Activity 7.5 Trigonometric Models

Overview:

There are two activities – one in-class activity and one out-of-class project – associated with trigonometric models. Activity 7.5a explores the graphical effects of adding periodic functions and how the graph of the sum is related to the graphs of the addends. It also has a question that requires students to write the equation that models the damped harmonic motion of shock absorbers.

Activity 7.5b was designed to be an out-of-class project.

Estimated Time Required:

Activity 7.5a should take about 30 minutes.

Activity 7.5b should take from 1 to 2 hours.

Technology:

Graphing calculator (optional for Activity 7.5a)

Activity 7.5b requires Internet access.

Prerequisite Concepts:

  • Sums of trigonometric function
  • Damped oscillations
  • Modeling periodic function

Discussion:

The point of Activity 7.5a is to expose students to some of the rich variety of functions that are possible by combining suitable elementary functions.

Obtaining the graphs of complicated functions is easy using graphing calculators. You can encourage some experimentation by having the students graph several examples fo f+g, where f(x) = A cos Bx and g(x) = C sin Dx, with A, B, C, and D integers that are reasonably close to zero.

Activity 7.5b has two parts: gathering data and modeling the data. In the gathering data part, students are asked to find the number of daylight hours at 20, 40 and 60 latitude for different days of the year. They are then asked to create a scatter plot of the data and find amplitude, period, phase shift, vertical shift, and write the equation that corresponds to each set of data. They are asked to compare the three graphs and note which features are the same and which are different and to describe why the various constant were the same or different. Finally, they are asked to describe the significance of June 21, March 21, September 21, and December 21.

Activity 7.5a Trigonometric Models

1.Given and , let .

a.) Determine the amplitude and period of both f and g.

b.) Graph f and g on the same set of axes.

c.) Do you think h is a periodic function? If so, what do you think its period and amplitude will be?

d.) Now graph h. Does it behave the way you predicted?

2.Consider the family of functions .

a.) What will the graph of f look like if g is a constant function? Why?

b.) What will the graph of f look like if g is a linear function? Why?

c.) Graph at least 1 example of each of the types of functions in parts (a) and (b).

3.Shock absorbers have to oscillate (in order to absorb the shock of going over bumps). But the oscillations also have to dissipate (decrease in amplitude) so that the vehicle will stop bouncing around.

a.) Brand X shock absorbers oscillate, completing 4 cycles in one minute. Write a function that shows this behavior.

b.) Just after going over a bump, the amplitude of the oscillation is 2 cm. The amplitude decreases so that after 2 minutes, the amplitude is only 1/8 cm. Assuming that the amplitude decreases exponentially, write a formula for a function that shows this behavior.

c.) Combine your answers to parts (a) and (b) to find and graph a formula for a function that shows the dissipating oscillations.

Activity 7.5b Trigonometric Models

You can calculate the sunrise, sunset, twilight, moonrise, moonset and twilight times for any location if you know the longitude and latitude. Longitude is in degrees, minus (-) for East of Greenwich (Europe), plus (+) for West (as in USA). Latitude is in degrees, plus (+) for North of the equator, minus (-) for South. In this activity we will be looking at how the number of hours of daylight changes throughout a year.

There are two parts to this project, gathering data and modeling the data. **You will need Internet access for the data-gathering portion and a graphing calculator for the modeling portion.

Gathering Data:

You will need to complete the following tables below by finding the number of hours of daylight at the specified Latitude (see tables below). Feel free to share data with your classmates. Use the daylight calculator below to complete the tables:

20 North Latitude
Date / Day of Year / Hours of Daylight / Date / Day of Year / Hours of Daylight
1/1 / 1 / 7/14 / 195
1/15 / 15 / 7/29 / 210
1/30 / 30 / 8/13 / 225
2/14 / 45 / 8/28 / 240
3/1 / 60 / 9/12 / 255
3/16 / 75 / 9/21 / 264
3/21 / 80 / 9/27 / 270
3/31 / 90 / 10/12 / 285
4/15 / 105 / 10/27 / 300
4/30 / 120 / 11/11 / 315
5/15 / 135 / 11/26 / 330
5/30 / 150 / 12/11 / 345
6/14 / 165 / 12/21 / 355
6/21 / 172 / 12/26 / 360
6/29 / 180
40 North Latitude
Date / Day of Year / Hours of Daylight / Date / Day of Year / Hours of Daylight
1/1 / 1 / 7/14 / 195
1/15 / 15 / 7/29 / 210
1/30 / 30 / 8/13 / 225
2/14 / 45 / 8/28 / 240
3/1 / 60 / 9/12 / 255
3/16 / 75 / 9/21 / 264
3/21 / 80 / 9/27 / 270
3/31 / 90 / 10/12 / 285
4/15 / 105 / 10/27 / 300
4/30 / 120 / 11/11 / 315
5/15 / 135 / 11/26 / 330
5/30 / 150 / 12/11 / 345
6/14 / 165 / 12/21 / 355
6/21 / 172 / 12/26 / 360
6/29 / 180
60 North Latitude
Date / Day of Year / Hours of Daylight / Date / Day of Year / Hours of Daylight
1/1 / 1 / 7/14 / 195
1/15 / 15 / 7/29 / 210
1/30 / 30 / 8/13 / 225
2/14 / 45 / 8/28 / 240
3/1 / 60 / 9/12 / 255
3/16 / 75 / 9/21 / 264
3/21 / 80 / 9/27 / 270
3/31 / 90 / 10/12 / 285
4/15 / 105 / 10/27 / 300
4/30 / 120 / 11/11 / 315
5/15 / 135 / 11/26 / 330
5/30 / 150 / 12/11 / 345
6/14 / 165 / 12/21 / 355
6/21 / 172 / 12/26 / 360
6/29 / 180

Modeling:

Follow the steps below using your graphing calculator:

1.Enter the day of the year in L1.

2.Enter the number of hours of daylight at 20 Latitude in L2.

3.Graph the scatter plot of L1 and L2 using the big dot. Sketch the plot on the grid below. Be sure to label and number your axes.

4.Find the amplitude, period, phase shift, and vertical shift of this graph.

Amplitude:Period:

Phase Shift: Vertical Shift:

5.Write the equation of the function that models this data:

Enter this equation into Y1 and graph the equation with the scatter plot. (Your equation should “fit” the scatter plot.)

6.Enter the number of hours of daylight at 40 Latitude in L3.

7.Graph the scatter plot of L1 and L3 using the + as the mark. Sketch the plot on the grid below. Be sure to label and number your axes. Use the same window as in part 3.

8.Find the amplitude, period, phase shift, and vertical shift of this graph.

Amplitude:Period:

Phase Shift: Vertical Shift:

9.Write the equation of the function that models this data:

Enter this equation into Y2 and graph the equation with the scatter plot. (Your equation should “fit” the scatter plot.)

10.Enter the number of hours of daylight at 60 Latitude in L4.

11.Graph the scatter plot of L1 and L4 using the small dot. Sketch the plot on the grid below. Be sure to label and number your axes. Use the same window as in part 3.

12.Find the amplitude, period, phase shift, and vertical shift of this graph.

Amplitude:Period:

Phase Shift: Vertical Shift:

13.Write the equation of the function that models this data:

Enter this equation into Y3 and graph the equation with the scatter plot. (Your equation should “fit” the scatter plot.)

14.Graph all three scatter plots and equations in the same viewing window. Sketch the plot in the window below.

15.Compare these graphs and answer the following questions: What features are the same? What features are different? What parameters (A, B, h, k) have the same values? Why is this true?

16.What is the significance of June 21, March 21, September 21, and December 21?