
下表总结了所有关键变换，供你在 IB 考前快速复习。
| Transformation | 变换 | Formula | 公式 | Graphical Effect | 图像效果 |
| Vertical translation | 垂直平移 | y = f(x) + k | Shift up (k > 0) or down (k < 0) | 向上（k > 0）或向下（k < 0）平移 |
| Horizontal translation | 水平平移 | y = f(x - h) | Shift right (h > 0) or left (h < 0) | 向右（h > 0）或向左（h < 0）平移 |
| Vertical stretch/compression | 垂直伸缩 | y = a·f(x) | Stretch (|a| > 1) or compress (0 < |a| < 1) vertically | 垂直拉伸（|a| > 1）或压缩（0 < |a| < 1） |
| Horizontal stretch/compression | 水平伸缩 | y = f(bx) | Compress (|b| > 1) or stretch (0 < |b| < 1) by 1/|b| | 以 1/|b| 的因子压缩（|b| > 1）或拉伸（0 < |b| < 1） |
| Reflection in x-axis | 关于 x 轴反射 | y = -f(x) | Flip upside down | 上下翻转 |
| Reflection in y-axis | 关于 y 轴反射 | y = f(-x) | Mirror left-right | 左右镜像 |
| Absolute value of function | 函数绝对值 | y = |f(x)| | Reflect negative parts above the x-axis | 将负值部分反射到 x 轴上方 |
| Function of absolute value | 绝对值函数 | y = f(|x|) | Mirror the right side to the left | 将右侧镜像到左侧 |
Master these transformations and their algebraic connections, and you will handle even the most demanding graph-related questions in IB Mathematics with confidence.
掌握这些变换及其代数联系，你就能自信地应对 IB 数学中最具挑战性的图像相关问题。
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