📚 Stretching Graphs | 图形的拉伸变换
Understanding graph stretches is essential for mastering function transformations in IGCSE Edexcel Mathematics. A stretch changes the shape of a graph by pulling it away from or squeezing it towards one of the axes, while the overall pattern remains recognisable. This article explores vertical and horizontal stretches, how to interpret them from equations, and how to sketch transformed graphs accurately.
理解图形拉伸是掌握IGCSE Edexcel数学中函数变换的关键。拉伸通过将图形远离或挤向坐标轴来改变其形状，但整体形态仍然可辨。本文将探讨垂直和水平拉伸，如何从方程中识别它们，以及如何准确地绘制变换后的图形。
1. Introduction to Graph Transformations | 图形变换简介
cted to recognise and sketch stretched graphs given in the forms y = a f(x) and y = f(bx).
图形变换包括平移、反射和拉伸。平移在不改变形状的情况下移动图形，反射将图形翻转到轴的另一侧，而拉伸则通过缩放与x轴或y轴的距离来实际改变形状。在IGCSE中，你需要能够识别并绘制给定形式为y = a f(x)和y = f(bx)的拉伸图形。
Stretches can be vertical or horizontal, and they correspond to multiplying the function output or the input variable. A solid understanding of these two types forms the basis for tackling more complex transformation problems.
拉伸可以是垂直的或水平的，它们分别对应乘以函数输出或输入变量。牢固掌握这两种类型是解决更复杂变换问题的基础。
2. Vertical Stretch: y = a f(x) | 垂直拉伸：y = a f(x)
A vertical stretch multiplies all y-coordinates of the graph by a scale factor a. The equation takes the form y = a f(x), where a is a positive constant. The x-coordinates remain unchanged, so every point (x, y) on the original graph moves to (x, a y).
垂直拉伸将图形上所有点的y坐标乘以比例因子a。方程形式为y = a f(x)，其中a是正常数。x坐标保持不变，因此原图上的每一个点(x, y)移动到(x, a y)。
If the original graph crosses the x-axis, those points stay fixed because their y-coordinate is zero. The overall effect is that the graph is pulled away from the x-axis for a > 1, or compressed towards it for 0 < a < 1.
如果原图与x轴相交，这些点保持不变，因为它们的y坐标为零。总体效果是，当a > 1时图形被拉离x轴，当0 < a < 1时被压向x轴。
3. Effect of a > 1 on Vertical Stretch | a > 1 时的垂直拉伸效果
la vertically by a factor of 3. The vertex at (0,0) stays the same, but a point like (1,1) becomes (1,3).
当a > 1时，图形变得更陡更高。例如，若f(x) = x²，则y = 3 x² 将基本抛物线垂直拉伸3倍。位于(0,0)的顶点保持不变，但像(1,1)这样的点变成(1,3)。
y = 3 f(x) or y = 3 x²
This transformation is often described as 'stretch parallel to the y-axis, scale factor 3'. In exams, you should clearly specify the direction and the scale factor when describing the stretch.
这种变换通常被描述为“平行于y轴拉伸，比例因子3”。在考试中，描述拉伸时应明确指明方向和比例因子。
4. Effect of 0 < a < 1 on Vertical Compression | 0 < a < 1 时的垂直压缩效果
alves every y-coordinate. The graph becomes flatter and closer to the x-axis. Using f(x) = |x|, the point (2, 2) would move to (2, 1).
当0 < a < 1时，图形被垂直压扁。例如，y = ½; f(x) 将每个y坐标减半。图形变得更平，更靠近x轴。以f(x) = |x|为例，点(2, 2)会移动到(2, 1)。
This is still a stretch, but the scale factor is between 0 and 1, sometimes called a compression. In many mark schemes, 'stretch scale factor ½ parallel to the y-axis' is accepted.
这仍然是一种拉伸，但比例因子介于0和1之间，有时称为压缩。在许多评分标准中，“平行于y轴拉伸，比例因子½”是可以接受的。
5. Horizontal Stretch: y = f(bx) | 水平拉伸：y = f(bx)
A horizontal stretch is given by y = f(bx) rdinates inversely: each point (x, y) on y = f(x) maps to (x/b, y) on the stretched graph.
水平拉伸由y = f(bx)给出，其中b是正常数。这里输入x在代入函数之前乘以b。这会反向改变x坐标：y = f(x)上的每个点(x, y)映射到拉伸图上的(x/b, y)。
Important: The effect on the graph is opposite to the value of b. If b > 1, the graph is compressed towards the y-axis. If 0 < b < 1, the graph is stretched away from the y-axis. Many students find this counter-intuitive, so careful practice is needed.
重要提示：图形上产生的效果与b的取值相反。若b > 1，图形被压向y轴。若0 < b < 1，图形被拉离y轴。很多学生觉得这有悖直觉，因此需要仔细练习。
6. Effect of b > 1 on Horizontal Compression | b > 1 时的水平压缩效果
xample, if f(x) = sin x°, then y = sin 2x° completes a full cycle in 180° instead of 360°.
对于y = f(2x)，图形在水平方向上被压缩为原来的½，因子为2。原先在(2, y)的点变成(1, y)。图形变窄。例如，若f(x) = sin x°，则y = sin 2x° 在180°内完成一个完整周期，而不是360°。
y = f(3x) → horizontal compression, factor ⅓
When describing this transformation, say 'stretch parallel to the x-axis, scale factor ½' (since x-coordinates are multiplied by 1/b). The key is to identify the horizontal stretch factor as 1/b.
描述这种变换时，要说“平行于x轴拉伸，比例因子½”（因为x坐标乘以1/b）。关键是要将水平拉伸因子识别为1/b。
7. Effect of 0 < b < 1 on Horizontal Stretch | 0 < b < 1 时的水平拉伸效果
ed by 2 before the function, so the graph is stretched horizontally by a factor of 2. A point (4, y) on the original becomes (8, y), effectively moving it further from the y-axis.
当b介于0和1之间时，图形变宽。y = f(½ x) 意味着在函数作用前x被除以2，所以图形在水平方向拉伸为原来的2倍。原图上的点(4, y)变成(8, y)，有效地将它移离y轴更远。
ac12; x° stretches the standard sine wave so that the period becomes 720°. This can be tricky to sketch, but remembering to divide x-coordinates by b makes it manageable.
对于三角函数图，y = sin ½; x° 将标准正弦波拉伸，使周期变为720°。这绘制起来可能比较棘手，但记住用b除x坐标便使其易于处理。
8. Combining Stretches with Other Transformations | 拉伸变换与其他变换的组合
a horizontal translation left by 1, followed by a vertical stretch of factor 2. The order matters: the stretch is applied to the translated function.
图形问题常常综合了拉伸和平移。例如，y = 2 f(x + 1) 先水平向左平移1个单位，再垂直拉伸2倍。顺序很重要：拉伸作用于平移后的函数。
(3(x - ⅔)). This reveals a horizontal stretch of factor ⅓, followed by a translation right by ⅔. Never apply transformations haphazardly; always follow the standard order.
如果拉伸位于括号内，如y = f(3x - 2)，应先因式分解：y = f(3(x - ⅔))。这揭示了一个水平拉伸因子⅓，然后向右平移⅔。绝不能随意应用变换；必须遵循标准顺序。
- For y = a f(bx + c) + d, rewrite as y = a f(b(x + c/b)) + d.
- First handle the inner horizontal stretch (1/b), then the horizontal translation (-c/b), then the vertical stretch (a), and finally the vertical translation (d).
- 对于 y = a f(bx + c) + d，重写为 y = a f(b(x + c/b)) + d。
- 首先处理内部水平拉伸(1/b)，接着水平平移(-c/b)，然后是垂直拉伸(a)，最后垂直平移(d)。
9. Identifying Stretches from Equations | 从方程识别拉伸变换
(2x)³ is the cube of 2x, which is f(2x), indicating a horizontal compression by factor ½.
给定一个方程，要识别出乘法因子是影响y输出还是x输入。与母函数进行比较。例如，y = ¼; x³ 是对f(x) = x³ 应用了垂直拉伸因子¼。而y = (2x)³ 是2x的立方，即f(2x)，表示水平压缩因子½。
ntal stretch (compression) by factor ⅓ because it can be written as f(3x) with f(x) = 1/x.
仔细区分y = 3/x 与 y = 1/(3x)。前者是对y = 1/x的垂直拉伸3倍，而后者可写成f(3x)，f(x) = 1/x，是水平压缩因子⅓。
10. Applying Stretches to Specific Functions | 对特定函数的拉伸应用
atic functions: y = a (x - p)² + q. The a value gives a vertical stretch with scale factor |a| a is negative, there is also a reflection in the x-axis, but the stretch magnitude remains |a|. The vertex (p, q) stays fixed under pure vertical stretch if translated form is used.
二次函数：y = a (x - p)² + q。a值代表垂直拉伸，比例因子为|a|。若a为负，还涉及关于x轴的反射，但拉伸倍数仍为|a|。当使用顶点式时，纯垂直拉伸下顶点(p, q)保持不变。
Trigonometric functions: y = a sin bx° or y = a cos bx°. The amplitude is |a| stretch of factor 1/b. These two stretches occur independently and can be combined freely.
三角函数：y = a sin bx° 或 y = a cos bx°。振幅为|a|，表示垂直拉伸。周期为360°/b，反映了水平拉伸因子1/b。这两种拉伸独立发生且可自由组合。
| Original | sin x° | cos x° | tan x° |
| Vertical stretch by a | a sin x° | a cos x° | a tan x° |
| Horizontal stretch by 1/b | sin bx° | cos bx° | tan bx° |
11. Sketching Stretched Graphs | 绘制拉伸图形
f every key point (intercepts, turning points, asymptotes) by a, keeping x the same. Plot the new points and draw a smooth curve.
先用铅笔轻轻画出基础图形。然后确定拉伸类型。对于垂直拉伸y = a f(x)，将每个关键点（截距、转折点、渐近线）的y坐标乘以a，x坐标保持不变。描出新点并画出光滑曲线。
vide the x-coordinates of key points by b and keep y-coordinates unchanged. With trigonometric graphs, mark the new intercepts and maxima/minima based on the altered period and amplitude. Always label the axes with the new scale.
对于水平拉伸y = f(bx)，将关键点的x坐标除以b，y坐标保持不变。对于三角函数图，根据变化后的周期和振幅标注新的截距和最大/最小值。务必在坐标轴上标出新的刻度。
Double-check that invariant points (like those on the axis perpendicular to the stretch direction) remain correctly placed. If a translation is also present, apply the stretch before the translation as per the correct order.
再次检查不变量点（如垂直于拉伸方向的轴上的点）是否正确就位。若同时存在平移，应按照正确顺序先拉伸后平移。
12. Common Mistakes and Tips | 常见错误与提示
A frequent error is confusing vertical and horizontal stretches. Remember: if the multiplier is outside the function, it affects y (vertical). If inside the argument, it affects x (horizontal). Think 'outside – y, inside – x'.
一个常见错误是混淆垂直和水平拉伸。记住：如果乘数在函数外部，它影响y（垂直）。如果在参数内部，它影响x（水平）。可以想成“外 – y，内 – x”。
scale factor 2 parallel to the x-axis would be incorrect; the correct factor is ½. Always state 1/b.
另一个错误是忘记了水平拉伸中的倒数效应。将y = f(2x)描述为平行于x轴拉伸因子2是不正确的；正确因子是½。始终说明为1/b。
then the translation. Practise with simple functions first, then build up complexity gradually. Use graph plotting software or your calculator to check sketches.
学生有时也会弄混复合变换的顺序。对于y = a f(bx + c) + d，先提取b，再进行平移。先用简单函数练习，然后逐步增加复杂性。使用绘图软件或计算器来检查草图。
Finally, when working with trigonometric graphs, do not forget that the period changes only with b, not with a. Keep amplitude and period calculations separate to avoid confusion.
最后，在处理三角函数图时，不要忘记周期只随b变化，而与a无关。将振幅和周期的计算分开，以避免混淆。
Published by TutorHao | IGCSE Edexcel 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