IB Mathematics: Stretches of Function Graphs | IB数学：函数图像的伸缩变换

📚 IB Mathematics: Stretches of Function Graphs | IB数学：函数图像的伸缩变换

Stretches are one of the most fundamental transformations in IB Mathematics. When a function f(x) is multiplied by a constant or its input is multiplied by a constant, the resulting graph is a stretch (or compression) of the original. Understanding stretches is essential for analysing trigonometric, exponential, and polynomial functions, and appears regularly in both AA and AI examinations.

伸缩变换是IB数学中最基本的变换之一。当函数f(x)乘以一个常数，或定义域内的自变量乘以一个常数时，所得图像即为原图像的拉伸或压缩。理解伸缩变换对于分析三角函数、指数函数和多项式函数至关重要，并且在AA和AI的考试中经常出现。

1. Vertical Stretch: y = k·f(x) | 垂直伸缩：y = k·f(x)

A vertical stretch is produced by multiplying the entire function by a positive constant k. The transformed function is g(x) = k·f(x), where k > 0.

垂直伸缩是通过将整个函数乘以一个正数常数k来实现的。变换后的函数为g(x) = k·f(x)，其中k > 0。

y = k·f(x), k > 0

If k > 1, the graph is stretched vertically away from the x-axis. Every y-coordinate is multiplied by k, while every x-coordinate remains unchanged. For example, if the original point is (a, b), the new point is (a, kb).

若k > 1，图像沿y轴方向远离x轴拉伸。每个y坐标乘以k，而x坐标保持不变。例如，若原点为(a, b)，则新点为(a, kb)。

If 0 < k < 1, the graph is compressed vertically towards the x-axis. For instance, y = ½·f(x) compresses the height of every point to half its original value.

若0 < k < 1，图像沿y轴方向向x轴压缩。例如，y = ½·f(x)将每点的高度压缩为原来的一半。

Key observation: the x-intercepts (where f(x) = 0) do not change under a vertical stretch, because zero multiplied by any k is still zero.

关键观察：在垂直伸缩下，x轴交点（即f(x) = 0处）不变，因为零乘以任何k仍为零。

k > 1: stretch away from the x-axis | 远离x轴拉伸
0 < k < 1: compress towards the x-axis | 向x轴压缩
x-coordinates unchanged | x坐标不变
x-intercepts invariant | x轴交点不变

2. Horizontal Stretch: y = f(k·x) | 水平伸缩：y = f(k·x)

A horizontal stretch is created by multiplying the input variable by a positive constant k. The transformed function is g(x) = f(kx), where k > 0.

水平伸缩是通过将自变量乘以正数常数k来实现的。变换后的函数为g(x) = f(kx)，其中k > 0。

y = f(k·x), k > 0

Here the effect is reversed compared to a vertical stretch. If k > 1, the graph is compressed towards the y-axis by a factor of 1/k. If 0 < k < 1, the graph is stretched away from the y-axis by a factor of 1/k.

这里的效果与垂直伸缩相反。若k > 1，图像沿x轴方向向y轴压缩至原来的1/k。若0 < k < 1，图像沿x轴方向远离y轴拉伸至原来的1/k。

For example, y = f(2x) means every x-coordinate is halved. A point (a, b) on the original graph moves to (a/2, b). Conversely, y = f(x/2) stretches the graph horizontally by a factor of 2.

例如，y = f(2x)意味着每个x坐标减半。原图像上的点(a, b)移动到(a/2, b)。反之，y = f(x/2)将图像水平拉伸2倍。

Key observation: the y-intercept (at x = 0) remains unchanged under a horizontal stretch, since f(0) is independent of the coefficient of x.

关键观察：在水平伸缩下，y轴截距（x = 0处）不变，因为f(0)与x的系数无关。

k > 1: compress towards the y-axis | 向y轴压缩
0 < k < 1: stretch away from the y-axis | 远离y轴拉伸
y-coordinates unchanged | y坐标不变
y-intercept invariant | y轴截距不变

3. Distinguishing Vertical and Horizontal Stretches | 区分垂直伸缩与水平伸缩

Students often confuse the two types of stretches. The simplest way to distinguish them is to ask: what is being multiplied?

学生经常混淆这两种伸缩。最简单的区分方法是问：被乘的是什么？

If the output f(x) itself is multiplied by a constant, it is a vertical stretch affecting y-coordinates. If the input x is multiplied by a constant, it is a horizontal stretch affecting x-coordinates.

如果被乘的是函数值f(x)本身，则为垂直伸缩，影响y坐标。如果被乘的是自变量x，则为水平伸缩，影响x坐标。

Transformation \| 变换	Effect \| 效果	Affects \| 影响
y = k·f(x)	Vertical stretch/compress \| 垂直拉伸/压缩	y-coordinates \| y坐标
y = f(kx)	Horizontal stretch/compress \| 水平拉伸/压缩	x-coordinates \| x坐标

Another helpful memory aid: vertical stretches act “outside” the function parentheses, while horizontal stretches act “inside” the function parentheses.

另一个有用的记忆技巧：垂直伸缩作用在函数括号”外部”，而水平伸缩作用在函数括号”内部”。

4. Effect on Key Features | 对函数关键特征的影响

A vertical stretch affects:
– Range: the range of y = kf(x) is multiplied by k
– Maxima and minima: the maximum and minimum values are scaled by k
– Amplitude: for sinusoidal functions, the amplitude becomes k times the original

垂直伸缩影响：
– 值域：y = kf(x)的值域乘以k
– 最大值和最小值：最大值与最小值按k缩放
– 振幅：对正弦型函数，振幅变为原来的k倍

A horizontal stretch affects:
– Domain: the domain of y = f(kx) is compressed by factor 1/k
– Period: for sinusoidal functions, the period becomes 2π/k
– Frequency: the frequency is multiplied by k

水平伸缩影响：
– 定义域：y = f(kx)的定义域压缩至原来的1/k
– 周期：对正弦型函数，周期变为2π/k
– 频率：频率乘以k

For y = sin(kx): Period = 2π/k | 对y = sin(kx)：周期 = 2π/k

For example, y = sin(3x) has period 2π/3, which means it completes one full cycle three times faster than y = sin(x).

例如，y = sin(3x)的周期为2π/3，意味着它完成一个完整周期的速度是y = sin(x)的三倍。

5. Combined Stretches | 伸缩变换的组合

When vertical and horizontal stretches are applied simultaneously, the transformation takes the form:

当垂直伸缩和水平伸缩同时施加时，变换形式为：

y = k·f(mx)

Here k controls the vertical stretch and m controls the horizontal stretch independently. For example, y = 3f(2x) means each point (a, b) maps to (a/2, 3b).

其中k控制垂直伸缩，m控制水平伸缩，二者相互独立。例如，y = 3f(2x)意味着每个点(a, b)映射到(a/2, 3b)。

The order of applying stretches matters only when adding translations. Stretches of the same type commute: a vertical stretch of 2 followed by a vertical stretch of 3 is equivalent to a single vertical stretch of 6. However, when a horizontal translation is involved, the order becomes significant.

在涉及平移时，伸缩的顺序才变得重要。同类型伸缩可交换：先垂直拉伸2倍再拉伸3倍，等价于一次垂直拉伸6倍。然而，当涉及水平平移时，顺序就变得关键。

6. Transformations: Stretch vs. Translation Order | 伸缩与平移的先后顺序

Consider the function y = f(2x – 4). Should we stretch first or translate first?

考虑函数y = f(2x – 4)。我们应该先伸缩还是先平移？

Method 1: Factor out the coefficient of x first.

方法1：先提取x的系数。

y = f(2x – 4) = f(2(x – 2))

This reveals the correct order: first compress horizontally by a factor of 1/2, then translate 2 units to the right. This is because the expression inside the parentheses is 2(x – 2): the factor 2 compresses the graph, and the term (x – 2) shifts it right by 2.

这揭示了正确的顺序：先将图像水平压缩至原长的1/2，然后向右平移2个单位。因为括号内表达式为2(x – 2)：因子2压缩图像，而(x – 2)将其向右平移2个单位。

Method 2 (incorrect shortcut): Translating first and then stretching produces y = f(2x – 4) if you translate right by 4 then compress by 1/2? Let us verify: translate f(x) right by 4 gives f(x – 4). Compress by 1/2 gives f(2x – 4). Wait, this is also correct!

方法2（另一种正确路径）：先平移后伸缩：将f(x)向右平移4个单位得到f(x – 4)，再压缩1/2得到f(2x – 4)。等等，这也是正确的！

Both methods are valid. The key is consistency: either you factor first and apply stretch then translation, or you translate by the full amount first and then stretch. The horizontal translation amount depends on the order chosen. Always factor the coefficient of x to avoid confusion in the first method; for the second, remember the translation value is the one before stretching.

两种方法都有效。关键是保持一致：要么先提取公因子，先伸缩后平移；要么先按完整量平移，然后再伸缩。水平平移量取决于所选顺序。在第一种方法中，务必提取x的系数以避免混淆；第二种方法中，记住平移量是拉伸之前的值。

7. Matrix Representation of Stretches | 伸缩变换的矩阵表示

For students studying Mathematics: Analysis and Approaches HL, stretches can be elegantly represented using matrix transformations on the coordinate plane.

对于学习分析与方法HL的学生，伸缩变换可以用平面坐标上的矩阵变换优雅地表示。

A vertical stretch by factor k and a horizontal stretch by factor 1/m correspond to the diagonal matrix:

垂直拉伸k倍且水平拉伸1/m倍对应的对角矩阵为：

[ 1/m 0 ]
[ 0 k ]

The transformation maps (x, y) to (x’, y’) via:

该变换将(x, y)映射到(x’, y’)：

x’ = (1/m)·x, y’ = k·y

For example, y = 2f(3x) corresponds to the matrix [[1/3, 0], [0, 2]]. Note that the horizontal factor and vertical factor appear as the diagonal entries in order.

例如，y = 2f(3x)对应的矩阵为[[1/3, 0], [0, 2]]。注意水平因子和垂直因子按顺序出现在对角线上。

This matrix approach is particularly useful when combining stretches with rotations or reflections, and appears in the transformations topic of the HL syllabus.

这种矩阵方法在将伸缩与旋转、反射组合时特别有用，在HL课程大纲的变换专题中会涉及。

8. Worked Example: Identifying Stretch Factors | 例题：识别伸缩因子

Problem: The graph of y = f(x) passes through the point (4, 6). The transformed graph is y = 3f(2x). Find the coordinates of the corresponding point on the transformed graph.

题目：函数y = f(x)的图像经过点(4, 6)。变换后的图像为y = 3f(2x)。求变换后图像上对应点的坐标。

Solution:

解答：

Step 1: The input x is multiplied by 2, so the x-coordinate is divided by 2: 4 ÷ 2 = 2.

第一步：自变量x乘以2，因此x坐标除以2：4 ÷ 2 = 2。

Step 2: The output f(x) is multiplied by 3, so the y-coordinate is multiplied by 3: 6 × 3 = 18.

第二步：函数值f(x)乘以3，因此y坐标乘以3：6 × 3 = 18。

Step 3: The corresponding point is (2, 18).

第三步：对应点为(2, 18)。

(4, 6) → (2, 18)

Common mistake: students sometimes multiply x by 2 instead of dividing. Remember: y = f(2x) compresses the graph, so x-values must be reduced.

常见错误：学生有时会将x乘以2而不是除以2。记住：y = f(2x)压缩图像，因此x值必须减小。

9. Worked Example: Writing the Transformed Function | 例题：写出变换后的函数

Problem: The function f(x) = √x is transformed by a horizontal stretch of factor 3 followed by a vertical compression of factor 1/2. Write down the equation of the transformed function.

题目：函数f(x) = √x 先经过水平拉伸3倍，再经过垂直压缩1/2。写出变换后函数的方程。

Solution:

解答：

Step 1: Horizontal stretch of factor 3 means the x-coordinate is multiplied by 3. In function notation, this is achieved by dividing x by 3: f(x/3).

第一步：水平拉伸3倍意味着x坐标乘以3。在函数表达式中，这通过将x除以3来实现：f(x/3)。

f(x/3) = √(x/3)

Step 2: Vertical compression of factor 1/2 means multiplying the output by 1/2:

第二步：垂直压缩1/2意味着将函数值乘以1/2：

g(x) = ½·√(x/3)

Therefore, the transformed function is g(x) = ½·√(x/3) = √(x/3) / 2.

因此，变换后的函数为g(x) = ½·√(x/3) = √(x/3) / 2。

Note: a horizontal stretch of factor 3 corresponds to dividing the input by 3, i.e., replacing x with x/3. This is a common point of confusion.

注意：水平拉伸3倍对应于除以3，即将x替换为x/3。这是常见的混淆点。

10. Exam Tips and Common Pitfalls | 考试技巧与常见陷阱

Tip 1: Always ask yourself whether the multiplier is inside or outside the function parentheses. Outside = vertical; inside = horizontal.

技巧1：始终问自己乘数在函数括号内还是括号外。括号外 = 垂直；括号内 = 水平。

Tip 2: For horizontal stretches, the factor is reciprocal. y = f(3x) is a compression by 1/3, not a stretch by 3.

技巧2：对于水平伸缩，因子取倒数。y = f(3x)是压缩至1/3，而不是拉伸3倍。

Tip 3: When translations and stretches are combined, always factor the coefficient of x first to determine the correct order.

技巧3：当平移与伸缩组合时，务必先提取x的系数以确定正确的变换顺序。

Pitfall 1: Applying the stretch factor to the wrong coordinate. A vertical stretch changes y, never x.

陷阱1：将伸缩因子应用到错误的坐标上。垂直伸缩改变y，绝不改变x。

Pitfall 2: Forgetting that y = f(2x – 4) must be rewritten as f(2(x – 2)) before identifying the translation.

陷阱2：忘记y = f(2x – 4)必须改写为f(2(x – 2))之后才能确定平移量。

Pitfall 3: Regarding y = f(kx) with 0 < k < 1, students often think it is a stretch. In fact, it is a horizontal stretch by factor 1/k > 1, so yes, it is a stretch — but less obviously, y = f(2x) is a compression. Always compute the reciprocal for the actual stretch factor.

陷阱3：对于y = f(kx)当0 < k < 1时，学生常认为是压缩。实际上，这是拉伸因子为1/k > 1的水平拉伸。不太直观的是，y = f(2x)是压缩。计算实际拉伸因子时务必取倒数。

11. Summary: The Stretch Checklists | 总结：伸缩变换检查清单

Vertical stretch y = kf(x):

垂直伸缩 y = kf(x)：

Multiply y-coordinate by k | y坐标乘以k
x-coordinate unchanged | x坐标不变
k > 1: stretch upward | k > 1：向上拉伸
0 < k < 1: compress downward | 0 < k < 1：向下压缩
Amplitude (for trig functions) multiplied by k | 振幅（三角函数）乘以k

Horizontal stretch y = f(kx):

水平伸缩 y = f(kx)：

Divide x-coordinate by k | x坐标除以k
y-coordinate unchanged | y坐标不变
k > 1: compress horizontally | k > 1：水平压缩
0 < k < 1: stretch horizontally | 0 < k < 1：水平拉伸
Period (for trig functions) divided by k | 周期（三角函数）除以k

Mastering these patterns will help you quickly and accurately solve transformation problems in the IB examinations. Practise with a variety of functions — trigonometric, exponential, quadratic — to internalise the rules.

掌握这些规律将帮助你在IB考试中快速准确地解决变换类问题。请用各种函数——三角函数、指数函数、二次函数——多加练习，将这些规则内化于心。

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Transformation \| 变换	Effect \| 效果	Affects \| 影响
y = k·f(x)	Vertical stretch/compress \| 垂直拉伸/压缩	y-coordinates \| y坐标
y = f(kx)	Horizontal stretch/compress \| 水平拉伸/压缩	x-coordinates \| x坐标

IB Mathematics: Stretches of Function Graphs | IB数学：函数图像的伸缩变换