Modeling with sin
For this section, use Slide 2: "Modeling the Ferris Wheel Dataset (sin)" of the Exploring Periodic Functions Desmos File. You’ll find the data from the Ferris Wheel plotted in red, along with a basic periodic model of the form f(x) = a sin(b(x - h)) + k.
1 Use the sliders to estimate the best periodic fit.
2 The peaks are at feet, troughs are at feet, midline is at feet and the amplitude is feet
3 The period of the data is minutes. If $$\displaystyle \mbox{period} = {2\pi \over\displaystyle \mbox{frequency}}$$, what is the frequency? cycles per minute
4 Adjust the slider for horizontal shift to find the best fit, then write your model below in Function and Pyret notation. Express h in terms of π.
Function Notation |
f(x) = amplitude × sin( frequency (x - horizontal shift) ) + vertical shift |
|
fun f(x): ( * sin( * (x - ))) + end |
Translating from sin to cos
For this section, advance to Slide 3: "Translating from sin to cos" of the Exploring Periodic Functions Desmos File. You’ll see a function f(x) defined here graphed in red, which uses cos instead of sin.
5 Adjust the sliders so that the function g perfectly overlaps the function f. What is the value of a? b? k?
6 What was the value of h, expressed as a decimal? What was the value of h, expressed a fraction of π?
7 Change the definition of p in Desmos to math row 1 below and adjust the definition of q to match the new curve. Complete the second row by changing the definition of p in Desmos again and adjust q again.
Function using sin |
Function using cos |
Vertical Shift k |
p(x) = 10 sin(1⋅(x - 0)) + 2 |
q(x) = |
|
p(x) = |
q(x) = |
8 Do you think that all basic cosine functions can be expressed as sine functions? Why or why not?
Modeling with cos
For this section, advance to Slide 4: "Modeling the Ferris Wheel Dataset (cos)" of the Exploring Periodic Functions Desmos File.
9 Translate your sin-based model to a cos-based one. Express the horizontal shift in terms of π.
Function Notation |
g(x) = amplitude × cos( frequency (x - horizontal shift) ) + vertical shift |
|
fun g(x): ( * cos( * (x - ))) + end |
These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). 
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