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指数函数的水平与竖直平移：从母函数到变换后函数

指数函数的水平与竖直平移：从母函数到变换后函数
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2024-04-08 15:55
向量函数变换函数图像
这个范例演示如何对指数函数母图进行水平移位与竖直移位，并据此写出平移后的新函数表达式。核心数学概念包括：指数函数 y=a·b^x（或 y=a·e^x 形式）的参数含义、水平平移（x→x−h 或 x→x+h）与竖直平移（y→y+k）的图形效果，以及用函数记号/代数关系识别变换后的函数。作图与交互要点来自构造摘要：通过 RandomUniform、Shuffle、Take、Element 生成题目参数与随机种子；用 Translate、Vector 表示平移向量与变换；借助 Asymptote 设定并显示指数函数渐近线；用 Intersect、Segment、Polygon 辅助标注关键点/线段与可视化结构；使用 Textfield、SelectedIndex、Corner 与 onboardingTextOffset 来提供输入框、选项与引导文本；条件判断 If 与交互对象（showanswer、showsolution、btnValidate、btnNew、showhint、tries、countCorrect）用于验证学生答案并呈现反馈。适合9–12年级学生用于函数变换识别练习，也可作为课堂演示或自学测验的交互资源。

GeoGebra 命令及参数

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.3513288554133347, 0.7948306045490805, 0.630092788766771, 0.09439542691709646, 0.7852024620150003, 0.6665376341339869, 0.5699671841562188, 0.24012199373492638, 0.8637805245124694, 0.08785833468869286, 0.44153601639483575, 0.16421085649828637, 0.4308267675680974, 0.5411244254626862, 0.2880187442285113, 0.3316458516994189, 0.04573402410943486, 0.1748004821183743, 0.41277587497294005, 0.6770733013427496, 0.1260328797579236, 0.8722298317610926, 0.8152164882118385, 0.5512860631616803, 0.8255619293406712, 0.8085752184173688, 0.025846084076791377, 0.18704123234882364, 0.8465831495641459, 0.14117869808029113, 0.9732256607136217, 0.5181025683199033, 0.2261469752439717, 0.12582050522863983, 0.07247790291947201, 0.7812487764802283, 0.07622122021518243, 0.033301068216008846, 0.7140290667453745, 0.5365255278801965, 0.8934939576435216, 0.6773999707892325, 0.3125153813230163, 0.7962439209825842, 0.6883449332215922, 0.2365848131161663, 0.5192110313604574, 0.5281400501340272, 0.6532510076193698, 0.5178487856685339, 0.9669488778459061, 0.8827101085828115, 0.7888507243892445, 0.18174849538978735, 0.5443091658553205, 0.6890222824074053, 0.9210390159573988, 0.0630952115773782, 0.1758407330256809, 0.9772818689520718, 0.02479759470131719, 0.4418205881887233, 0.1772572402514549, 0.8600873222285432, 0.7899873828257541, 0.23329379584198423, 0.5889163845457557, 0.6376654451960931, 0.6910392314671455, 0.7168107131050525, 0.9074426719934164, 0.7954103513863395, 0.29637459615903033, 0.3151062173290358, 0.8840941853192839, 0.5947755991070007, 0.034472814567535676, 0.4865591850053309, 0.7994022329992151, 0.7909816178761417, 0.9717722934410394, 0.265265837552714, 0.7267275137182303, 0.6530990154870339, 0.10305266275830192, 0.7423502997862352, 0.7926790498100168, 0.40385645499896744, 0.9475907129053986, 0.5447615865701813, 0.4098328278724489, 0.20799601297957987, 0.35421746876780824, 0.5053759111353929, 0.726864624368723, 0.16916092573026423, 0.9354247863068216, 0.7404773332852942, 0.42307647877908006, 0.966150620290243, 0.854879403543547, 0.8976347101537169, 0.8583493116020401, 0.579914407017659, 0.03409829983748836, 0.4586226869382642, 0.5057945168569734, 0.8128807178573046, 0.5926234474515693, 0.07835339249537238, 0.1328045018907437, 0.7621298275170507, 0.1618056039996985, 0.2817911212384236, 0.9568182845582216, 0.6875186800341042, 0.4152656787243638, 0.2265157243437538, 0.5449200257395702, 0.5151525710074281, 0.5392484981631999, 0.8289813015331442, 0.31137950597020375, 0.04557373839783843, 0.5634184860290352, 0.477585138673989, 0.43442037540695716, 0.7246988696348894, 0.5271670551391427, 0.1854176023590368, 0.5728683519951794, 0.2938867594099253, 0.828211767374742, 0.46000721979586334, 0.40488295838699917, 0.373554957388864, 0.6591629777648523, 0.11650586541820049, 0.7447178896217643, 0.19248870867495307, 0.7238178830772269, 0.7145775674822615, 0.2593819258713156, 0.5023973580204484, 0.868213581608559, 0.04776003865013425, 0.24518807670411394, 0.13437335487794114, 0.28192931435827906, 0.3872614355573916, 0.24448099023721742, 0.9706958359684774, 0.10277509167725514, 0.6100360488611456, 0.3753232787209039, 0.7818954782708568, 0.3428372304863721, 0.7535505403609188, 0.341598767313794, 0.8651595394008118, 0.785873575895136, 0.18690840843644918, 0.17785622551946598, 0.5960052769335367, 0.4752593923350076, 0.6099436773727398, 0.5022697845575405, 0.7806875172609469, 0.03808201883392005, 0.8185258569642648, 0.1023321411067003, 0.012030139790984561, 0.8245660009491218, 0.5056362537476095, 0.021325081821110453, 0.49628427957284676, 0.3227819248471848, 0.4637817079468291, 0.4092263996805353, 0.46321442797967016, 0.5765516165770184, 0.365005470933323, 0.4295153614449439, 0.6659017968002766, 0.14960788770533714, 0.3501350760343593, 0.3033291680487885, 0.5607285294337738, 0.821402088681551, 0.04897652328428288, 0.6946003414405885, 0.5616682018294237, 0.9594735822680267, 0.9479186475808071, 0.07417039500527417, 0.8963579389251224, 0.050004201951754985, 0.7708560526471414, 0.49742780694892413, 0.9844648190293982}
  2. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  3. Shuffle(Sequence[t, t, -5, 5, 1] \ {0}) → allH, {-3, 5, 3, -1, -2, 1, 4, 2, -4, -5}
  4. Element(allH, Mod[countQuestion, Length[allH]] + 1) → numH
  5. Shuffle(Sequence[t, t, -3, 3, 1] \ {0}) → allK, {2, 1, 3, -3, -1, -2}
  6. Element(allK, Mod[countQuestion, Length[allK]] + 1) → numK
  7. Shuffle({-3, -2, 2, 3}) → allA, {3, -3, 2, -2}
  8. Shuffle(Sequence[t, t, 2, 8, 1]) → allB, {4, 6, 2, 5, 3, 8, 7}
  9. Element(allA, Mod[countQuestion, Length[allA]] + 1) → numA
  10. Element(allB, Mod[countQuestion, Length[allB]] + 1) → numB
  11. Translate(given, Vector[(numH, numK)]) → final
  12. Textfield(inputB) → InputBoxB
  13. SelectedIndex(ddMenu) → is
  14. Corner(5) → layoutAppSize
  15. onboardingTextOffset(Corner[onboardingText, 1], Corner[onboardingText, 3], onboardingCase, Corner[1], Corner[3], Corner[5]) → OnboardingTextOffset
  16. Textfield(inputA) → InputBoxA
  17. Textfield(inputC) → InputBoxC
  18. Polygon(P1, Mirror[P1, yAxis], Mirror[P1, (0, 0)], Mirror[P1, xAxis]) → peep, p1, f, g, h
  19. Segment((-10.85, 0), (10.85, 0)) → fakeX
  20. Segment((0, -10.85), (0, 10.85)) → fakeY
  21. Translate(horTrans, Vector[(0, ((aniSli - 1) * numK))]) → verTrans
  22. If(aniSli < 1, Translate[GI, Vector[((aniSli * numH), 0)]], Translate[GI, Vector[(numH, 0)]]) → GIH
  23. Translate(GIH, Vector[(0, ((aniSli - 1) * numK))]) → GIV
  24. Vector(GI, GIH) → HVect
  25. Vector(GIH, GIV) → VVect
  26. Asymptote(verTrans) → ao
  27. Vector(GI, GIV) → transVec
  28. Segment(GI, GIH) → xComp
  29. Segment(GIH, GIV) → yComp
  30. Intersect(final, If[numA < 0, y = -10, y = 11], (-2.2075187496394193, 11)) → PLG
  31. Intersect(studF, If[3 ≤ is ≤ 5 ∧ numA < 0 ∨ is ≟ 2 ∧ inputA < 0, y = -10, y = 11], (11, 11)) → PLS

AI 绘图提示语参考

绘制一个指数函数的图形，并通过水平和垂直平移变换，生成一个新的指数函数图形。首先绘制原始指数函数，然后通过水平平移数H个单位和垂直平移数K个单位，得到变换后的指数函数图形。注意观察平移前后图形的变化，并尝试根据图形确定变换后的函数表达式。
以上提示语为根据原始绘图命令整理而来，仅供参考。