📚 Transformation of Graphs | 图像变换
In IB Mathematics, mastering graph transformations is essential for understanding how functions behave when their equations are altered. Rather than plotting numerous points, you can apply a set of rules to translate, stretch, compress, or reflect the graph of a parent function. This article provides a systematic guide to all standard transformations, the correct order of operations, and practical tips for sketching transformed graphs efficiently and accurately.
在IB数学中，掌握图像变换对于理解函数方程改变后图像如何变化至关重要。你不必逐点描画，而是可以应用一组规则对母函数的图像进行平移、伸缩或反射。本文系统讲解所有标准变换规则、正确的操作顺序以及高效、准确绘制变换后图像的实用技巧。
1. Understanding the Parent Function | 理解母函数
the parent function. Common parent functions in the IB syllabus include f(x) = x², f(x) = √x, f(x) = 1/x, f(x) = sin x, f(x) = eˣ asymptotes of these basic graphs allows you to predict the new graph after transformation.
每次变换都从一个已知的基本图像开始，这个基本图像被称为母函数。IB课程中常见的母函数包括 f(x) = x²、f(x) = √x、f(x) = 1/x、f(x) = sin x、f(x) = eˣ 和 f(x) = ln x。熟悉这些基本图像的形状、关键点和渐近线，你就能预判变换后的新图像。
When applying transformations, you essentially map each point (x, y) on the parent graph to a new point (x', y') according to specific rules. This point-mapping approach is a powerful tool for sketching.
应用变换时，本质上是在根据特定规则将母函数图像上的每个点 (x, y) 映射到一个新的点 (x', y')。这种点映射方法是作图的强大工具。
2. Vertical Translations: y = f(x) + k | 垂直平移：y = f(x) + k
the graph vertically. If k > 0, the graph moves up by k units; if k < 0, it moves down by |k| units. The transformation can be written as (x, y) → (x, y + k).
给函数加上一个常数 k，图像会垂直移动。若 k > 0，图像向上移动 k 个单位；若 k < 0，图像向下移动 |k| 个单位。该变换可表示为 (x, y) → (x, y + k)。
2 is shifted down by 2. This does not affect the shape or the x-coordinates of any points.
例如，从 f(x) = x² 开始，g(x) = x² + 3 的图像是向上平移 3 个单位的抛物线，而 h(x) = x² − 2 则是向下平移 2 个单位。这种变换不改变图像形状，也不改变任何点的 x 坐标。
3. Horizontal Translations: y = f(x − h) | 水平平移：y = f(x − h)
ally. If h > 0, the graph moves to the right by h units; if h < 0, it moves to the left by |h| units. The notation (x, y) → (x + h, y) helps clarify the direction.
将函数中的 x 替换为 (x − h) 会使图像水平移动。若 h > 0，图像向右移动 h 个单位；若 h < 0，图像向左移动 |h| 个单位。使用 (x, y) → (x + h, y) 这种记法有助于理清方向。
(x + 2) = (x − (−2)), h = −2, so the shift is two units to the left. Practice with f(x) = √x: g(x) = √x − 4 shifts the square root curve 4 units right.
一个常见的误解是认为 f(x + 2) 会使图像向右移动；然而，因为 (x + 2) = (x − (−2))，h = −2，所以实际上是向左移动两个单位。用 f(x) = √x 练习一下：g(x) = √x − 4 将平方根曲线向右平移了 4 个单位。
4. Vertical Stretches and Compressions: y = a f(x) | 垂直伸缩：y = a f(x)
ltiplying the entire function by a constant a causes a vertical stretch or compression. If |a| > 1, the graph stretches away from the x-axis; if 0 < |a| < 1, it compresses toward the x-axis. Points transform as (x, y) → (x, a y).
用常数 a 乘以整个函数会导致垂直拉伸或压缩。若 |a| > 1，图像沿竖直方向远离 x 轴拉伸；若 0 < |a| < 1，图像向 x 轴压缩。点的变换为 (x, y) → (x, a y)。
mes g(x) = 3 sin x, tripling the amplitude, while h(x) = (1/2) sin x halves the amplitude.
如果 a 为负数，还会同时发生关于 x 轴的反射，这将在后续小节中讨论。例如，f(x) = sin x 变成 g(x) = 3 sin x，振幅变为原来的三倍；而 h(x) = (1/2) sin x 则使振幅减半。
5. Horizontal Stretches and Compressions: y = f(bx) | 水平伸缩：y = f(bx)
Replacing x with bx inside the function produces a horizontal stretch or compression. If |b| > 1, the graph compresses horizontally toward the y-axis (looks squeezed); if 0 < |b| < 1, it stretches away from the y-axis. The point mapping is (x, y) → (x/b, y).
将函数中的 x 替换为 bx 会产生水平伸缩。若 |b| > 1，图像沿水平方向朝 y 轴压缩（看起来被挤压）；若 0 < |b| < 1，图像则远离 y 轴拉伸。点的映射关系为 (x, y) → (x/b, y)。
period doubles to 4π. Always check the effect on key x-coordinates to confirm your sketch.
请注意，水平变换常常让人觉得与直觉相反。对于 y = sin(2x)，周期减半为 π；对于 y = sin(x/2)，周期加倍为 4π。务必检查关键 x 坐标上的效果以确认草图。
6. Reflections Across the Axes | 关于坐标轴的反射
→ (x, −y). The second negates the x-input, reflecting across the y-axis: (x, y) → (−x, y).
两种基本的反射是 y = −f(x) 和 y = f(−x)。前者将 y 值取反，图像关于 x 轴对称：(x, y) → (x, −y)。后者将 x 的输入取反，图像关于 y 轴对称：(x, y) → (−x, y)。
(−x) = −f(x), so a reflection in the y-axis looks identical to a reflection in the x-axis.
同时进行这两种反射，即 y = −f(−x)，等价于关于原点旋转 180°。对于像 f(x) = x³ 这样的奇函数，f(−x) = −f(x)，所以关于 y 轴的反射看起来与关于 x 轴的反射相同。
7. The General Transformation Form | 标准变换通式
)) + k, where each parameter has a specific role. Here, a controls vertical stretch/compression and reflection in the x-axis; b controls horizontal stretch/compression and reflection in the y-axis; h is the horizontal shift; k is the vertical shift.
所有IB图像变换都可以用通式 y = a f(b(x − h)) + k 来概括，其中每个参数都有特定的作用。a 控制垂直伸缩以及关于 x 轴的反射；b 控制水平伸缩以及关于 y 轴的反射；h 是水平平移量；k 是垂直平移量。
t be rewritten as f(2(x + 3)) to identify the correct horizontal shift: h = −3, and b = 2.
关键是要将自变量写成 (x − h) 的形式放在函数内部。例如，f(2x + 6) 必须重写为 f(2(x + 3))，才能正确识别水平平移量：h = −3，b = 2。
8. Order of Transformations | 变换的顺序
is: first, horizontal translation by h; second, horizontal stretch/compression by 1/b; then vertical stretch/compression by a; and finally vertical translation by k. Alternatively, you can follow the “inside first, then outside” ions affecting x first (with stretch before shift, being careful), then those affecting y.
当应用多重变换时，顺序至关重要。针对 y = a f(b(x − h)) + k 的推荐顺序是：首先，水平平移 h；其次，水平伸缩（因子为 1/b）；然后垂直伸缩（因子为 a）；最后垂直平移 k。也可以遵循“先内后外”规则：先处理影响 x 的变换（注意伸缩在平移之前，需谨慎），再处理影响 y 的变换。
− 3)² + 1 by shifting right 3, then vertically stretching by 2, and finally shifting up 1.
一个典型错误是在垂直伸缩之前进行垂直平移，这会导致点的位置错误。为避免失误，始终让一个测试点按顺序经历所有变换。例如，将 f(x) = x² 变为 g(x) = 2(x − 3)² + 1：先向右平移 3，然后垂直拉伸为 2 倍，最后向上平移 1。
9. Worked Examples of Combined Transformations | 复合变换实例
Consider the base function f(x) = √x. Describe and sketch the transformation to g(x) = −2√3 − x actor 1 (none), vertical stretch by 2, reflection in x-axis (a = −2), vertical shift up 4.
考虑母函数 f(x) = √x。描述并绘制到 g(x) = −2√3 − x + 4 的变换。首先，改写为 g(x) = −2√(−(x − 3)) + 4。由此可以看出：关于 y 轴反射（b = −1），水平右移 3，水平压缩因子 1（无变化），垂直拉伸 2 倍，关于 x 轴反射（a = −2），垂直上移 4。
Following the correct order: start with (0,0). Reflect in y-axis: (0,0) unchanged. Shift right 3: (3,0). Vertical stretch by 2 and reflect in x-axis: (3,0). Shift up 4: (3,4) is the new starting point. The graph opens downward from (3,4).
遵循正确的顺序：从 (0,0) 开始。关于 y 轴反射：(0,0) 不变。向右平移 3：(3,0)。垂直拉伸 2 倍并关于 x 轴反射：(3,0)。向上平移 4：(3,4) 成为新的起点。图像从 (3,4) 开始向下延伸。
10. Transformations of Trigonometric Functions | 三角函数的变换
n is y = A sin(B(x − C)) + D, where A is the amplitude, B affects the period (period = 2π/|B|), C is the horizontal shift (phase shift), and D is the vertical shift (principal axis).
在IB考试中，经常考查三角函数的图像变换。正弦函数的一般形式为 y = A sin(B(x − C)) + D，其中 A 为振幅，B 影响周期（周期 = 2π/|B|），C 为水平平移量（相位移动），D 为垂直平移量（主轴线）。
os(3(x + π/3)) − 1. The amplitude is 2, period is 2π/3, phase shift is −π/3 (left π/3), and the graph is shifted down 1 unit. Knowing how to extract these features from the equation saves time in exams.
对于 y = 2 cos(3x + π) − 1，重写为 y = 2 cos(3(x + π/3)) − 1。振幅为 2，周期为 2π/3，相位移动为 −π/3（左移 π/3），图像下移 1 个单位。从方程中提取这些特征可以在考试中节省时间。
11. Transformations of Exponential and Logarithmic Functions | 指数函数与对数函数的变换
Exponential functions of the form y = aˣ can be shifted and stretched like any other function. A common transformation is y = eᵏˣ ns, but be mindful of its vertical asymptote at x = 0, which moves with horizontal shifts.
形如 y = aˣ 的指数函数可以像其他函数一样进行平移和伸缩。常见变换如 y = eᵏˣ，会水平压缩或拉伸增长率。自然对数 y = ln x 经历类似的变换，但要注意它在 x = 0 处的垂直渐近线，该渐近线会随水平平移而移动。
For instance, y = 3 ⋅ 2ˣ⁻⁴ + 5 takes the base 2ˣ by 5. The horizontal asymptote moves from y = 0 to y = 5, which is crucial for sketching.
例如，y = 3 ⋅ 2ˣ⁻⁴ + 5 将底函数 2ˣ 右移 4，垂直拉伸 3 倍，再上移 5。水平渐近线从 y = 0 移动到 y = 5，这对绘图至关重要。
12. Using Transformations to Find Equations from Graphs | 利用变换从图像求方程
IB exam questions often present a transformed graph and ask for its equation. Identify the parent function first. Then, determine the transformations by comparing key features: the new position of a maximum, minimum, intercept, or asymptote. Use this to deduce h, k, a, and b.
IB考试题目通常会给出一个变换后的图像，要求写出其方程。首先要识别出母函数。然后，通过比较关键特征来确定变换：极值点、截距或渐近线的新位置。由此推断出 h、k、a 和 b。
ion is y = (x − 2)² − 3. Always check for possible reflections when the graph is inverted.
对于一个顶点在 (2, −3) 且经过 (4, 1) 的抛物线，从 f(x) = x² 开始。平移为右移 2、下移 3：y = a(x − 2)² − 3。代入 (4,1) 求得 a = 1。方程为 y = (x − 2)² − 3。当图像倒置时，一定要检查是否存在反射。
13. Common Pitfalls and Exam Tips | 常见错误与考试技巧
Many students confuse the direction of horizontal translations. Remember: f(x + 2) shifts left. Another frequent mistake is forgetting to factorise the coefficient of x inside the function before identifying h. Also, do not apply the vertical stretch after the vertical shift. Always apply stretches/compressions and reflections before translations in the vertical direction.
许多学生容易混淆水平平移的方向。记住：f(x + 2) 向左平移。另一个常见错误是在识别 h 之前忘记将 x 的系数提取出来。此外，不要在垂直平移之后进行垂直伸缩。在竖直方向上，一定要先进行伸缩/反射，再进行平移。
For sketching, mark the new positions of a few key points rather than trying to redraw the entire curve from memory. Using dashed lines for asymptotes and indicating clearly labelled axes will make your graphs more accurate and earn full marks on Paper 2 and Internal Assessment tasks.
在绘图时，标出几个关键点的新位置，而不是试图凭记忆重画整个曲线。使用虚线表示渐近线并清晰地标注坐标轴，能使你的图像更加准确，在试卷二和内部评估作业中拿到满分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图