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指数函数的平移变换：识别基底与平移量

指数函数的平移变换：识别基底与平移量
几何画板及绘图内容加载中，请稍等……
2023-10-18 17:17
函数变换滑块动画函数图像
该范例展示如何把“父函数”指数函数进行平移后，图像方程与形状随参数变化的对应关系。核心数学概念包括指数函数的母函数、底数对增长/衰减的影响、水平平移（由指数中的平移量决定）与竖直平移（由函数整体加减决定），并通过函数记号与图像联动来帮助学生区分变量含义。作图与交互要点：利用 Mirror 对象做对称/基准构型，借助 Sequence、Point、Polygon、Segment、circlePixels 及 circle 等建立坐标参考与图形元素；同时用 Element/子对象集合组织关键点与构件；用 Segment、Polygon 等形成可视化区域；通过多对象文本与标注（text、labelX、labelY、textBase、textNotDef）强化“底数”“平移方向”的解释；交互部分结合滑块与按钮（btnShow、btnReset）以及展示控制对象（showParent、onboarding、pulsing、pulsingObject）引导学生观察参数变化下的父函数与变换后函数关系。适合9-12年级用于函数变换识别、图像与代数形式的互译练习，也可作为课堂探究或自学材料的交互式演示。

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, {17, 5, 11, -6, 13, -18, 12, -12, 4, 15, 2, 10, -17, 1, -9, 0, 3, -16, -14, 16, -7, -8, -4, 18, -13, -11, 14, -2, 7, -1, 9, -3, 8, -15, -5, 6, -10}
  8. Element(allA, Mod[countQuestion, Length[allA]] + 1) → valA
  9. circlePixels(A, 7 + (7 * pulsing), Corner[1], Corner[3], Corner[5]) → pulsingObject
  10. Segment(AO1, AO2) → fakeAO
  11. Sequence((k, 0), k, -10, 10, 1) → xGrid
  12. Sequence((0, k), k, -10, 10, 1) → yGrid
  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

绘制一个坐标网格，范围从-10到10，包含水平和垂直的网格线。在网格上放置一个由四个点构成的四边形，这些点通过关于坐标轴的对称变换得到，形成一个对称的图形。在网格中随机选择一个点，并围绕它绘制一个动态变化的圆形，其大小会随时间脉冲变化。同时，在坐标轴上绘制两条线段，分别表示水平方向和垂直方向的参考线，并添加一条连接两个特定点的线段。整个图形用于展示指数函数平移变换的视觉效果，包含可交互的按钮和滑块来控制显示。
以上提示语为根据原始绘图命令整理而来，仅供参考。