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绝对值方程的根 by cemccourseware

绝对值方程的根 by cemccourseware
几何画板及绘图内容加载中，请稍等……
2019-03-12 04:24
函数解析几何导数极限最值优化
本范例演示了绝对值方程的根的图形表示。涉及的核心数学概念包括绝对值函数、方程求解、根的统计特性、线性回归、极限和逻辑。作图要点包括使用Random、Element、Expand、RemoveUndefined、Derivative、Intersect、Mirror、Union、Zip、Point、Line、Length、Polygon、Segment、LaTeX、CountIf等GeoGebra命令和对象。适合高中阶段学习函数和方程的学生使用。

GeoGebra 命令及参数

  1. Random(-5, 5) → hVal
  2. Random(1, 3) → kVal
  3. Element(lineSlopes, Random[1, 4]) → slope
  4. Expand((aVal * (x - hVal)^(2)) + kVal) → f
  5. RemoveUndefined({Intersect[f, xAxis]}) → fRoots
  6. Derivative(f) → f'
  7. Intersect(f, f'(x) = slope) → TangentPoint
  8. Mirror(TangentPoint, x = hVal) → alternateTangent
  9. Union(fRoots, {Fvertex, TangentPoint, alternateTangent}) → FkeyPts
  10. Zip((0, y(A) - (slope * x(A))), A, FkeyPts) → keyYIntercepts
  11. Union(Sequence[(0, i), i, -10, 10, 0.5], keyYIntercepts) → allSliderVals
  12. Point(allSliderVals) → B
  13. Line(A, B) → g
  14. RemoveUndefined(Union[{Intersect[f, g]}, {Intersect[(-f(x)), g]}]) → allIntersectPts
  15. RemoveUndefined(Zip[If[y(A) ≥ 0, A], A, allIntersectPts]) → positiveIntPts
  16. Length(positiveIntPts) → numIntersectionPts
  17. Polygon({(vbLeft, vbBottom), (vbLeft, vbTop), (vbRight, vbTop), (vbRight, vbBottom)}) → viewbox
  18. Zip(If[IsInRegion[A, viewbox], true], A, positiveIntPts) → visibleIntPts
  19. Segment((0, floor(vbBottom + 3)), (0, floor(vbTop))) → yAxisVerticalSlider
  20. LaTeX(Expand[f]) → Lblf
  21. LaTeX("\text{\textbf{Note: } \\The number of roots of \(\left\vert ax^2 + bx +c \right\vert =px+q\) \\is equal to the number of intersection points of \\\(y=\left\vert ax^2 + bx + c\right\vert\) and \(y=px+q\).}") → txtNote
  22. CountIf(x ≟ true, visibleIntPts) → numVisible
  23. LaTeX(If[numIntersectionPts ≟ numVisible, "", (("Only " * If[numVisible ≟ 3, "three are", If[numVisible ≟ 2, "two are", "one is"]]) * " visible on the graph.")]) → fbNumVisible
  24. LaTeX(("There " * If[numIntersectionPts ≟ 0, "are no intersection points.", If[numIntersectionPts ≟ 1, "is one intersection point.", (("are " * If[numIntersectionPts ≟ 2, "two", If[numIntersectionPts ≟ 3, "three", "four"]]) * " intersection points. ")]])) → fbNumIntPoints
  25. LaTeX((((("\text{" * fbNumIntPoints) * "\\") * fbNumVisible) * "}")) → txtNumIntPts
  26. RemoveUndefined(Zip[If[IsInRegion[A, viewbox], A], A, positiveIntPts]) → allIntersectionsInRegion
  27. LaTeX((((("\text{\textbf{Question: }" * MainInstruc) * "\\\textbf{Instructions: }") * SpecInstruc) * "}")) → allInstruc
  28. Segment(Element[legendAnchors, 1], Element[legendAnchors, 1] + (3, 0)) → fxLegend
  29. Segment(Element[legendAnchors, 2], Element[legendAnchors, 2] + (3, 0)) → lineLegend
  30. LaTeX((("\vphantom{\dfrac{1}{2}}y=\left\vert " * Lblf) * " \right\vert")) → lblGraph

AI 绘图提示语参考

画一个绝对值方程的根的图形，展示绝对值函数与线性函数的交点。首先绘制绝对值函数图像，再绘制与之相交的直线。标记所有交点，并突出显示在视图范围内的交点。计算并显示交点数量，以及视图范围内的交点数量。最后，在左侧添加函数表达式和说明文字，解释交点数量与方程根的关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。