📚 Investigation 3 - Stretches | 探究3 - 拉伸变换
In this investigation, you will explore how stretching a graph vertically or horizontally transforms its shape, equation, and key features. Stretches are fundamental transformations in function theory, allowing us to scale graphs relative to the axes. By the end, you will be able to identify, apply, and combine stretches with confidence, and recognise how they alter the coordinates of every point on a curve.
在本探究中，你将探索垂直或水平拉伸图像如何改变其形状、方程和关键特征。拉伸是函数理论中的基本变换，可以使图像相对于坐标轴进行缩放。完成本探究后，你将能够自信地识别、应用和组合拉伸变换，并理解它们如何改变曲线上每一个点的坐标。
1. Setting the Scene | 场景设定
tercepts at x = 1 and x = 3, and its vertex lies at (2, −1). The graph is a smooth, upward‑ parabola. You should sketch it on graph paper or visualise it on a GDC before proceeding.
我们从一个简单但内涵丰富的母函数开始：f(x) = x² − 4x + 3。这个二次函数可以因式分解为 f(x) = (x − 1)(x − 3)，由此可知 x 轴截距位于 x = 1 和 x = 3，而顶点位于 (2, −1)。其图像是一条光滑的开口向上的抛物线。在继续之前，你应当在方格纸上画出草图，或在 GDC 上观察它。
In IB Mathematics, a stretch ertical stretches of the form y = a · f(x) and horizontal stretches of the form y = f(bx).
在 IB 数学中，拉伸被定义为图像只沿一个方向（平行于坐标轴）进行的放大（或缩小）。与平移不同，拉伸会改变图形的比例，而不仅仅是位置。我们将研究两类拉伸：形如 y = a · f(x) 的垂直拉伸，以及形如 y = f(bx) 的水平拉伸。
2. Vertical Stretch: y = a · f(x) | 垂直拉伸：y = a · f(x)
) = 2 f(x) = 2x² − 8x + 6. Multiplying the entire function by the factor 2 changes every y‑coordinate by the same scale factor, while x‑coordinates remain unchanged. The new parabola still crosses the x‑ ) = 0 then 2·0 = 0. However, the vertex has moved from (2, −1) to (2, −2). The graph looks “taller” and narrower.
考虑变换 g(x) = 2 f(x) = 2x² − 8x + 6。将整个函数乘以因数 2，会以相同的比例因子改变每一个 y 坐标，而 x 坐标保持不变。新的抛物线仍然在 x = 1 和 x = 3 处穿过 x 轴，因为若 f(x) = 0，则 2·0 = 0。然而，顶点已从 (2, −1) 移动到 (2, −2)。图像看起来“更高”且更窄。
In general, a vertical stretch with scale factor a (a > 0) is expressed as:
一般而言，比例因子为 a（a > 0）的垂直拉伸表示为：
y = a · f(x)
- a > 1: the graph is stretched away from the x‑axis (expansion).
- 0 < a < 1: the graph is compressed towards the x‑axis.
- a = 1: no change.
- If a is negative, there is also a reflection in the x‑axis combined with a stretch, but our focus here is on positive stretch factors.
- a > 1：图像远离 x 轴拉伸（扩张）。
- 0 < a < 1：图像向 x 轴压缩。
- a = 1：无变化。
- 若 a 为负数，则同时包含关于 x 轴的反射以及拉伸，但本探究的重点是正拉伸因子。
The mapping for any point (x, y) on the original graph becomes (x, a·y). This explains why horizontal intercepts are invariant under a pure vertical stretch.
原图像上任意一点 (x, y) 的映射变为 (x, a·y)。这就解释了为何在纯垂直拉伸下，水平截距保持不变。
3. Horizontal Stretch: y = f(bx) | 水平拉伸：y = f(bx)
retch to f(x) using h(x) = f(0.5x) = (0.5x)² − 4(0.5x) + 3 = 0.25x² − 2x + 3. Here every x‑ ffect is that the graph is stretched horizontally by a factor of 1/0.5 = 2 away from the y‑axis. The x‑intercepts are no longer at 1 and 3; instead they move to x = 2 and x = 6.
现在对 f(x) 施加水平拉伸：h(x) = f(0.5x) = (0.5x)² − 4(0.5x) + 3 = 0.25x² − 2x + 3。这里每一个 x 坐标在代入函数之前都先除以 0.5（即等价地乘以 2）。其效果是图像沿水平方向以因子 1/0.5 = 2 远离 y 轴拉伸。x 轴截距不再是 1 和 3，而是移动到了 x = 2 和 x = 6。
More generally, a horizontal stretch is described by:
更一般地，水平拉伸由以下形式描述：
y = f(bx), b > 0
- b > 1: the graph is compressed towards the y‑axis by a factor of 1/b.
- 0 < b < 1: the graph is stretched away from the y‑axis by a factor of 1/b.
- b > 1：图像向 y 轴压缩，压缩因子为 1/b。
- 0 < b < 1：图像远离 y 轴拉伸，拉伸因子为 1/b。
ches in the alternative form y = f(x / k), where k is the stretch factor parallel to the x‑ horizontal compression by factor 1/2. It is crucial to be comfortable with both notations.
许多 IB 题目会用另一种形式 y = f(x / k) 表示水平拉伸，其中 k 是平行于 x 轴的拉伸因子。例如，y = f(x / 2) 给出因子为 2 的水平拉伸，而 y = f(2x) 则给出因子为 1/2 的水平压缩。熟练运用这两种记号至关重要。
4. Point Mapping and Invariant Features | 点映射与不变特征
is the key to mastering stretches. For y = a · f(x), the transformation can be written as:
理解坐标映射是掌握拉伸的关键。对于 y = a · f(x)，变换可写为：
(x , y) → (x , a·y)
For y = f(bx), the mapping is:
而 y = f(bx) 的映射为：
(x , y) → (x/b , y)
This reveals that the y‑ is invariant under a horizontal stretch because x/b remains 0 when x = 0. Similarly, the x‑axis intercepts are invariant under a vertical stretch.
这揭示了 y 轴截距（x = 0）在水平拉伸下保持不变，因为当 x = 0 时，x/b 仍为 0。类似地，x 轴截距在垂直拉伸下保持不变。
f(0.5x), the original (1,0) maps to (2,0) and (3,0) maps to (6,0). The vertex (2, −1) becomes (4, −1) under the horizontal stretch – notice the y‑coordinate remains −1.
我们以 f(x) = x² − 4x + 3 进行验证。在垂直拉伸 g(x) = 2 f(x) 下，截距 (1,0) 和 (3,0) 保持不动。在水平拉伸 h(x) = f(0.5x) 下，原 (1,0) 映射为 (2,0)，(3,0) 映射为 (6,0)。顶点 (2, −1) 在水平拉伸下变为 (4, −1)——注意 y 坐标保持为 −1。
5. Stretch Factors and Equation Forms | 拉伸因子与方程形式
k: x is replaced by 3x, so to obtain the same output as the original at a given x, the input must be one‑third of the original. Graphically, points are pulled towards the y‑axis, and the horizontal stretch factor is 1/3.
常见的混淆点在于水平拉伸中的倒数关系。如果你看到 y = f(3x)，请思考：x 被替换成了 3x，因此要获得与原函数在某一给定 x 处相同的输出，输入必须变为原来的三分之一。从图形上看，点被拉向 y 轴，水平拉伸因子为 1/3。
Below is a comparison table to help memorise the link between equation and geometric effect:
下表可帮助记忆方程与几何效果之间联系：
| Equation (b>0) | Effect on graph | Stretch factor |
|---|---|---|
| y = f(2x) | Horizontal compression towards y‑axis | 1/2 |
| y = f(x/3) | Horizontal stretch away from y‑axis | 3 |
| y = 4 f(x) | Vertical stretch away from x‑axis | 4 |
| y = (1/3) f(x) | Vertical compression towards x‑axis | 1/3 |
the axis parallel to which the stretch occurs and the scale factor clearly. For instance, “a horizontal stretch with scale factor 2, parallel to the x‑axis” for y = f(x/2).
在 IB 考试中描述变换时，务必清楚地说明拉伸所平行的坐标轴以及比例因子。例如，对于 y = f(x/2)，应描述为“平行于 x 轴、比例因子为 2 的水平拉伸”。
6. Effect on Key Points and Shape | 对关键点与形状的影响
factors. Starting again from f(x) = x² − 4x + 3, we build a table of transformed vertices:
我们来分析在不同拉伸因子下，抛物线的顶点、截距以及整体宽度如何变化。仍然以 f(x) = x² − 4x + 3 为起点，我们构建一个变换顶点的表格：
| Transformation | New Equation | Vertex | x‑intercepts |
|---|---|---|---|
| Original f | x² − 4x + 3 | (2, −1) | 1, 3 |
| Vert. stretch ×2 | 2x² − 8x + 6 | (2, −2) | 1, 3 |
| Horiz. stretch ×2 (b=0.5) | 0.25x² − 2x + 3 | (4, −1) | 2, 6 |
| Horiz. compression ×1/2 (b=2) | 4x² − 8x + 3 | (1, −1) | 0.5, 1.5 |
Notice that a vertical stretch multiplies the y‑ keeping the axis of symmetry x = 2 unchanged. A horizontal stretch changes the vertex’s x‑coordinate and the distance between intercepts, but leaves the minimum y‑value untouched.
请注意，垂直拉伸会将顶点及所有点的 y 坐标相乘，而对称轴 x = 2 保持不变。水平拉伸会改变顶点的 x 坐标以及截距之间的距离，但最小 y 值不受影响。
= 2x² − 8x + 6, the leading coefficient is 2, while in h(x) = 0.25x² − 2x + 3, it is 0.25.
抛物线的形状也会发生变化。垂直拉伸因子 a > 1 使其更陡峭；水平拉伸因子 k > 1 使其更宽。这些变化可通过观察 x² 项系数看到：g(x) = 2x² − 8x + 6 的首项系数为 2，而 h(x) = 0.25x² − 2x + 3 的首项系数为 0.25。
7. Combining Vertical and Horizontal Stretches | 垂直与水平拉伸的组合
We can apply both a vertical and a horizontal stretch to the same function. Consider:
我们可以对同一个函数同时施加垂直和水平拉伸。考虑：
k(x) = 3 f(2x)
This means first replace x by 2x (horizontal compression by factor 1/2), then multiply the whole result by 3 (vertical stretch by factor 3). The order of these two stretches does not matter because they affect different coordinates independently.
这意味着先将 x 替换为 2x（因子为 1/2 的水平压缩），然后将整个结果乘以 3（因子为 3 的垂直拉伸）。这两种拉伸的顺序无关紧要，因为它们独立地影响不同的坐标。
Using our parent parabola, the combined transformation gives:
利用我们的母抛物线，组合变换给出：
k(x) = 3[ (2x)² − 4(2x) + 3 ] = 3(4x² − 8x + 3) = 12x² − 24x + 9
The vertex originally (2, −1) maps as follows: under horizontal compression (2x), x‑coordinate becomes 2/2 = 1; under vertical stretch ×3, y‑coordinate becomes 3·(−1) = −3. So the new vertex is (1, −3). The x‑intercepts become 0.5 and 1.5.
原顶点 (2, −1) 的映射如下：在水平压缩（2x）下，x 坐标变为 2/2 = 1；在垂直拉伸 ×3 下，y 坐标变为 3·(−1) = −3。所以新顶点为 (1, −3)。x 轴截距变为 0.5 和 1.5。
When giving a description of a combined transformation, it is standard practice to mention the horizontal stretch first and then the vertical stretch (though, as noted, order does not affect the final image as long as no translation is involved).
在描述组合变换时，惯例是先提水平拉伸，再提垂直拉伸（不过如前所述，只要不含平移，顺序并不影响最终图像）。
8. Stretches and Translations: Order Matters | 拉伸与平移：顺序至关重要
Once a translation enters the picture, the order of transformations becomes critical. Compare these two sequences on f(x):
一旦平移加入进来，变换的顺序就变得至关重要。比较以下两种对 f(x) 的变换序列：
A: Vertical stretch factor 2, then translate up by 3 units. This yields g(x) = 2 f(x) + 3.
序列 A：先垂直拉伸因子 2，再向上平移 3 个单位，得到 g(x) = 2 f(x) + 3。
factor 2. This yields h(x) = 2 ( f(x) + 3 ) = 2 f(x) + 6, a completely different function.
序列 B：先向上平移 3 个单位，再垂直拉伸因子 2，得到 h(x) = 2 ( f(x) + 3 ) = 2 f(x) + 6，这是一个完全不同的函数。
he argument of f reveals the true horizontal shift: f(2(x + 2)) represents a horizontal compression by factor 1/2 followed by a translation of 2 units to the left. Always rewrite expressions involving both a horizontal stretch and a horizontal translation in the form f( b(x − h) ) to identify the shift correctly.
同样的敏感性也出现在水平拉伸与水平平移的组合中。例如，y = f(2x + 4) 与 y = f(2(x + 2))。将 f 的自变量因式分解可揭示真实的水平位移：f(2(x + 2)) 表示先进行因子为 1/2 的水平压缩，再向左平移 2 个单位。务必将同时包含水平拉伸与水平平移的表达式写成 f( b(x − h) ) 的形式，以便正确识别位移。
9. Stretch Factors and Periodicity of Trigonometric Functions | 拉伸因子与三角函数的周期性
tan x. For the sine curve, a vertical stretch y = A sin x changes the amplitude from 1 to |A|, while a horizontal stretch y = sin(bx) changes the period from 2π to 2π/|b|.
拉伸对周期函数（如 sin x、cos x 和 tan x）有着格外鲜明的影响。对于正弦曲线，垂直拉伸 y = A sin x 会将振幅从 1 变为 |A|，而水平拉伸 y = sin(bx) 则将周期从 2π 变为 2π/|b|。
factor 2; reminding themselves of the mapping (x , y) → (x/2 , 3y) helps avoid this error.
例如，y = 3 sin(2x) 结合了因子 ×3 的垂直拉伸（振幅为 3）和因子为 1/2 的水平压缩（周期为 π）。学生看到因数 2 时，常误以为周期是 4π；提醒自己映射关系 (x , y) → (x/2 , 3y) 有助于避免这种错误。
hile a horizontal stretch y = tan(bx) changes the distance between asymptotes from π to π/|b| vestigating these effects graphically strengthens your overall understanding of stretches.
正切函数的反应则不同：垂直拉伸会改变分支的陡峭程度，但不改变垂直渐近线的位置；而水平拉伸 y = tan(bx) 将相邻渐近线间的距离从 π 变为 π/|b|。通过图像研究这些效应，能加深你对拉伸的整体理解。
10. Applying Stretches to Inverse and Composite Functions | 将拉伸应用于反函数与复合函数
functions and composite functions. If you have y = f⁻¹(x), what does y = f⁻¹(2x) represent? n the inverse corresponds to a vertical stretch on the original function and vice versa. This interplay is a rich topic for investigation.
拉伸同样适用于反函数和复合函数。若已知 y = f⁻¹(x)，那么 y = f⁻¹(2x) 表示什么？
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