Transforming Graphs of Functions | 函数图像变换规律总结

📚 Transforming Graphs of Functions | 函数图像变换规律总结

Understanding how transformations affect the graph of a function is one of the most powerful tools in IB Mathematics. Whether you are studying Analysis & Approaches or Applications & Interpretation, a solid grasp of translations, reflections, stretches, and absolute value transformations can dramatically improve your speed and accuracy on exams.

理解图像变换如何影响函数图形，是IB数学中最强大的工具之一。无论你学习的是分析与方法（AA）还是应用与解释（AI），扎实掌握平移、反射、伸缩和绝对值变换，都能显著提升你在考试中的速度和准确率。

1. Vertical & Horizontal Translations | 垂直与水平平移

A translation shifts the graph of a function without changing its shape. The two fundamental rules are straightforward: for a positive constant \(k\) (we do not use LaTeX here — instead we write \(k\) simply as k), the graph of y = f(x) + k moves the curve upward by k units, while y = f(x) − k moves it downward by k units. For horizontal shifts, y = f(x − h) moves the graph to the right by h units, and y = f(x + h) moves it to the left by h units. Notice the counterintuitive direction for horizontal shifts: subtracting inside the function moves right.

平移变换在不改变函数图形形状的前提下移动整个图形。两条基本规则非常直观：对于正常数k，y = f(x) + k 将曲线向上移动k个单位，而 y = f(x) − k 将其向下移动k个单位。对于水平平移，y = f(x − h) 将图形向右移动h个单位，y = f(x + h) 则向左移动h个单位。注意水平平移的方向是反直觉的：在函数内部做减法时，图形向右移动。

The key point to remember is that vertical translations affect the output (y-values) directly, while horizontal translations affect the input (x-values). For example, if f(x) = x², then f(x − 3) = (x − 3)² has its vertex at (3, 0), not at (−3, 0).

需要记住的关键点是：垂直平移直接作用于输出值（y值），而水平平移作用于输入值（x值）。例如，若 f(x) = x²，则 f(x − 3) = (x − 3)² 的顶点在 (3, 0)，而不是 (−3, 0)。

Summary: y = f(x) + k → shift up k; y = f(x) − k → shift down k; y = f(x − h) → shift right h; y = f(x + h) → shift left h.

总结：y = f(x) + k → 上移k；y = f(x) − k → 下移k；y = f(x − h) → 右移h；y = f(x + h) → 左移h。

2. Reflections: x-axis & y-axis | 反射变换：关于x轴与y轴

Reflections produce a mirror image of the original graph. The rule y = −f(x) reflects the graph of y = f(x) across the x-axis. Every point (x, y) on the original curve becomes (x, −y). This means all positive y-values become negative, and all negative y-values become positive. A function and its reflection across the x-axis will have opposite signs for every non-zero output.

反射变换产生原图形的镜像。规则 y = −f(x) 将 y = f(x) 的图形关于x轴反射。原曲线上的每个点 (x, y) 变为 (x, −y)。这意味着所有正的y值变为负的，所有负的y值变为正的。一个函数与其关于x轴的反射图像，在每个非零输出处符号相反。

The rule y = f(−x) reflects the graph across the y-axis. Every point (x, y) becomes (−x, y). This transformation preserves the y-values but swaps the x-values. For even functions, such as f(x) = x² or f(x) = cos(x), the reflection across the y-axis produces an identical graph because these functions are symmetric about the y-axis.

规则 y = f(−x) 将图形关于y轴反射。每个点 (x, y) 变为 (−x, y)。此变换保留y值但交换x值。对于偶函数，如 f(x) = x² 或 f(x) = cos(x)，关于y轴的反射产生完全相同的图形，因为这些函数关于y轴对称。

An important exam trick: when a graph is symmetric about the y-axis, f(−x) = f(x) for all x in the domain. When a graph is symmetric about the origin (odd functions), f(−x) = −f(x). Recognizing these symmetries can save you from performing lengthy calculations.

一个重要的考试技巧：当图形关于y轴对称时，对所有定义域内的x，f(−x) = f(x)。当图形关于原点对称时（奇函数），f(−x) = −f(x)。识别这些对称性可以避免冗长的计算。

3. Vertical & Horizontal Stretches | 垂直与水平伸缩

Stretches (also called dilations or scalings) change the size of the graph while preserving its overall shape. The vertical stretch y = a·f(x), where a > 1, stretches the graph away from the x-axis by a factor of a. Every point (x, y) becomes (x, a·y). If 0 < a < 1, the graph is compressed toward the x-axis. For example, y = 2f(x) makes the graph twice as tall at every x-value, while y = ½f(x) flattens the graph.

伸缩变换（也称缩放变换）改变图形的大小，同时保持其整体形状。垂直伸缩 y = a·f(x)，其中 a > 1，将图形沿远离x轴的方向拉伸a倍。每个点 (x, y) 变为 (x, a·y)。若 0 < a < 1，则图形向x轴压缩。例如，y = 2f(x) 使图形在每个x值处变为原来的两倍高，而 y = ½f(x) 使图形变得平坦。

Horizontal stretches follow the rule y = f(x/a). When a > 1, the graph stretches away from the y-axis by a factor of a; every point (x, y) becomes (a·x, y). When 0 < a < 1, the graph compresses toward the y-axis. Again, the direction is counterintuitive: dividing x by a value greater than 1 stretches horizontally. A common student mistake is to assume that y = f(2x) stretches the graph by a factor of 2 — in fact, it compresses it by a factor of 2.

水平伸缩遵循规则 y = f(x/a)。当 a > 1 时，图形沿远离y轴的方向拉伸a倍；每个点 (x, y) 变为 (a·x, y)。当 0 < a < 1 时，图形向y轴压缩。同样，方向是反直觉的：将x除以大于1的值会水平拉伸。学生常犯的错误是认为 y = f(2x) 将图形拉伸2倍——实际上，它是将图形压缩为原来的1/2。

Transformation	Effect on Graph	Point Mapping
y = a·f(x), a > 1	Vertical stretch by factor a	(x, y) → (x, a·y)
y = a·f(x), 0 < a < 1	Vertical compression by factor a	(x, y) → (x, a·y)
y = f(x/a), a > 1	Horizontal stretch by factor a	(x, y) → (a·x, y)
y = f(x/a), 0 < a < 1	Horizontal compression by factor a	(x, y) → (a·x, y)

Key insight: a ‘stretch’ outside the function (multiplying f(x)) affects y-values vertically; a ‘stretch’ inside the function (multiplying x) affects x-values horizontally.

关键洞察：函数外部的乘法（乘以 f(x)）影响y值，产生垂直伸缩；函数内部的乘法（乘以x）影响x值，产生水平伸缩。

4. Combining Transformations | 组合变换与变换顺序

In IB exams, you will frequently encounter multiple transformations applied to a single function. The order in which transformations are applied matters significantly, especially when combining horizontal and vertical transformations. A reliable strategy is to follow the standard order: horizontal stretch/compression first, then horizontal translation; vertical stretch/compression first, then vertical translation. Equivalently, apply transformations in the same order as the algebraic operations appear when reading from the inside out.

在IB考试中，你经常会遇到对同一个函数施加多种变换的情况。变换的顺序至关重要，尤其是在组合水平变换和垂直变换时。一个可靠的策略是遵循标准顺序：先进行水平伸缩，再进行水平平移；先进行垂直伸缩，再进行垂直平移。等价地，按照代数运算从内到外出现的顺序依次施加变换。

Consider the function g(x) = 2f(3x − 6) + 1. To graph g from f, you should first factor the inside: g(x) = 2f(3(x − 2)) + 1. Then apply in this order: (1) horizontal compression by a factor of 3; (2) horizontal translation right by 2 units; (3) vertical stretch by a factor of 2; (4) vertical translation up by 1 unit. If you reverse the first two steps, you will obtain a different graph.

考虑函数 g(x) = 2f(3x − 6) + 1。要从f得到g的图形，首先对内层进行因式分解：g(x) = 2f(3(x − 2)) + 1。然后按以下顺序操作：(1) 水平压缩为原来的1/3；(2) 向右水平平移2个单位；(3) 垂直拉伸2倍；(4) 向上垂直平移1个单位。如果颠倒前两步，得到的图形将会不同。

A useful way to verify your transformation order is to track a single point. Take a known point on y = f(x), such as (0, f(0)), and apply each transformation step by step. If the final coordinates match what you expect from direct substitution into g(x), your order is correct.

验证变换顺序的一个有效方法是追踪一个关键点。取 y = f(x) 上的一个已知点，如 (0, f(0))，逐步施加每个变换。如果最终坐标与直接代入 g(x) 得到的结果一致，说明你的顺序是正确的。

5. Absolute Value Transformations | 绝对值变换

Absolute value transformations are a favourite topic in IB exams because they combine algebraic reasoning with visual intuition. The transformation y = |f(x)| reflects any portion of the graph that lies below the x-axis upward, above the x-axis. All negative y-values become positive; positive y-values remain unchanged. Graphically, you keep everything above the x-axis and mirror the parts below it across the x-axis.

绝对值变换是IB考试中的热门考点，因为它将代数推理与直观图形相结合。变换 y = |f(x)| 将图形中位于x轴下方的部分向上翻折到x轴上方。所有负的y值变为正的；正的y值保持不变。图形上，保留x轴上方的部分，并将x轴下方的部分关于x轴镜像翻折。

The transformation y = f(|x|) is different: it keeps the portion of the graph for x ≥ 0 unchanged, and then reflects that right-hand portion across the y-axis to create the left-hand side. The original left-hand portion is discarded. For example, if f(x) = x − 2, then f(|x|) = |x| − 2, which is a V-shaped graph with vertex at (0, −2).

变换 y = f(|x|) 则不同：它保留 x ≥ 0 部分的图形不变，然后将这一右侧部分关于y轴反射，生成左侧部分。原左侧部分被丢弃。例如，若 f(x) = x − 2，则 f(|x|) = |x| − 2，这是一个顶点在 (0, −2) 的V形图形。

Here is a common exam trap: for y = |f(x)|, the function becomes non-negative everywhere, so any roots (x-intercepts) of the original function remain roots. But for y = f(|x|), the function is even (symmetric about the y-axis), regardless of whether the original f was even. Practice distinguishing these two cases with simple functions such as f(x) = x² − 4 or f(x) = x³.

这里有一个常见的考试陷阱：对于 y = |f(x)|，函数处处非负，因此原函数的任何零点（x截距）仍然是零点。但对于 y = f(|x|)，函数一定是偶函数（关于y轴对称），无论原函数f是否为偶函数。用简单函数如 f(x) = x² − 4 或 f(x) = x³ 练习区分这两种情况。

6. Tracking Key Points & Asymptotes | 关键点与渐近线的追踪

When applying transformations, it is often unnecessary to redraw the entire graph from scratch. Instead, track a small set of key features: intercepts, turning points, endpoints (for restricted domains), and asymptotes. Each of these features transforms according to the same rules as the points on the graph.

在施加变换时，通常没有必要从头重画整个图形。相反，追踪一小组关键特征即可：截距、转折点、端点（对于有限定义域）和渐近线。每个特征都按照图形上点的相同规则进行变换。

For vertical transformations (y = a·f(x) + k), the x-coordinates of all key points remain unchanged, and only the y-coordinates are scaled and shifted. For horizontal transformations (y = f(x/a − h)), the y-coordinates remain unchanged, and only the x-coordinates are transformed. Vertical asymptotes transform horizontally; horizontal asymptotes transform vertically.

对于垂直变换（y = a·f(x) + k），所有关键点的x坐标保持不变，只有y坐标被缩放和平移。对于水平变换（y = f(x/a − h)），y坐标保持不变，只有x坐标被变换。垂直渐近线做水平变换；水平渐近线做垂直变换。

Consider the rational function f(x) = 1/x, which has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. Under the transformation y = 2f(x − 3) + 1, the vertical asymptote moves to x = 3, and the horizontal asymptote moves to y = 1. The transformed function is 2/(x − 3) + 1, and its asymptotes are immediately visible.

考虑有理函数 f(x) = 1/x，它有垂直渐近线 x = 0 和水平渐近线 y = 0。在变换 y = 2f(x − 3) + 1 下，垂直渐近线移到 x = 3，水平渐近线移到 y = 1。变换后的函数为 2/(x − 3) + 1，其渐近线一目了然。

Rule: vertical asymptotes follow horizontal transformations; horizontal asymptotes follow vertical transformations.

规则：垂直渐近线跟随水平变换；水平渐近线跟随垂直变换。

7. Inverse Functions & Image | 反函数与对应图像

The graph of the inverse function f⁻¹(x) is obtained by reflecting the graph of y = f(x) across the line y = x. Every point (x, y) on the original function corresponds to the point (y, x) on the inverse. This reflection swaps roots and y-intercepts: if f(a) = b, then f⁻¹(b) = a. The domain of f becomes the range of f⁻¹, and vice versa.

反函数 f⁻¹(x) 的图形可通过将 y = f(x) 的图形关于直线 y = x 反射得到。原函数上的每个点 (x, y) 对应反函数上的点 (y, x)。这种反射交换零点和y截距：若 f(a) = b，则 f⁻¹(b) = a。f 的定义域变为 f⁻¹ 的值域，反之亦然。

For a function to have an inverse that is also a function, the original function must be one-to-one (horizontal line test). If f is not one-to-one, you can restrict its domain to make it invertible. In transformation terms, reflecting a non-one-to-one function across y = x produces a curve that fails the vertical line test.

要使函数的反函数也是函数，原函数必须是一一对应的（水平线检验）。若f不是一一对应的，可以通过限制定义域使其可逆。从变换的角度看，将非一一对应函数的图形关于 y = x 反射，得到的曲线无法通过垂直线检验。

In IB exams, questions on inverse graphs often ask you to identify the graph of f⁻¹ from a set of options, or to find the intersection point of f and f⁻¹. Recall that any point where a function and its inverse intersect must lie on the line y = x, since if f(a) = b and f⁻¹(a) = b, then both (a, b) and (b, a) must satisfy the same relationship, forcing a = b.

在IB考试中，关于反函数图形的问题常常要求你从一组选项中指出 f⁻¹ 的图形，或求f与f⁻¹的交点。回顾：函数与其反函数的任何交点必定位于直线 y = x 上，因为若 f(a) = b 且 f⁻¹(a) = b，则 (a, b) 和 (b, a) 必须同时满足同一关系，从而迫使 a = b。

8. Exam Tips & Common Mistakes | 考试技巧与常见错误

Through years of marking IB exams, examiners repeatedly identify the same types of errors in transformation questions. Being aware of these pitfalls is the first step toward avoiding them. Here is a concise list of what to watch out for.

通过多年IB考试的阅卷经验，考官反复发现变换题型中相同的错误类型。意识到这些陷阱是避免它们的第一步。以下是需要警惕的简明清单。

Confusing the direction of horizontal shifts: y = f(x + 2) shifts left, not right. Always think about what value of x makes the inside zero.

混淆水平平移的方向：y = f(x + 2) 向左移而非向右移。始终思考使括号内为零的x值。
Treating y = f(2x) as a stretch instead of a compression. Remember: multiplying x by a number greater than 1 compresses the graph horizontally.

将 y = f(2x) 误认为是拉伸而非压缩。记住：将x乘以大于1的数会水平压缩图形。
Applying vertical and horizontal transformations in the wrong order when both are present. Always factor the inside first, then work from the inside out.

当垂直和水平变换同时存在时，以错误的顺序施加。始终先对内层因式分解，然后由内向外操作。
Forgetting to transform asymptotes. Even if an asymptote is not drawn, you lose marks for not indicating its new position.

忘记变换渐近线。即使渐近线未画出来，不标明其新位置也会失分。
Confusing y = |f(x)| with y = f(|x|). Test with x = −1 to see the difference immediately.

混淆 y = |f(x)| 与 y = f(|x|)。用 x = −1 代入即可立即看出区别。

One of the most effective revision techniques is to build a transformation table for a simple parent function, such as f(x) = 1/x or f(x) = x², and then systematically apply each transformation type. Draw the resulting graphs side by side and annotate the coordinates of key points. This creates lasting visual memory that will serve you well under exam pressure.

最有效的复习技巧之一是为一个简单的母函数（如 f(x) = 1/x 或 f(x) = x²）建立变换表格，然后系统地施加每种变换类型。将得到的图形并排画出，并标注关键点的坐标。这将建立持久的视觉记忆，在考试压力下对你大有帮助。

Mastering graph transformations is not merely a matter of memorising rules — it is about developing an intuition for how algebraic changes to a function manifest as geometric changes to its graph. With regular practice, you will be able to visualise transformations instantly, saving precious time in the exam hall and avoiding costly errors.

掌握图像变换不仅仅是记住规则——更在于培养一种直觉：函数的代数变化如何体现为其图形的几何变化。通过定期练习，你将能够立即在脑海中可视化变换过程，在考场中节省宝贵时间，避免严重的失分错误。

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Transforming Graphs of Functions | 函数图像变换规律总结