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Graph Transformations of Sine and Cosine Functions (Radians)

Graph Transformations of Sine and Cosine Functions (Radians)
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2019-08-06 06:03
FunctionTrigonometryTransformation
This example demonstrates the graph transformations of sine and cosine functions in radians. Key mathematical concepts include function transformations, periodicity, and phase shift. By adjusting parameters, observe the changes in the function graphs. GeoGebra commands such as Shuffle, Element, and Sort are used for dynamic transformations. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Shuffle(Sequence[i, i, 0.5, 4, 0.5]) → aValueOptions, {1, 3.5, 3, 2, 4, 1.5, 2.5, 0.5}
  2. Shuffle({3, 3 / 2, 3 / 4}) → bValueOptions, {3, 0.75, 1.5}
  3. Element(bValueOptions, 1) → bVal
  4. Element({1 / 2, 1 / 3, 3 / 4, -3 / 4, -1 / 2, -1 / 3}, Random[1, 6]) → hValue
  5. Sort(RemoveUndefined[Zip[If[A ≤ (4 * π) - period ∧ A ≥ (-2 * π), A], A, Sequence[(hValue * π) + ((i * 2) * π / abs(bVal)), i, -10, 10]]]) → validStartingPositions
  6. If(Defined[IndexOf[minUserXVal, validStartingPositions]], minUserXVal, (hValue * π)) → startingXVal
  7. If(isSinFunc, sin(x), cos(x)) → parentFunc
  8. LaTeX(("y=" * parentFunc)) → lblParent
  9. UnicodeToLetter(10003) → cmark
  10. UnicodeToLetter(10007) → xmark
  11. If(isSinFunc, (aVal * sin((bVal * (x - (hValue * π))))) + kValue, (aVal * cos((bVal * (x - (hValue * π))))) + kValue) → fTranslated
  12. Sequence(startingXVal + (i * quarterPeriod), i, 0, 4) → correctXvals
  13. Zip((A, fTranslated(A)), A, correctXvals) → correctPoints
  14. Flatten(Zip[Sequence[(A, j), j, -5, 5, 0.5], A, validStartingPositions]) → gridSpaceExtra
  15. NumToFracNoPhantom(bVal) → bValFmt
  16. If(bVal ≟ 1, "", If[bVal ≟ -1, "-", bValFmt]) → bValFmtNoOnes
  17. NumToFracNoPhantom(aVal) → aValFmt
  18. If(aVal ≟ 1, "", If[aVal ≟ -1, "-", aValFmt]) → aValFmtNoOnes
  19. NumToFracNoPhantom(kValue) → kValFmt
  20. Sort(userPtLst) → userPtsSorted
  21. If(hValue < 0, "left", "right") → shiftDirection
  22. Polygon({(left, bottom), (left, top), (right, top), (right, bottom)}) → viewbox
  23. Flatten(Sequence[Sequence[(i, j), i, (-2 * π), (4 * π), π / 12], j, -6, 6, 0.5]) → gridSpace
  24. LaTeX((((((((((("\text{\(y=" * aValFmtNoOnes) * If[isSinFunc, "\sin", "\cos"]) * "(") * bValFmtNoOnes) * "(x") * If[hValue < 0, "+", "-"]) * FractionText[abs(hValue)]) * "\pi ))") * If[kValue < 0, kValFmt, ("+" * kValFmt)]) * "\)}")) → lblFx
  25. LaTeX(((("phase shift is \(" * FractionText[abs(hValue)]) * "\pi\) to the ") * shiftDirection)) → fbPhaseShift
  26. Point(Union[gridSpace, {Element[homeBase, 1]}]) → B2
  27. Point(Union[gridSpace, {Element[homeBase, 3]}]) → B4
  28. Point(Union[gridSpace, {Element[homeBase, 4]}]) → B5
  29. Point(Union[gridSpace, {Element[homeBase, 5]}]) → B6
  30. Point(Union[gridSpace, {Element[homeBase, 2]}]) → B3
  31. If(kValue < 0, "down", "up") → hShiftDirection
  32. If(kValue > 0, (0, kValue), (0, kValue + 0.75)) → axisLblPt
  33. LaTeX((("\text{\(y=" * kValFmt) * "\)}")) → lblAxis
  34. LaTeX((((("\text{\textbf{Question: }" * QUESTION) * "\\\textbf{Instructions: }") * INSTRUCTIONS) * "}")) → allInstruc
  35. Sequence((i, bottom), i, (-2 * π), (4 * π), π / 12) → vAnchors
  36. Sequence((right, i), i, -6, 6, 0.25) → hAnchors
  37. Point(vAnchors) → vGuide1
  38. Point(vAnchors) → vGuide2
  39. Sequence(((4 * π) + 0.2, i), i, 5, 3.5, -1.5) → legendAnchors
  40. Point(hAnchors) → hGuide1
  41. Point(hAnchors) → hGuide2
  42. Point(hAnchors) → hGuide3
  43. Segment(Element[legendAnchors, 1], Element[legendAnchors, 1] + (1, 0)) → legend1
  44. Segment(Element[legendAnchors, 2], Element[legendAnchors, 2] + (1, 0)) → legend2
  45. Sequence((i * π), i, -2, 4, 1 / 12) → xVals
  46. Sequence(If[i ≟ 0, "0", If[i ≟ 1, "\pi", If[i ≟ -1, "-\pi", (FracText[i] * "\pi")]]], i, -2, 4, 1 / 12) → activeXTxt
  47. LaTeX(If[aVal < 0, "Since \(a\lt0\), there is a reflection in the \(x\)-axis.", " "]) → fbReflectionX
  48. Sequence(Segment[(i, bottom), (i, top)], i, (-2 * π), (4 * π), π / 4) → vGrid
  49. LaTeX((((((((((((("\text{From the formula \(" * lblFx) * "\), ") * fbAmplitude) * ", \\") * fbPeriod) * ", ") * fbPhaseShift) * ", and the vertical\\displacement is ") * fbVertical) * ". ") * fbReflectionX) * "}")) → fbComplete