Modeling with Circular Functions
Chad Crumley
Essential
Questions: What are some
real-world applications that use circular functions? What is the difference between a model of best fit and a
regression? Using data, give a
sine or cosine equation that matches the graph of the model from the data.
Student
Activity
For this
activity students will need a graphing calculator with a sinusoidal regression
capability. Each student should do
their own work and the teacher should demonstrate how to input the data, how to
make scatterplots using technology, and use the regression feature. Before showing students the regression
equation, allow them to predict their own estimate using the scatterplot. Allow ample time for completion and
discussion especially if tables, scatterplots, and regression analysis have not
been discussed earlier.
The table
below shows the number of hours of daylight in Seattle, Washington, as a
function of the day of the year.
|
Date |
Day
of the year |
Number
of hours of daylight |
|
15-Jan |
15 |
8.88 |
|
15-Feb |
46 |
10.3 |
|
15-Mar |
74 |
11.85 |
|
15-Apr |
105 |
13.62 |
|
15-May |
135 |
15.1 |
|
15-Jun |
166 |
15.95 |
|
15-Jul |
196 |
15.58 |
|
15-Aug |
227 |
14.27 |
|
15-Sep |
258 |
12.6 |
|
15-Oct |
288 |
10.92 |
|
15-Nov |
319 |
9.33 |
|
15-Dec |
349 |
8.48 |
1.
Using
the graphing calculator, enter the day of the year as x and the number of hours of
daylight as y.
2.
Use
the technology to make a scatterplot.
3.
Predict
a sine equation to fit the scatterplot by estimating the period, amplitude,
phase shift, and vertical shift.
Graph your estimate on the same graph as the scatterplot.
4.
Perform
a sine regression and compare the period, amplitude, phase shift, and vertical
shift to your estimates. Graph the
regression along with your estimate and the scatterplot.
5.
Is
the regression a better fit and does it make sense?
6.
Using
the estimate equation and the regression, find the date on which the number of
hours of daylight is the greatest (the summer solstice) and find the number of
hours of daylight on that day.
7.
Find
the percent error between these two.
The following two labs
are similar to the Tuning Fork lab earlier in the unit. These labs allow the student to work extensively
with sinusoidal data. Then they
curve fit the data to a sine or cosine function using information learned
throughout the unit. The labs
below are collaborative and work best for groups of 3 or 4.
For a Pendulum lab, click here.
For a Slinky lab, click here.
Student
Activity
This activity is a
collaborative activity and works best with a group of 2 or 3.
From an almanac or other
weather reference, find the times of sunrise and sunset for various dates in a
particular city for a particular year.
1.
Make
a scatterplot of date versus time of sunrise, and date versus time of
sunset. Sketch curves of good fit
through the data. Find an equation
to model each set of data.
2.
How
would Daylight Savings Time affect a graph of time of sunset or sunrise?
3.
From
the equations of sunset and sunrise in relation to date, how could you
determine an equation for the number of hours of daylight in relation to the
date?
4.
How
does the latitude or longitude of a city affect the time of sunset or
sunrise? Collect other data and
supply theoretical evidence to support your answer.
For additional support and resources check the following references.
References
Adcock, Rebecca. (2006) Beginners Trigonometry in 6
Lessons. The University of Georgia: EMT 6690
Hungerford,
Thomas W. (2000) Contemporary
Precalculus: A Graphing
Approach. Chicago:
Harcourt, Inc. 3rd Ed.
Senk S.
et al. (1998) Functions, Statistics and Trigonometry. The University of Chicago School
Mathematics Project. Scott Foresman Addison Wesley.