Graphing Multiple Periods of f(x)=cos 2x

GeoGebra Commands & Parameters

1. LaTeX(steps) → counter
2. LaTeX("\text{Move the slider to solve the problem.}") → text0
3. Corner(1) → A
4. Corner(3) → B
5. If(choose ≟ 1, Simplify[(a * sin((b * x)))], Simplify[(a * cos((b * x)))]) → f
6. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → Problem
7. LaTeX("\text{Graph several periods of }\textcolor{#000099}{f(x)}.") → text1
8. If(choose ≟ 1, ((-π)) / ((2 * b)), 0) → left
9. If(choose ≟ 1, π / ((2 * b)), π / b) → right
10. If(choose ≟ 1, "-", "") → posneg
11. If(choose ≟ 1, LaTeX[(("-\frac{\pi}{" * Simplify[(2 * b)]) * "}")], "0") → leftText
12. If(choose ≟ 1, LaTeX[(("\frac{\pi}{" * Simplify[(2 * b)]) * "}")], LaTeX[(("\frac{\pi}{" * Simplify[b]) * "}")]) → rightText
13. LaTeX("\text{Solve for the inverse function }\textcolor{#006633}{g(x)}\text{ algebraically.}") → text11
14. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → text14
15. LaTeX("\text{Switch }\textcolor{#000099}{x}\text{ and }\textcolor{#000099}{f(x)}.") → text15
16. If(choose ≟ 1, "sin", "cos") → functext
17. LaTeX((("\text{Take the inverse " * functext) * " of both sides.}")) → text21
18. LaTeX("\text{Simplify and use symmetry of equality.}") → text23
19. If(b ≠ 1, LaTeX[Simplify[1 / b]], "") → invCoeff
20. Mirror(h, y = x) → Inverse
21. LaTeX((((((("\textcolor{#006633}{g(x)=\frac{" * functext) * "^{-1}\left(\frac{x}{") * a) * "}\right)}{") * b) * "}}")) → text26
22. LaTeX((((((("\textcolor{#006633}{" * b) * "g(x)=") * functext) * "^{-1}\left(\frac{x}{") * a) * "}\right)}")) → text24
23. LaTeX((((((((((("\textcolor{#006633}{" * functext) * "^{-1}\left(\frac{x}{") * a) * "}\right)=") * functext) * "^{-1}\left(") * functext) * "(") * b) * "g(x))\right)}")) → text22
24. LaTeX((((("\textcolor{#006633}{\frac{x}{" * a) * "}=") * functext) * "(g(x))}")) → text20
25. LaTeX("\text{Rename }\textcolor{#000099}{f(x)}\text{ as }\textcolor{#006633}{g(x)}.") → text17
26. LaTeX((((((("\textcolor{#006633}{x=" * a) * "") * functext) * "(") * b) * "g(x))}")) → text18
27. LaTeX((((((("\textcolor{#006633}{g(x)=" * invCoeff) * "") * functext) * "^{-1}\left(\frac{x}{") * a) * "}\right)}")) → invlabel
28. LaTeX((("\text{Divide both sides by }\textcolor{#006633}{" * b) * "}.")) → text25
29. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → restrictlabel
30. LaTeX((((((("\textcolor{#000099}{x=" * a) * " ") * functext) * "(") * b) * "f(x))}")) → text16
31. If(choose ≟ 1, "sine", "cosine") → name
32. If(choose ≟ 1, (((-3) * a / 4), asin((((-3) * a / 4)) / a) / b), (((-3) * a / 4), acos((((-3) * a / 4)) / a) / b)) → C
33. LaTeX((("\text{The trigonometric function }\textcolor{#000099}{f(x) = " * f) * "}\text{ is not fully invertible because of its }\cr\text{periodic nature. Viewing several periods of the function is the first step toward }\cr\text{identifying a restricted domain on which the function is invertible.}")) → text2
34. LaTeX("\text{Restrict the domain until there is a one-to-one correspondence. }") → text3
35. LaTeX("\text{Determine the range of }\textcolor{#000099}{f(x)}\text{ within the restricted domain.}") → text8
36. LaTeX((((((((("\text{The inverse function of }\textcolor{#000099}{f(x)=" * f) * "}\text{ is }\textcolor{#006633}{g(x)=") * invCoeff) * "") * functext) * "^{-1}\left(\frac{x}{") * a) * "}\right)}.")) → text27
37. LaTeX("\text{Determine the domain and range of the inverse function }\textcolor{#006633}{g(x)}.") → text28
38. LaTeX("\text{The domain and range of an inverse function are the range and (restricted) }\cr\text{domain of the original function.}") → text29
39. LaTeX((((((((((((((((((((((((("\text{For the original function } \textcolor{#000099}{f(x) = " * f) * "} \text{, the restricted domain is } \textcolor{#000099}{\left[") * leftText) * ",") * rightText) * "\right]} \cr\text{and the range is }\textcolor{#000099}{[-") * a) * ",") * a) * "]} \text{. }\cr\text{Therefore, for the inverse function }\textcolor{#006633}{g(x)=") * invCoeff) * "") * functext) * "^{-1}\left(\frac{x}{") * a) * "}\right)} \text{, the domain is } \cr \textcolor{#006633}{[–") * a) * ", ") * a) * "]} \text{ and the range is } \textcolor{#006633}{\left[") * leftText) * ",") * rightText) * "\right]}.")) → text30
40. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
41. LaTeX("\text{Look at the graph of }\textcolor{#000099}{f(x)}\text{ to find the widest possible }\textcolor{#000099}{x}\text{-interval containing }\cr\textcolor{#000099}{x = 0}\text{ for which no }\textcolor{#000099}{y}\text{-value is repeated.}") → text4
42. LaTeX((((((("\text{For the function }\textcolor{#000099}{f(x) = " * f) * "}\text{, this restricted domain is }\textcolor{#CC0000}{\left[") * leftText) * ",") * rightText) * "\right]}.")) → text5
43. LaTeX((((("\text{Notice that multiplying the " * name) * " function by }\textcolor{#000099}{") * a) * "}\text{ does not affect the}\cr\text{restricted domain.}")) → text6
44. LaTeX("\text{Adjust the window properties to display the graph only for }\textcolor{#000099}{x}\text{-values in the }\cr\text{restricted domain. Make sure you do not change the }\textcolor{#000099}{y}\text{-axis window settings.}") → text7
45. LaTeX("\text{Look at the graph of }\textcolor{#000099}{f(x)}\text{ on its restricted domain, and identify the range of }\cr\textcolor{#000099}{y}\text{-values }\text{that are visible for the function.}") → text9
46. LaTeX((((((((((((((("\text{For the function }\textcolor{#000099}{f(x) = " * f) * "}\text{ on the restricted domain }\textcolor{#000099}{\left[") * leftText) * ",") * rightText) * "\right]}\text{, }\cr\textcolor{#000099}{y}\text{-values on the visible part of the function range from }\textcolor{#000099}{–") * a) * "}\text{ to }\textcolor{#000099}{") * a) * "}\text{.}\cr\text{Therefore, the range of }\textcolor{#000099}{f(x)}\text{ within the restricted domain is }\textcolor{#000099}{\left[–") * a) * ", ") * a) * "\right]}.")) → text10
47. LaTeX((("\text{Divide both sides by }\textcolor{#006633}{" * a) * "}.")) → text19
48. LaTeX("\text{Original function }\textcolor{#000099}{f(x)}") → text13
49. LaTeX("\text{Recall that you can find the inverse of a function }\textcolor{#000099}{f(x)}\text{ by switching the domain }\cr\text{and function variables, renaming the function as }\textcolor{#006633}{g(x)}\text{, and then solving for }\textcolor{#006633}{g(x)}.") → text12