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余弦函数cos4x在[-270°,270°]内完整周期数探究

余弦函数cos4x在[-270°,270°]内完整周期数探究
几何画板及绘图内容加载中，请稍等……
2014-05-13 21:22
函数三角函数滑块动画函数图像
所属专题 CCGPS AA Unit 5 Lesson 2
本范例演示余弦函数 g(x)=cos4x 在限定区间 [-270°,270°] 上的图像，并统计其中包含多少个完整周期。核心概念包括周期函数、角频率、周期长度、度数与弧度切换、定义域限制以及函数图像绘制；由于 cos4x 的周期为 90°，在 540° 的区间内可观察到 6 个完整周期。作图要点：使用 LaTeX 与 FractionText 呈现公式和分数，用 If、Min、Max 控制显示条件与边界，用 Sequence 和 TableText 生成计数或表格文本，通过 stepMax、stepper、steps、counter、toggle、button1 等对象实现步进计数与交互切换，并用 domdeg、domrad、fullper、fplotdeg、fplotrad、xmin、xmax、ymin 等对象组织定义域、周期和绘图范围。适合高中代数、三角函数图像与周期分析的教学演示、课堂探究和自主练习。

GeoGebra 命令及参数

  1. LaTeX(steps) → counter
  2. LaTeX("\text{Identify the period of }\textcolor{#000099}{g(x).}") → text1
  3. LaTeX("\text{This will be needed to calculate the number of complete cycles of }\textcolor{#000099}{g(x)}\text{ that}\cr\text{exist in the restricted domain.}") → text2
  4. LaTeX("\text{Set the argument of the function equal to the period of the parent cosine}\cr\text{function, }\textcolor{#000099}{\cos(x)}\text{, and solve for }\textcolor{#000099}{x}.") → text3
  5. FractionText(permult) → permulttext
  6. LaTeX((((("\text{The period of the function }\textcolor{#000099}{g(x) = " * g) * "}\text{ is }\textcolor{#000099}{") * permulttext) * "\pi}.")) → text5
  7. FractionText(dlrad) → dlradtext
  8. FractionText(drrad) → drradtext
  9. If(toggle ≟ 1, LaTeX[(("\text{The period of }\textcolor{#000099}{g(x)}\text{, converted to degrees, is }\textcolor{#000099}{" * perdeg) * "^{\circ}}.")], LaTeX[(((("\text{The domain of }\textcolor{#000099}{g(x)}\text{, converted to radians, is }\textcolor{#000099}{\left[" * dlradtext) * "\pi,") * drradtext) * "\pi\right]}.")]) → text9
  10. If(toggle ≟ 1, LaTeX["\text{In order to determine the number of completed cycles, calculate the total}\cr \text{number of degrees within the domain of the function, and then divide this}\cr\text{number by the period measured in degrees.}"], LaTeX["\text{In order to determine the number of completed cycles, calculate the total}\cr \text{number of radians within the domain of the function, and then divide this}\cr\text{number by the period measured in radians.}"]) → text11
  11. LaTeX("\text{Domain:}") → text12
  12. LaTeX("\text{Period:}") → text14
  13. LaTeX("\frac{Domain}{Period}:") → text16
  14. If(toggle ≟ 1, LaTeX[(((((("\textcolor{#000099}{" * dr) * "^{\circ}-\left(") * dl) * "^{\circ}\right)=") * domdeg) * "^{\circ}}")], LaTeX[(((((("\textcolor{#000099}{" * drradtext) * "\pi-\left(") * dlradtext) * "\pi\right)=") * domrad) * "\pi}")]) → text13
  15. If(toggle ≟ 1, LaTeX[(("\textcolor{#000099}{" * perdeg) * "^{\circ}}")], LaTeX[(("\textcolor{#000099}{" * permulttext) * "\pi}")]) → text15
  16. LaTeX("\text{Verify the result by graphing.}") → text19
  17. If(toggle ≟ 1, dl - perdeg / 4, ((dlrad - permult / 4) * π)) → xmin
  18. If(toggle ≟ 1, dr + perdeg / 4, ((drrad + permult / 4) * π)) → xmax
  19. Min(0 - a / 4, c - a - a / 4) → ymin
  20. Max(0 + a / 4, c + a + a / 4) → ymax
  21. LaTeX((((((((("\text{The number of complete cycles of }\textcolor{#000099}{g(x)=" * g) * "}\text{ on the}\cr\text{restricted domain }\textcolor{#000099}{[") * dl) * "^{\circ},") * dr) * "^{\circ}]}\text{ is }\textcolor{#000099}{") * fullper) * "}.")) → text18
  22. If(toggle ≟ 1, Sequence[(dl + (n * perdeg), fplotdeg(dl + (n * perdeg))), n, 0, fullper, 1], Sequence[(((dlrad + (n * permult)) * π), fplotrad(((dlrad + (n * permult)) * π))), n, 0, fullper, 1]) → list1
  23. Sequence(Vector[Element[list1, n], Element[list1, n + 1]], n, 1, Length[list1] - 1) → list2
  24. LaTeX((((((("\textcolor{#000099}{g(x)=" * g) * "}\text{ on }\textcolor{#000099}{\left[") * dl) * "^{\circ},") * dr) * "^{\circ}\right]}")) → Problem
  25. LaTeX("\text{Convert the period to degrees or convert the domain to radians.}") → text6
  26. If(toggle ≟ 1, LaTeX["\text{Converting period to degrees}"], LaTeX["\text{Converting domain to radians}"]) → toggleText
  27. LaTeX("\text{Since the domain is given in degrees and the period is found in radians, one}\cr\text{or the other must be converted so that the units match before any comparison can be made.}") → text7
  28. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text20
  29. If(toggle ≟ 1, LaTeX[(((((("\textcolor{#000099}{\text{Greatest integer less than or equal to }\left(\frac{" * domdeg) * "^{\circ}}{") * perdeg) * "^{\circ}}\right)=") * fullper) * "}")], LaTeX[(((((("\textcolor{#000099}{\text{Greatest integer less than or equal to }\left(\frac{" * domrad) * "\pi}{") * permult) * "\pi}\right)=") * fullper) * "}")]) → text17
  30. LaTeX((((("\text{Determine the number of complete cycles }\textcolor{#000099}{g(x)}\text{ makes over the domain }\textcolor{#000099}{[" * dl) * "^{\circ}, ") * dr) * "^{\circ}]}.")) → text10
  31. If(toggle ≟ 1, LaTeX[(((((("\textcolor{#000099}{1}\text{ radian}\textcolor{#000099}{=\frac{180^{\circ}}{\pi}}\text{, so }\textcolor{#000099}{" * permulttext) * "\pi}\text{ radians is }\textcolor{#000099}{") * permulttext) * "\pi\cdot\frac{180^{\circ}}{\pi}=") * perdeg) * "^{\circ}}.")], LaTeX[(((((((((("\textcolor{#000099}{1^{\circ}=\frac{\pi\text{ radians}}{180}}\text{, so }\cr \textcolor{#000099}{" * dl) * "^{\circ}}\text{ is }\textcolor{#000099}{") * dl) * "^{\circ}\frac{\pi}{180^{\circ}}=") * dlradtext) * "\pi}\cr \text{and}\cr \textcolor{#000099}{") * dr) * "^{\circ}\frac{\pi}{180^{\circ}}=") * drradtext) * "\pi}.")]) → text8
  32. TableText({{LaTeX[(("\textcolor{#000099}{" * b) * "x = 2\pi}")], LaTeX["\text{Set the argument equal to the period of }\textcolor{#000099}{\cos(x)}."]}, {LaTeX[(("\textcolor{#000099}{x = \frac{2}{" * b) * "}\pi}")], LaTeX["\text{Solve for }\textcolor{#000099}{x}."]}, {LaTeX[(("\textcolor{#000099}{x = " * permulttext) * "\pi}")], LaTeX["\text{Simplify.}"]}}) → text4