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 $$\displaystyle \text{period} = {2\pi \over\displaystyle \text{frequency}}$$, 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 pi.

Function Notation

f(x) = amplitude × sin( frequency (x - horizontal shift) ) + vertical shift


Pyret Notation

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 blue, which uses cos instead of sin.

5 Adjust the sliders so that the function q perfectly overlaps the function p. 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 pi?

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 pi.

Function Notation

g(x) = amplitude × cos( frequency (x - horizontal shift) ) + vertical shift


Pyret Notation

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). CCbadge Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.