Function Graph Transformations | 函数图像变换

📚 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。

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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 \((0, 3)\)。

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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 \((2, 0)\)。

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3. Vertical Stretch and Compression | 垂直伸缩

Multiplying a function by a positive constant \(a\) produces a vertical stretch or compression: \(y = a \cdot f(x)\). 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：\(y = a \cdot f(x)\)。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, \(y = 3\sin x\) 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 \(y = 3\sin x\) 将正弦波垂直拉伸 3 倍，振幅从 1 变为 3。

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4. Horizontal Stretch and Compression | 水平伸缩

For \(y = f(bx)\), the graph is horizontally stretched or compressed by a factor of \(\frac{1}{b}\) (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 \(\frac{1}{b}\) 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 \(b = \frac{1}{2}\), the graph appears wider. This concept is critical for understanding the period of sine and cosine functions: \(y = \sin(2x)\) has period \(\pi\), not \(2\pi\).

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 \(b = \frac{1}{2}\)，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 \(\pi\)，而非 \(2\pi\)。

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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, \(y = -e^x\) is the mirror image of the exponential growth curve \(y = e^x\) about the horizontal axis. The \(x\)-intercepts stay the same, but the range is negated: if \(f(x) \in [0, \infty)\), then \(-f(x) \in (-\infty, 0]\).

例如，\(y = -e^x\) 是指数增长曲线 \(y = e^x\) 关于水平轴的镜像。\(x\) 轴截距保持不变，但值域取相反数：若 \(f(x) \in [0, \infty)\)，则 \(-f(x) \in (-\infty, 0]\)。

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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 \(y = 2^x\) yields \(y = 2^{-x} = (\frac{1}{2})^x\), 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 \(y = 2^x\) 反射得到 \(y = 2^{-x} = (\frac{1}{2})^x\)，即衰减型指数函数。

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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 \(x \ge 0\) 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 \(x \ge 0\) 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

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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。

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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, \(\frac{2\pi}{|B|}\) is the period, \(C\) is the horizontal phase shift, and \(D\) is the vertical shift. For example, \(y = 2\sin(3x – \frac{\pi}{2}) + 1\) can be rewritten as \(2\sin(3(x – \frac{\pi}{6})) + 1\), revealing: amplitude 2, period \(\frac{2\pi}{3}\), phase shift right by \(\frac{\pi}{6}\), and vertical shift up by 1.

其中 \(|A|\) 为振幅，\(\frac{2\pi}{|B|}\) 为周期，\(C\) 为水平相位移，\(D\) 为垂直位移。例如，\(y = 2\sin(3x – \frac{\pi}{2}) + 1\) 可改写为 \(2\sin(3(x – \frac{\pi}{6})) + 1\)，由此可看出：振幅为 2，周期为 \(\frac{2\pi}{3}\)，相位向右平移 \(\frac{\pi}{6}\)，垂直向上平移 1。

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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 \(\frac{1}{2}\). | 忘记 \(y = f(2x)\) 是将图像以 \(\frac{1}{2}\) 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。

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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。

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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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