CCGPS AA 5.2.3 Example 3: Write an Equation to Describe the Graphed Function

GeoGebra Commands & Parameters

1. LaTeX(steps) → counter
2. Segment(A, B) → period
3. Segment(C, A) → amplitude
4. LaTeX("\text{Choose to use either the sine or cosine form.}") → text1
5. If(cmult > 0, LaTeX["\text{ to the right}"], If[cmult < 0, LaTeX["\text{ to the left}"], ""]) → cosshiftText
6. If(sincmult > 0, LaTeX["\text{ to the right}"], If[sincmult < 0, LaTeX["\text{ to the left}"], ""]) → sinshiftText
7. LaTeX("\text{Determine the value of a.}") → text7
8. LaTeX("\text{Determine the value of b.}") → text10
9. LaTeX("\text{Determine the value of c.}") → text16
10. LaTeX("\text{The variable c gives the horizontal shift.}") → text17
11. Vector(E, A) → u
12. Vector(F, D) → v
13. LaTeX("\text{Determine the value of d.}") → text24
14. LaTeX("\text{The variable d gives the vertical shift.}") → text25
15. If(d > 0, " up", If[d < 0, " down", ""]) → vertshiftText
16. If(d > 0, "+", If[d < 0, "-", ""]) → vertshiftSign
17. LaTeX((("\text{The value of d, the vertical shift, is " * d) * ".}")) → text27
18. Vector(((3 * period / 8), 0), ((3 * period / 8), d)) → w
19. LaTeX(("f(x)=" * f)) → text32
20. LaTeX((((((((("f(x)=(" * a) * ")\cos\left[(") * b) * ")(x-") * FractionText[cmult]) * "\pi)\right]+(") * d) * ")")) → text31
21. LaTeX((((((((("f(x)=(" * a) * ")\sin\left[(") * b) * ")(x-") * FractionText[cmult - 0.25]) * "\pi)\right]+(") * d) * ")")) → text35
22. LaTeX(("f(x)=" * g)) → text36
23. Min(0 - a, d - (2 * a)) → ymin
24. Max(0 + a, d + (2 * a)) → ymax
25. LaTeX((((((FractionText[cmult + 2 / b] * "\pi-") * FractionText[cmult]) * "\pi=") * FractionText[2 / b]) * "\pi")) → text13
26. LaTeX("\text{Move the slider to solve the problem.}") → text0
27. LaTeX("\text{This periodic function can be described by a sine function or a cosine function. }\cr\text{Either function will work.}") → text2
28. LaTeX((((((("\text{There is an identifiable rising midpoint at x = }" * FractionText[sincmult]) * "\pi. \cr\text{Thus, this function can be described as a sine function }\cr\text{that has been shifted }") * FractionText[abs(sincmult)]) * "\pi\text{ units}") * sinshiftText) * ".")) → text6
29. LaTeX((((((("\text{There is an identifiable maximum at x = }" * FractionText[cmult]) * "\pi. \cr\text{Thus, this function can be described as a cosine function }\cr\text{that has been shifted } ") * FractionText[abs(cmult)]) * "\pi\text{ units}") * cosshiftText) * ".")) → text4
30. LaTeX("\text{The variable a is the amplitude, or how far the function rises above its midline.}") → text8
31. LaTeX("\text{The period (the length of one cycle of the curve) is equal to }\frac{2\pi}{b}.") → text11
32. LaTeX((((("\text{One cycle of this curve starts at x =}" * FractionText[cmult]) * "\pi\text{ and ends at x = }") * FractionText[cmult + 2 / b]) * "\pi. \cr\text{Subtract to find the length of one cycle of the curve.}")) → text12
33. LaTeX((("\text{The period of the function is }" * FractionText[2 / b]) * "\pi. \cr\text{Substitute this value into the formula for the period of the}\cr\text{function and solve for b.}")) → text14
34. LaTeX("\text{The value of the horizontal shift depends on whether the curve}\cr\text{is viewed as a cosine function or as a sine function.}") → text18
35. LaTeX("\text{In either case, shifts to the right are designated as positive }\cr\text{and shifts to the left are designated as negative.}") → text19
36. LaTeX("\text{Substitute a, b, c, and d into the general form of the cosine or sine function.}") → text28
37. LaTeX("\text{Substitute the known values into the general form of the }\cr\text{cosine function and simplify.}") → text29
38. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → text37
39. LaTeX((((((("\text{The midline is y = }" * d) * "\text{, and the function rises }") * a) * " \text{ unit(s) above its midline. }\cr\text{Therefore, the value of a, the amplitude, is }") * a) * ".")) → text9
40. TableText({{LaTeX["\text{period}=\frac{2\pi}{b}"], LaTeX["\text{Formula for the period}"]}, {LaTeX[(FractionText[2 / b] * "\pi=\frac{2\pi}{b}")], LaTeX["\text{Substitute for the period.}"]}, {LaTeX[(("\left(" * FractionText[2 / b]) * "\pi\right)b=2\pi")], LaTeX["\text{Multiply both sides by b.}"]}, {LaTeX[(("b=\frac{2\pi}{" * FractionText[2 / b]) * "\pi}")], LaTeX["\text{Divide both sides by the period.}"]}, {LaTeX[("b=" * b)], LaTeX["\text{Simplify.}"]}}) → text15
41. LaTeX("\text{As a cosine function:}") → text20
42. LaTeX((("\text{}c=" * FractionText[cmult]) * "\pi")) → text21
43. LaTeX("\text{As a sine function:}") → text22
44. LaTeX((("\text{}c=" * FractionText[cmult - 0.25]) * "\pi")) → text23
45. LaTeX((((((((((("\text{The midline is y = }" * d) * "\text{; thus, the function has been shifted") * vertshiftText) * " ") * abs(d)) * " units (") * vertshiftSign) * "") * d) * ").}")) → text26
46. LaTeX("\text{f(x) = a cos [b(x – c)] + d}") → text30
47. LaTeX("\text{Or, substitute the known values into the general form of the }\cr\text{sine function and simplify.}") → text33
48. LaTeX("\text{f(x) = a sin [b(x – c)] + d}") → text34
49. LaTeX("\text{For cosine form:}") → text3
50. LaTeX("\text{For sine form:}") → text5