CCGPS AA 5.2.2 Example 4: Graphing a Cosine Function

GeoGebra Commands & Parameters

1. Zip(FractionText[a], a, mins) → minlist
2. Zip(LaTeX[(a * "\pi")], a, minlist) → minlistfull
3. Zip(((u * π), c - abs(a)), u, mins) → minpts
4. Zip(LaTeX[((((((((a * "\cos\left((") * b) * ")\left(") * u) * "\right)\right)+(") * c) * ")=") * y(v))], u, minlistfull, v, minpts) → yminlist
5. Zip(((u * π), c + abs(a)), u, maxes) → maxpts
6. Zip(FractionText[a], a, maxes) → maxlist
7. Zip(LaTeX[(a * "\pi")], a, maxlist) → maxlistfull
8. Zip(LaTeX[((((((((a * "\cos\left((") * b) * ")\left(") * u) * "\right)\right)+(") * c) * ")=") * y(v))], u, maxlistfull, v, maxpts) → ymaxlist
9. LaTeX(steps) → counter
10. Simplify((b * x)) → argum
11. FractionText(dl) → domleft
12. FractionText(dr) → domright
13. If(domleft ≟ "-1", LaTeX["-\pi"], LaTeX[(domleft * "\pi")]) → domLeft
14. If(domright ≟ "1", LaTeX["\pi"], LaTeX[(domright * "\pi")]) → domRight
15. LaTeX((((((("\textcolor{#000099}{f(x)=" * f) * "}\text{ over }\textcolor{#000099}{\left[") * domLeft) * ",") * domRight) * "\right]}")) → Problem
16. LaTeX("\text{Identify the amplitude of the function.}") → text1
17. LaTeX("\text{This will help determine the y-axis scale.}") → text2
18. LaTeX("\text{Determine the period of the function.}") → text4
19. LaTeX("\text{Identify any other coefficients or terms that would affect the shape of the graph.}") → text8
20. LaTeX("\text{This is necessary to determine if the graph is translated horizontally or vertically, or if}\cr\text{other mathematical terms affect the graph’s shape.}") → text9
21. LaTeX("\text{This will determine how many cycles of the function show in the given domain.}") → text5
22. LaTeX("\text{Draw and label the axes for your graph.}") → text12
23. LaTeX("\text{The x-axis scale is determined by the given restricted domain.}") → text13
24. LaTeX((((("\text{The x-axis scale will cover at least from }\textcolor{#000099}{" * dl) * "\pi}\text{ to }\textcolor{#000099}{") * dr) * "\pi}.")) → text14
25. LaTeX("\text{The y-axis scale is determined by the amplitude and the vertical shift, if any.}") → text15
26. LaTeX((((("\text{The y-axis scale should cover at least from }\textcolor{#000099}{y = " * Simplify[c - a]) * "}\text{ to }\textcolor{#000099}{y = ") * Simplify[c + a]) * "}.")) → text16
27. Zip(FractionText[a], a, zeros) → zerolist
28. Zip(LaTeX[(a * "\pi")], a, zerolist) → zerolistfull
29. Zip(((a * π), c), a, zeros) → zeropts
30. TableText({zerolistfull}) → zerosnq
31. LaTeX("\text{Determine what values of the restricted domain are the maximum and minimum of the}\cr\text{function value(s). List the points corresponding to these extremes, and plot them on the}\cr\text{graph.}") → text21
32. TableText({maxlistfull}) → maxnq
33. TableText({minlistfull}) → minnq
34. LaTeX("\text{Sketch a smooth curve over the given restricted domain.}") → text46
35. LaTeX((((("\text{The cosine argument is }\textcolor{#000099}{" * argum) * "}\text{. Solve }\textcolor{#000099}{") * argum) * "=2\pi}\text{ for }\textcolor{#000099}{x}\text{ to find the period.}")) → text6
36. LaTeX("\text{Determine the values of the restricted domain for which the cosine value(s) equal 0.}\cr\text{List the points corresponding to these zeros, and plot them on the graph.}") → text17
37. LaTeX("\text{The cosine function is zero when its argument is }\textcolor{#000099}{\frac{\pi}{2}\pm n\pi}\text{ where }\textcolor{#000099}{n}\text{ is an integer, }\textcolor{#000099}{0,1,2,\ldots}.") → text18
38. LaTeX((((("\textcolor{#000099}{" * b) * "x=\frac{\pi}{2}\pm n\pi}\cr\textcolor{#000099}{x=") * zerosnq) * "}")) → text19
39. LaTeX("\text{The cosine function has maximums when its argument is }\textcolor{#000099}{\pm2n\pi}\text{ where }\textcolor{#000099}{n}\text{ is an integer.}") → text22
40. LaTeX("\text{The cosine function has minimums when its argument is }\textcolor{#000099}{\pi\pm2n\pi}\text{ where }\textcolor{#000099}{n}\text{ is an integer.}") → text25
41. TableText({ymaxlist}, "v") → ymaxtext
42. TableText({yminlist}, "v") → ymintext
43. Zip(LaTeX[((((((((a * "\cos\left((") * b) * ")\left(") * u) * "\right)\right)+(") * c) * ")=") * y(v))], u, zerolistfull, v, zeropts) → yzero
44. TableText({yzero}, "v") → yzerotext
45. FractionText(permult) → permulttext
46. LaTeX((((((((((((("\textcolor{#000099}{x=\frac{2}{" * b) * "}\pi}\text{ The period is }\textcolor{#000099}{") * permult) * "\pi}.\cr\text{The number of cycles in the domain is }\textcolor{#000099}{\frac{domain}{period}=\frac{") * domRight) * "-") * domLeft) * "}{") * permulttext) * "\pi}=") * cycles) * "}.")) → text7
47. LaTeX("\text{Move the slider to solve the problem.}") → text0
48. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text47
49. LaTeX((("\text{The corresponding y-values are: }\textcolor{#000099}{" * yzerotext) * "}")) → text20
50. LaTeX((("\text{The corresponding y-values are: }\textcolor{#000099}{" * ymintext) * "}")) → text27
51. LaTeX((((("\text{The coefficient of the cosine function is }\textcolor{#000099}{" * a) * "}\text{; therefore, the amplitude is }\textcolor{#000099}{") * a) * "}.")) → text3
52. LaTeX("\text{The cosine argument has no constant term, so there is no horizontal shift.}") → text10
53. If(c ≟ 0, LaTeX["\text{The function has no constant term, so there is no vertical shift.}"], If[c > 0, LaTeX[(((("\text{The constant term }\textcolor{#000099}{+" * c) * "}\text{ in the function means the cosine function will be shifted up by }\textcolor{#000099}{") * abs(c)) * "}.")], LaTeX[(((("\text{The constant term }\textcolor{#000099}{" * c) * "} \text{ in the function means the sine function will be shifted down by }\textcolor{#000099}{") * abs(c)) * "}.")]]) → text11
54. LaTeX((("\text{The corresponding y-values are: }\textcolor{#000099}{" * ymaxtext) * "}")) → text24
55. LaTeX((((("\textcolor{#000099}{" * b) * "x=\pi\pm 2n\pi} \to \textcolor{#000099}{x=") * minnq) * "}")) → text26
56. LaTeX((((("\textcolor{#000099}{" * b) * "x=\pm 2n\pi} \to \textcolor{#000099}{x=") * maxnq) * "}")) → text23