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检验直线与抛物线系统方程的交点解

检验直线与抛物线系统方程的交点解
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2024-06-13 03:47
函数解析几何函数图像
该范例通过图形演示“直线与抛物线的交点如何对应系统方程的解”，并引导学生判断某个点是否同时满足两条方程。核心数学概念包括：二元方程组、非线性关系（线与二次曲线）、函数图像的交点与“解”的等价关系，以及从数值代入验证解的思想。作图与交互要点体现在：范例使用随机参数生成多组方程情境（如 RandomUniform、Sequence、Flatten、KeepIf、First、Element 组合选择候选值），构造对应的多项式函数表达式（Polynomial），计算或筛选“可能顶点/可能解集合”（例如使用 RemoveUndefined、KeepIf、First、Element 等处理未定义与筛选），并在平面中生成关键点与线段/多边形辅助几何关系（Point、Segment、Polygon），随后通过文本与图形位置微调展示信息（Text、Translate、Rotate、Split、Append、PointToPixels、TextToUnicode/UnicodeToText 等），再将不匹配的候选项剔除（RemoveUndefined、Zip）。适合 9-12 年级学生用于代数系统方程或图形法求解练习，也可作为课堂探究题与随堂验证活动的交互资源。

GeoGebra 命令及参数

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.3853180500635086, 0.36786167154423083, 0.3314899008851462, 0.4495898113206599, 0.7062391257328886, 0.8344662421036378, 0.22918499003983894, 0.079953373668172, 0.5209780387007789, 0.2735134895943693, 0.659835476913766, 0.967471125139848, 0.5120405543162071, 0.1878068650045772, 0.6425084400793382, 0.05588990138577263, 0.4953026774237742, 0.7183547614533085, 0.10065415855108611, 0.12190724108639135, 0.46657363747713015, 0.09996257148366718, 0.04575802394282447, 0.43279546821414605, 0.37009188346240574, 0.5989118343857159, 0.042775727967683516, 0.057192155759809384, 0.16583040117625739, 0.9570412333769425, 0.9465084403228637, 0.13030071971067922, 0.24978184743771492, 0.9075109935373384, 0.7045874813545581, 0.05368030705223599, 0.2946271074230456, 0.5419369083262746, 0.3601895579299874, 0.47717719899975675, 0.047360163846397525, 0.6364216641360324, 0.42599246493367715, 0.49636025224367125, 0.7255501132793349, 0.4852904008168496, 0.17020696371094612, 0.8630713760975706, 0.2949656216956968, 0.8060545266884918, 0.20578439475788265, 0.14938727586437156, 0.7782699518674823, 0.3772818878654581, 0.6151008159092369, 0.6764119364169152, 0.8474144590598995, 0.5026433736480894, 0.4945569316821931, 0.8788794980173206, 0.59121552213518, 0.17943497813249068, 0.6666095909109567, 0.014545867254726947, 0.5143955823687387, 0.7383414722488786, 0.27076389491749814, 0.5537822050632526, 0.8251006613729599, 0.14311221126100332, 0.6000063344053986, 0.8096572620167068, 0.9045767386008041, 0.5141610333228498, 0.7550481961898011, 0.6421051872673396, 0.1452955256899183, 0.5588815063611642, 0.32557430822251276, 0.3615242998449608, 0.19236383877319874, 0.403686395897923, 0.5254561443006815, 0.8173375766516967, 0.07753449923805626, 0.9384331312550346, 0.167589414718326, 0.20566825401832536, 0.3855556790631602, 0.512990051095077, 0.7236547324624096, 0.657345154054686, 0.34895331555338516, 0.45822713658840786, 0.1346765629852131, 0.03294997941523795, 0.43963641285045596, 0.7096812373691975, 0.679768801613856, 0.055554419487152104, 0.7439471339324176, 0.751550892809333, 0.9029070120825964, 0.08416806433218435, 0.28194746681066213, 0.3715171526213953, 0.17095550768875775, 0.11172850542769515, 0.011842158323222383, 0.5031661504187364, 0.08050415798403343, 0.050713469981225764, 0.46648171904084434, 0.10250608003896008, 0.026333037675006634, 0.5771394263011324, 0.4308726965472317, 0.2582256671768807, 0.34052455602580245, 0.5798355879431913, 0.24754152170663296, 0.19193233304528523, 0.9583809508039988, 0.07644130859383882, 0.1504559449119155, 0.46463887430359946, 0.7153818447069626, 0.8605739414379827, 0.078003319807534, 0.7392325448742443, 0.7484292888627293, 0.6260493825170392, 0.23125675745817487, 0.7655663819455766, 0.01458028148917782, 0.8639105565775459, 0.9254306477034889, 0.6951532007752785, 0.06135351479645739, 0.3963295345693092, 0.09003439723565104, 0.8862227885725366, 0.9845254900616076, 0.2281976794035199, 0.9667777165460659, 0.24711896049190493, 0.6820926670050901, 0.4924026634559002, 0.8437637722602729, 0.7366679560032785, 0.11677929936887324, 0.878415370678109, 0.3699261634972961, 0.33424136660083476, 0.014107825381487528, 0.29586926568250166, 0.6550229543932176, 0.5697657205264075, 0.7286260452090852, 0.9153137796234563, 0.3677965636653318, 0.25489798055142204, 0.7773979888279198, 0.7072580433404932, 0.08813947641953399, 0.300990563244363, 0.06731741385395901, 0.22799466545844002, 0.8982774600418089, 0.5173110772999231, 0.6411907332136443, 0.2265040415381253, 0.0930874194717725, 0.31426121748167213, 0.43686313211333155, 0.9109870785692942, 0.3407136820198115, 0.5110562870537999, 0.03392463886871422, 0.7362605431830888, 0.23162763117929844, 0.38548122743686586, 0.5437345464157332, 0.6643716152654024, 0.13719637585395444, 0.6249301011143797, 0.29064089345735833, 0.5003493655701663, 0.057683094064709575, 0.5329662399111396, 0.7253483731991004, 0.3730081800848912, 0.5043968200605479, 0.8641065491962815, 0.1854010076935218, 0.6471546642323245, 0.8714417624398099, 0.9214374458311377, 0.5326978623373068, 0.7784216917295964}
  2. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  3. If(-4 + floor((currentRandomSeeds(2) * 8)) ≠ 0, -4 + floor((currentRandomSeeds(2) * 8)), 1) → yV
  4. Element(coeffsA, Mod[floor((currentRandomSeeds(3) * 10)), Length[coeffsA]] + 1) → coeffA
  5. Sequence(xA + (0.5 * ii), ii, 1, Min[10 - (2 * xA), 6]) → xBPos, {-1.5, -1, -0.5, 0, 0.5, 1}
  6. Element(xBPos, xBind) → xB
  7. Flatten(Sequence[Sequence[(ii, jj), ii, -4, 4], jj, If[coeffA ≤ 0, -5 - coeffA, -4], If[coeffA ≥ 0, 5 - coeffA, 4]]) → PossibleVertices
  8. Element(PossibleVertices, floor((currentRandomSeeds(2) * Length[PossibleVertices])) + 1) → Vert
  9. Polynomial(g0) → g
  10. If(countQuestion ≟ 0, 2, If[currentRandomSeeds(1) ≤ 0.8, 2, 1]) → NumOfSols
  11. KeepIf(y(PP) ≟ g(x(PP)), PP, Flatten[Sequence[Sequence[(ii, jj), ii, -5, 5], jj, -5, 5]]) → PossibleSols
  12. Element(PossibleSols, 1 + floor((Length[PossibleSols] * currentRandomSeeds(4)))) → Sol1
  13. Element(PossibleSols \ {Sol1}, 1 + floor(((Length[PossibleSols] - 1) * currentRandomSeeds(4)))) → Sol2
  14. First({Sol1, Sol2}, NumOfSols) → sols
  15. Corner(5) → layoutAppSize
  16. PointToPixels(Corner[onboardingDisplayText, 2], Corner[1], Corner[3], Corner[5]) → layoutOnboarding
  17. onboardingTextOffset(Corner[onboardingText, 1], Corner[onboardingText, 3], onboardingCase, Corner[1], Corner[3], Corner[5]) → OnboardingTextOffset
  18. Polygon({(-5.5, -5.5), (5.5, -5.5), (5.5, 5.5), (-5.5, 5.5)}) → q1
  19. Flatten(Sequence[Sequence[y = (mm * x) + bb, mm, -5, 5], bb, -5, 5]) → Lines
  20. Zip(RemoveUndefined[{Intersect[g, ll]}], ll, Lines) → l2
  21. RemoveUndefined(Zip[If[Length[ll] ≟ 0, LL, NaN], ll, l2, LL, Lines]) → l3
  22. Element(l3, 1 + floor((currentRandomSeeds(1) * Length[l3]))) → j
  23. Polynomial(Element[{h0, h1, h2}, NumOfSols + 1]) → h
  24. Segment((0, -5.42), (0, 5.42)) → i
  25. Text("\math{x}", (5.75, 0.075), true, true, 0, 0) → labelx
  26. Text("\math{y}", (0, 5.9), true, true, 0, 0) → labely
  27. Translate(p, Vector[(Vector[Corner[1], Corner[4]] / 500 * 16)]) → p'
  28. Sequence(Segment[(ii, -0.05), (ii, 0.05)], ii, -5, 5, 0.5) → l1
  29. Rotate(i, 90°) → i'
  30. Text((((((((("(\textcolor{" * colorx) * "}{") * x(Sol2)) * "},\textcolor{") * colory) * "}{") * y(Sol2)) * "})"), Sol2, true, true, 0, If[y(Sol1) ≥ y(Sol2), -1, 1]) → labelP2
  31. Text(g) → gtext
  32. KeepIf(abs(y(PP)) ≤ 5, PP, Sequence[(ii, g(ii)), ii, -5, 5, 0.01]) → Points1
  33. KeepIf(abs(y(PP)) ≤ 5, PP, Sequence[(ii, h(ii)), ii, -5, 5, 0.01]) → Points2
  34. Sort(Join[{Points1, Points2}]) → Points
  35. Point(Points) → A
  36. Derivative(g) → f
  37. TextToUnicode(gtext) → gtextUnic
  38. UnicodeToText(Flatten[Zip[If[nn ≟ 120, TextToUnicode[xsubstitute], {nn}], nn, gtextUnic]]) → text2
  39. Text(h) → htext
  40. Split(htext, {"x"}) → htextSplit
  41. Append(Zip[tt + xsubstitute, tt, First[htextSplit, If[h(0) ≟ 0 ∧ Slope[h] ≠ 0, Length[htextSplit], Length[htextSplit] - 1]]], If[h(0) ≟ 0 ∧ Slope[h] ≠ 0, "", htextSplit(-1)]) → htextSubst
  42. UnicodeToText(Flatten[Zip[If[nn ≟ 120, TextToUnicode[xsubstitute2], {nn}], nn, gtextUnic]]) → gtextSubstFinal2
  43. Append(Zip[tt + xsubstitute2, tt, First[htextSplit, If[h(0) ≟ 0 ∧ Slope[h] ≠ 0, Length[htextSplit], Length[htextSplit] - 1]]], If[h(0) ≟ 0 ∧ Slope[h] ≠ 0, "", htextSplit(-1)]) → htextSubst2
  44. Text((((((((("(\textcolor{" * colorx) * "}{") * x(A)) * "},\textcolor{") * colory) * "}{") * y(A)) * "})"), A - (0, 0.15), true, true, 0, -1) → labelA
  45. circlePixels(A, 7 + (7 * pulsing), Cor1, Cor3, (800, 500)) → c

AI 绘图提示语参考

绘制一个坐标系，包含一条随机生成的直线和一条抛物线，它们的交点用点标出。在图中展示两条函数的解析式，并用不同颜色标注两个交点的坐标。背景为一个浅灰色边框的矩形，坐标轴上有刻度标记，并在轴上标注变量名。
以上提示语为根据原始绘图命令整理而来，仅供参考。