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指数函数伸缩变换：观察父函数的图像与方程变化

指数函数伸缩变换：观察父函数的图像与方程变化
几何画板及绘图内容加载中，请稍等……
2023-10-18 21:14
函数变换几何作图滑块动画函数图像
本范例通过对指数函数进行“伸缩/拉伸（dilation）”变换，让学习者观察从父函数到新函数时图像与方程的对应关系。核心数学概念包括：指数函数的父函数特征、底数（base）与伸缩因子（stretch）的作用、函数变换在图像上的几何效果，以及用函数符号/代数表达来刻画变换。作图与交互要点是利用 GeoGebra 的命令与对象构建可控参数：在构造摘要中使用了 Mirror（镜像/对称辅助构造）、Sequence（生成序列点）、Point（点）、Segment（线段）、Polygon（多边形形状辅助）、circle（圆形元素）、Shuffle（对象随机化或变换）、Element（元素引用）、VerticalText（竖排文本）、以及多类文本显示对象（如 textQuestion、textBase、textFun、textNotDef、labelX、labelY）。同时结合滑块与参数对象（如 k、h、A、B、C、f、par、showParent、onboarding、pulsing 等）实现对伸缩与父函数对照的动态展示，配合网格对象 bigGrid 提供读图参照。适合 9–12 年级的函数与变换单元，用于课堂探究、作业巩固或交互式复习：如识别变换对指数图像增长/衰减与“拉伸方向/幅度”的影响，并用函数记号写出变换后的表达式。

GeoGebra 命令及参数

  1. Mirror(P1, yAxis) → P2
  2. Mirror(P2, xAxis) → P3
  3. Sequence(Sequence[(j, k), j, -10, 10, 1], k, -10, 10, 1) → bigGrid, {{(-10, -10), (-9, -10), (-8, -10), (-7, -10), (-6, -10), (-5, -10), (-4, -10), (-3, -10), (-2, -10), (-1, -10), (0, -10), (1, -10), (2, -10), (3, -10), (4, -10), (5, -10), (6, -10), (7, -10), (8, -10), (9, -10), (10, -10)}, {(-10, -9), (-9, -9), (-8, -9), (-7, -9), (-6, -9), (-5, -9), (-4, -9), (-3, -9), (-2, -9), (-1, -9), (0, -9), (1, -9), (2, -9), (3, -9), (4, -9), (5, -9), (6, -9), (7, -9), (8, -9), (9, -9), (10, -9)}, {(-10, -8), (-9, -8), (-8, -8), (-7, -8), (-6, -8), (-5, -8), (-4, -8), (-3, -8), (-2, -8), (-1, -8), (0, -8), (1, -8), (2, -8), (3, -8), (4, -8), (5, -8), (6, -8), (7, -8), (8, -8), (9, -8), (10, -8)}, {(-10, -7), (-9, -7), (-8, -7), (-7, -7), (-6, -7), (-5, -7), (-4, -7), (-3, -7), (-2, -7), (-1, -7), (0, -7), (1, -7), (2, -7), (3, -7), (4, -7), (5, -7), (6, -7), (7, -7), (8, -7), (9, -7), (10, -7)}, {(-10, -6), (-9, -6), (-8, -6), (-7, -6), (-6, -6), (-5, -6), (-4, -6), (-3, -6), (-2, -6), (-1, -6), (0, -6), (1, -6), (2, -6), (3, -6), (4, -6), (5, -6), (6, -6), (7, -6), (8, -6), (9, -6), (10, -6)}, {(-10, -5), (-9, -5), (-8, -5), (-7, -5), (-6, -5), (-5, -5), (-4, -5), (-3, -5), (-2, -5), (-1, -5), (0, -5), (1, -5), (2, -5), (3, -5), (4, -5), (5, -5), (6, -5), (7, -5), (8, -5), (9, -5), (10, -5)}, {(-10, -4), (-9, -4), (-8, -4), (-7, -4), (-6, -4), (-5, -4), (-4, -4), (-3, -4), (-2, -4), (-1, -4), (0, -4), (1, -4), (2, -4), (3, -4), (4, -4), (5, -4), (6, -4), (7, -4), (8, -4), (9, -4), (10, -4)}, {(-10, -3), (-9, -3), (-8, -3), (-7, -3), (-6, -3), (-5, -3), (-4, -3), (-3, -3), (-2, -3), (-1, -3), (0, -3), (1, -3), (2, -3), (3, -3), (4, -3), (5, -3), (6, -3), (7, -3), (8, -3), (9, -3), (10, -3)}, {(-10, -2), (-9, -2), (-8, -2), (-7, -2), (-6, -2), (-5, -2), (-4, -2), (-3, -2), (-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (3, -2), (4, -2), (5, -2), (6, -2), (7, -2), (8, -2), (9, -2), (10, -2)}, {(-10, -1), (-9, -1), (-8, -1), (-7, -1), (-6, -1), (-5, -1), (-4, -1), (-3, -1), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (3, -1), (4, -1), (5, -1), (6, -1), (7, -1), (8, -1), (9, -1), (10, -1)}, {(-10, 0), (-9, 0), (-8, 0), (-7, 0), (-6, 0), (-5, 0), (-4, 0), (-3, 0), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (6, 0), (7, 0), (8, 0), (9, 0), (10, 0)}, {(-10, 1), (-9, 1), (-8, 1), (-7, 1), (-6, 1), (-5, 1), (-4, 1), (-3, 1), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (7, 1), (8, 1), (9, 1), (10, 1)}, {(-10, 2), (-9, 2), (-8, 2), (-7, 2), (-6, 2), (-5, 2), (-4, 2), (-3, 2), (-2, 2), (-1, 2), (0, 2), (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (7, 2), (8, 2), (9, 2), (10, 2)}, {(-10, 3), (-9, 3), (-8, 3), (-7, 3), (-6, 3), (-5, 3), (-4, 3), (-3, 3), (-2, 3), (-1, 3), (0, 3), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (7, 3), (8, 3), (9, 3), (10, 3)}, {(-10, 4), (-9, 4), (-8, 4), (-7, 4), (-6, 4), (-5, 4), (-4, 4), (-3, 4), (-2, 4), (-1, 4), (0, 4), (1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4), (7, 4), (8, 4), (9, 4), (10, 4)}, {(-10, 5), (-9, 5), (-8, 5), (-7, 5), (-6, 5), (-5, 5), (-4, 5), (-3, 5), (-2, 5), (-1, 5), (0, 5), (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (7, 5), (8, 5), (9, 5), (10, 5)}, {(-10, 6), (-9, 6), (-8, 6), (-7, 6), (-6, 6), (-5, 6), (-4, 6), (-3, 6), (-2, 6), (-1, 6), (0, 6), (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (7, 6), (8, 6), (9, 6), (10, 6)}, {(-10, 7), (-9, 7), (-8, 7), (-7, 7), (-6, 7), (-5, 7), (-4, 7), (-3, 7), (-2, 7), (-1, 7), (0, 7), (1, 7), (2, 7), (3, 7), (4, 7), (5, 7), (6, 7), (7, 7), (8, 7), (9, 7), (10, 7)}, {(-10, 8), (-9, 8), (-8, 8), (-7, 8), (-6, 8), (-5, 8), (-4, 8), (-3, 8), (-2, 8), (-1, 8), (0, 8), (1, 8), (2, 8), (3, 8), (4, 8), (5, 8), (6, 8), (7, 8), (8, 8), (9, 8), (10, 8)}, {(-10, 9), (-9, 9), (-8, 9), (-7, 9), (-6, 9), (-5, 9), (-4, 9), (-3, 9), (-2, 9), (-1, 9), (0, 9), (1, 9), (2, 9), (3, 9), (4, 9), (5, 9), (6, 9), (7, 9), (8, 9), (9, 9), (10, 9)}, {(-10, 10), (-9, 10), (-8, 10), (-7, 10), (-6, 10), (-5, 10), (-4, 10), (-3, 10), (-2, 10), (-1, 10), (0, 10), (1, 10), (2, 10), (3, 10), (4, 10), (5, 10), (6, 10), (7, 10), (8, 10), (9, 10), (10, 10)}}
  4. Point(bigGrid) → A
  5. Mirror(P3, yAxis) → P4
  6. Polygon(P1, P2, P3, P4) → peep, p1, p1', p1'', p1'''
  7. Shuffle(Sequence[t, t, -18, 18]) → allA, {9, 8, -7, -13, -4, -2, 18, 6, -18, 17, 0, -9, -16, -11, -1, 14, 10, 3, -5, 1, 16, 11, 4, -8, -10, 5, -15, 13, -12, 12, -3, -14, -6, -17, 2, 15, 7}
  8. Element(allA, Mod[countQuestion, Length[allA]] + 1) → valA
  9. circlePixels(A, 7 + (7 * pulsing), Corner[1], Corner[3], Corner[5]) → pulsingObject
  10. VerticalText("\textsf{Stretch}") → labelStretch
  11. Sequence((0, k), k, -10, 10, 1) → yTicks
  12. Sequence((k, 0), k, -10, 10, 1) → xTicks
  13. Segment((x(P1) + 0.05, 0), ((-x(P1)) - 0.05, 0)) → fakeX
  14. Segment((0, y(P1) + 0.05), (0, (-y(P1)) - 0.05)) → fakeY

AI 绘图提示语参考

绘制一个包含网格、点、多边形、圆、文本标签、坐标轴刻度线和虚线的图形。首先创建一个从-10到10的网格，标记一个点A在网格上。然后通过镜像操作生成四个点，构成一个多边形p1。添加一个圆和一个文本“Stretch”标注变换关系。绘制坐标轴刻度线xTicks和yTicks，以及虚线fakeX和fakeY表示对称轴。最后将所有元素组合展示，体现变换对图形的影响。
以上提示语为根据原始绘图命令整理而来，仅供参考。