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Roots of an Absolute Value Equation by cemccourseware

Roots of an Absolute Value Equation by cemccourseware
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2019-03-12 04:24
FunctionCoordinate GeometryDerivativeLimitOptimization
This example demonstrates the graphical representation of the roots of an absolute value equation. Key math concepts include absolute value functions, equation solving, statistical characteristics of roots, linear regression, limits, and logic. Construction highlights involve commands/objects like Random, Element, Expand, RemoveUndefined, Derivative, Intersect, Mirror, Union, Zip, Point, Line, Length, Polygon, Segment, LaTeX, CountIf in GeoGebra. Suitable for high school students learning functions and equations.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a graph showing the roots of an absolute value equation, highlighting intersections between an absolute value function and a linear function. Plot the absolute value function and the intersecting line. Mark all intersection points, emphasizing those within the visible range. Calculate and display the total number of intersection points and the number visible in the current view. Add labels for the function expression and explanatory text describing the relationship between intersection points and the roots of the equation.
The prompt above was compiled from the original drawing commands and is for reference only.