📚 Stretching Graphs | 图形拉伸
Graph stretching is a transformation that scales a function's graph either vertically or horizontally. In Edexcel A-Level Mathematics, you need to understand how multiplying the function or its input affects the shape, key points, and asymptotes of the graph. This topic builds on basic function notation and is often tested alongside translations and reflections.
图形拉伸是一种通过垂直或水平缩放来改变函数图像形状的变换。在Edexcel A-Level数学中，你需要理解对函数本身或其输入进行乘法运算如何影响图像的形状、关键点和渐近线。此主题建立在基本函数记号的基础上，常与平移和反射一起考查。
1. Introduction to Graph Stretches | 图形拉伸简介
given by y = a f(x), and horizontal stretch (or compression) given by y = f(bx). The factors a and b determine the scale and direction of the stretch. When |a| > 1 or |b| > 1, the graph is stretched away from the axis; when 0 < |a| < 1 or 0 < |b| < 1, it is compressed towards the axis.
拉伸变换改变了图形上各点到坐标轴的距离。主要分为两类：由 y = a f(x) 给出的垂直拉伸（或压缩），以及由 y = f(bx) 给出的水平拉伸（或压缩）。因子 a 和 b 决定了缩放的程度和方向。当 |a| > 1 或 |b| > 1 时，图形沿轴的方向被拉伸远离；当 0 < |a| < 1 或 0 < |b| < 1 时，图形被压缩靠近坐标轴。
2. Vertical Stretch: y = a f(x) with a > 1 | 垂直拉伸：a > 1
le x-intercepts remain unchanged because f(x) = 0 gives a × 0 = 0. The transformation can be described by the mapping (x, y) → (x, a y).
若 a > 1，原图像的每个 y 坐标都乘以 a，产生远离 x 轴的垂直拉伸。图像变高，而 x 截距保持不变，因为当 f(x) = 0 时 a × 0 = 0。该变换可用映射 (x, y) → (x, a y) 描述。
actor 2. The vertex stays at (0,0), but points like (1,1) become (1,2), making the parabola narrower from the y-axis perspective.
例如，若 f(x) = x²，则 y = 2 f(x) = 2x² 产生因子为 2 的垂直拉伸。顶点保持在 (0,0)，但像 (1,1) 这样的点变成 (1,2)，使抛物线从 y 轴方向看更窄。
3. Vertical Compression: 0 < a < 1 | 垂直压缩：0 < a < 1
Again, x-intercepts are invariant. The mapping is still (x, y) → (x, a y), but with a smaller multiplier.
当 0 < a < 1 时，图像被垂直压缩靠近 x 轴。y 值按小于 1 的因子缩放，使图像显得更扁平。同样，x 截距保持不变。映射仍为 (x, y) → (x, a y)，但乘数更小。
0.5 instead of 1. The period remains 2π, but the maximum and minimum values are now ±0.5.
Published by TutorHao | A-Level 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