Lesson 4

Lesson 4: Modelling Data with Sinusoidal Functions

 

Focus

 

left: Jupiterimages/Photos.com/Thinkstock, right: iStockphoto/Thinkstock

 

The Canadian lynx’s most important source of food is the snowshoe hare. When a predator has only a single significant source of food, the two populations follow a somewhat sinusoidal pattern as shown in the diagrams. Why does this cyclical pattern occur?

 

How reasonable is it to use a sinusoidal curve to model this data? Do you think this trend would continue for a long time?

 

 

 

In this lesson you will explore various ways of using sinusoidal functions to represent data. You will then use these functions to help interpret the data.

 

Lesson Outcomes

 

At the end of this lesson you will be able to

• determine a sinusoidal curve that best represents a set of data
• interpret data using a sinusoidal curve of best fit

Lesson Question

 

You will investigate the following question: How can sinusoidal data be represented using a sinusoidal function?

 

Assessment

 

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 4 Assignment (Download the Lesson 4 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms
• work under Project Connection

Materials and Equipment

• graphing software or graphing calculator

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

 

Before beginning this lesson, you should be able to

• draw a scatter plot given a set of points
• use technology to produce a scatter plot given a set of points
• draw a line or curve of best fit for a scatter plot
• interpolate and extrapolate information given a set of data points

Are You Ready?

 

Complete these questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. Fill in the blanks.
   1. The independent variable is located on the ___________ axis. This variable does not depend on the other variable.
   2. The dependent variable is located on the ___________ axis. This variable depends on the other variable.
   3. To estimate a value of a function between two known values is called ___________.
   4. To estimate a value of a function beyond given known values is called ___________.
   
   Answers
2. Julie gathered information about her age and height from the markings on the wall in her house.
   
   

   Age (yr)	1	2	3	4	5	6
   Height (cm)	78	81	90	98	107	120

   1. Label the horizontal and vertical axis.
   2. Draw a scatter plot.
   3. Draw a curve of best fit. (It may be a line or a smooth curve.)
   4. Describe the trend in the data.
   5. Predict how tall Julie was at 2.5 yr.
   6. How tall was she at 9 yr?
      
      
   Answers

If you answered the Are You Ready? questions without difficulty, move to Discover.

 

If you found the Are You Ready? questions difficult, complete Refresher.

Refresher

 

“Solving Using Trend Lines” reviews how lines of best fit can be used to help interpret data.

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“Fitting a Line to Data” shows how data can be plotted and given a line of best fit using Excel.

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Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

 

Try This 1

 

iStockphoto/Thinkstock

 

In this part of the world, summer days have more hours of daylight and winter days have fewer hours of daylight. But does the number of hours of daylight follow a particular pattern? Consider the following information.

 

Date	Hours of Daylight in Medicine Hat
January 31	9.1
February 28	10.7
March 31	12.7
April 30	14.5
May 31	15.8
June 30	16.1
July 31	15.1
August 31	13.3
September 30	11.5
October 31	9.7
November 30	8.2
December 31	7.9

1.  
   1. Explain any patterns you see from the chart about daylight hours.
   2. What do you think the data will be like in the following year?
2.  
   1. Create a scatter plot of this data with the month on the x-axis and the hours of daylight on the y-axis.
   2. Draw a curve of best fit for the data.
   3. How would you describe the shape of your curve of best fit? Explain.
   4. How well does the curve of best fit represent your points?
3. Assume the data followed the same pattern and continued for multiple years. Estimate the period, amplitude, and midline. What does each period, amplitude, and midline represent in the context of the given data?

Save your responses in your course folder.

 

Share 1

 

With a partner or in a group, discuss the following questions based on your graphs created in Try This 1.

1. Is it reasonable to use the curve of best fit to predict the hours of daylight on February 19 of this year? How about February 19 seven years from now?
2. When graphing the data, you assumed each month was the same length, but this is not true. Comment on how much you expect this simplification to affect your graph.

If required, save a record of your discussion in your course folder.

Explore

 

In Try This 1, you saw that it is possible to sketch a curve of best fit for sinusoidal data. You can then use the curve of best fit to predict the parameters of a sinusoidal equation. In the daylight example, both interpolation and extrapolation were reasonable because the data followed a sinusoidal pattern and you expect each year to look similar. Care needs to be taken when determining and interpreting a line or curve of best fit. You will explore this line or curve of best fit matter further in Try This 2.

 

Try This 2

 

A tide graph predicts the height of water at a particular location at particular times. Consider the following tide graph for Vancouver from October 7 to 11, 2011.

 

SOURCE: Tide, Currents, and Water Levels, (Fisheries and Oceans Canada, 2011), <http://www.tides.gc.ca/>; (09/02/2012).

1. Explain why a sinusoidal model is reasonable for this data.
2. Use Tides Model 1 to determine an equation that matches the data. Adjust the parameters a, b, c, and d until the graph overlaps the points as closely as possible. Record your equation.
   
   
    
   
3. Three points on the original scatter plot are (10, 1.9), (50, 3.0), and (100, 3.3). Calculate the y-value, to the nearest tenth, for the times of 10, 50, and 100 using your equation from question 2. When you’re making the calculations, be sure your calculator is in radian mode. Use a chart similar to the following one to organize the information. How close are the values predicted using an equation to the actual points?
   
   

   x-value	y-value Using Equation Found in Question 2	Actual y-value
   10		1.9
   50		3.0
   100		3.3

4. At midnight on January 6, 2012 (2184 h after midnight on October 7), the tide height was 1.9 m. How close does your equation predict this value?
5. Consider the tide graph from the same location from November 16 to 20, 2011. What characteristics of this graph are similar to a sine graph? What characteristics are different?
   
   
   SOURCE: Tide, Currents, and Water Levels, (Fisheries and Oceans Canada, 2011), <http://www.tides.gc.ca/>; (09/02/2012).
   
6. Use Tides Model 2 to try to model the data using an equation of the form
   
   
    
   
7. Is it reasonable to use an equation of the form to model this data? Explain.

Save your answers in your course folder.

 

Share 2

 

With a partner or in a group, discuss the following questions based on what you learned in Try This 2.

1. How can you decide when a sinusoidal curve should be used to model data?
2. Why are calculated values in question 3 of Try This 2 not the actual values that were given in the chart?
3. What advantage is there in using an equation to represent data?
4. Is it reasonable to use interpolation to predict values using your first graph? What about extrapolation? Explain.

If required, save a record of your discussion in your course folder.

In Try This 2, you saw that it is reasonable to use a sinusoidal curve of best fit to represent a set of data and make predictions for the October 7 to 11 data. However, you need to be careful about when to use a sinusoidal curve of best fit. Not all periodic data can be modelled nicely using a sinusoidal curve; the distribution of the scatter plot will help you decide if a sinusoidal model is reasonable.

 

Interpolation will usually yield a reasonable result, but you need to be very careful with extrapolation. The data pattern will need to be consistent to use extrapolation, and small errors in the model can create large extrapolation errors if the prediction is far outside the data range.

 

Self-Check 1

1. A clock with a pendulum sits above a counter. The height of the pendulum above the counter is measured at various time intervals.
   
   

   Time (s)	0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0
   Height (cm)	22.0	16.0	10.0	16.0	22.0	16.0	10.0	16.0	22.0

   1. Plot the data.
   2. Sketch a curve of best fit for the data.
   
   

   Answers

2. What is the period of the graph? What does it represent in this scenario? Answers
3. Use your graph to predict the height of the pendulum at
   1. 1.7 s
   2. 12.5 s
   
   Answers
4. How accurate do you expect your predictions from question 3 to be? Explain. Answers

Although creating a curve of best fit by hand can be done, it is often easier and more accurate to create a curve by using a graphing calculator or some other graphing technology. You can also use this technology to find the regression equation that best fits the data given. A regression is actually a statistical analysis that assesses the association between two variables. It is very difficult to calculate a regression by hand, so you can use technology to help.

 

Try This 3

 

Hagan is performing an experiment with a spring attached to a weight. She places a distance sensor 50 cm below the weight. She then lifts the weight 10 cm and turns on the distance sensor as she drops the weight.

 

hand: iStockphoto/Thinkstock

 

Hagan received the following information from the sensor.

 

Time (s)	Height (cm)
0	60.0
0.2	57.7
0.4	53.2
0.6	46.9
0.8	41.9
1.0	40.0
1.2	42.8
1.4	50.9
1.6	56.9
1.8	59.6
2.0	59.4

Time (s)	Height (cm)
2.2	53.2
2.4	46.0
2.6	41.7
2.8	39.8
3.0	43.2
3.2	51.5
3.4	58.1
3.6	59.8
3.8	58.2
4.0	51.4

 

If you are unfamiliar with using your calculator for data entry and/or data regression, you may need to consult the calculator’s manual or contact your teacher for assistance with this activity.

1. Make a scatter plot of the data using a graphing calculator.
2.  
   1. Use your calculator to determine a sinusoidal regression equation for the data. Record this equation.
   2. Graph the regression equation on the same grid as your scatter plot. How well does the regression equation match the data?
3. What is the period of this graph? Explain how you determined the period.
4.  
   1. Determine the height at 1.34 s.
   2. Determine the height at 200.0 s.
   3. Determine the first three points where the weight is at a height of 55.0 cm.

Save your responses in your course folder.

 

Share 3

 

With a partner or in a group, discuss the following questions based on your graph created in Try This 3.

1. How did your strategies compare for determining the measures in question 4?
2. How accurate do you expect your heights from questions 4.a. and 4.b. to be? Explain.

If required, save a record of your discussion in your course folder.

In Try This 3, you saw that it is possible to use a sinusoidal regression with a graphing calculator to quickly and fairly accurately find a curve of best fit for sinusoidal data. You may have also found that it is possible to determine points on the curve.

 

Consider the following. Suppose you have a regression equation of where x represents time and y represents height. If you want to determine the times when y = 4.5, it is possible to substitute y = 4.5 into the equation and solve as follows:

 

 

But this process is somewhat complicated. An easier strategy is to graph the equations and y = 4.5 and then find the x-value of the intersections.

 

 

From the intersections in the graph, you can see that possible x-values include 0.99, 7.92, 13.56, 20.49, and 26.13.

Connect

 

Lesson 4 Assignment

Lesson 4 Summary

 

 

Curve fitting is the general method for using a line or curve to estimate the relationship between two variables. Throughout Lesson 4 you used a curve of best fit for sinusoidal data, which is periodic and looks like a sine wave when graphed. The curve determines the specific parameters that make the sinusoidal function match your data as closely as possible. After finding the sinusoidal regression function, interpolation and extrapolation can be used. However, care must be taken to make sure the predictions are reasonable.