8/2/2019 Developing the Sine Function

1/4

The Sine Regression for Real-life Data 1

Learning Objectives

To make it clear to the students that trigonometry exists outside the classroom.

To become more proficient at creating trig functions to fit a set of data

Investigations Goals

Determine the sine regression to fit the data Based on the equation of a sinusoidal function, students will be able to identify Period,

Amplitude, Horizontal Shift, Vertical Shift,

Investigation

This investigation will explore the relationship between the day of the year and the amount of daylight

using data from the Old Farmers Almanac, 2008

Create a scatter plot of the following data using the day of the year as the independent variable and the

amount of daylight as the dependent variable.

Day of Month

2008

Day of Year Amount of Daylight

(min)

January 6th 6 553

January 20th 20 574

February 3rd 34 604

February 17th 48 640

March 2nd 62 678

March 16th 76 719

March 30th 90 759

April13th 104 798

April 27th 118 835May 11th 132 869

May 25th 146 895

June 8th 160 912

June 22nd 174 917

July 6th 188 909

July 20th 202 890

August 3rd 216 861

August 17th 230 828

August 31st 244 790

September 14th

258 751September 28th 272 711

October 12th 286 673

October 26th 300 634

November 9th 314 600

November 23rd 328 570

December 7th 342 551

December 21st 356 544

Regressions

1. Based on the shape of the graph, what type of regression will be appropriate? (Hint: You might

want to anticipate what the data might look like for the next year or two.) Explain your decisionby providing two aspects of the graph that are unique to this type of equation.

1

8/2/2019 Developing the Sine Function

2/4

2. Use your TI calculator and determine your regression equation. How well does it fit the data?

Explain.

3. Graph the regression equation over your data. Estimate the Period and Amplitude of the

regression graph.

In this activity, we will find the period, amplitude and the phase shift of a sinusoidal curve. Before we

can accomplish this task we must expand our view of the sinusoidal regression because we can only

see the curve for one year. Lets expand the viewing window by raising the x-max to 3 years or 1100

days. Now, we can see that the curve repeats for each calendar year.

Attributes of a Sine Graph Questions:

Period:

Now, lets discuss the period of a curve. The period is the measure of one length of a trigonometric

cycle. To find the period of our sine regression, we need to find two days that have the same length in

time.

4. The easiest values to locate on a periodic graph are the maximums and minimums. Use your

calculator to find the first two maximums or the first two minimums. Go to the CALC menu

and use maximum/minimums to find these x values.

5. Subtract the two consecutive x-values to find the period. What is the period? How do you know

that your answer is correct?

6. If you found the maximum values, find the minimum values or vice versa.

7. Why does the regression equation not give us the exact dates?

2

8/2/2019 Developing the Sine Function

3/4

The Sine Regression for Real-life Data 3

Amplitude:

Next, we want to investigate the amplitude of a sine curve. So, we will find the amplitude by saying

that it is half of the difference between the maximum and minimum of the graph.

1. Using the y values of the maximums and minimums that you found above, determine theamplitude of the graph. Use these two values to find the height of the curve and divide by 2.

What is the amplitude of the graph?

2. Now, look at the regression equation ( ) dcbxay ++= sin . How does your amplitude comparewith the amplitude in the equation?

Horizontal Shift:

Finally, lets consider the Horizontal shift. So, we need to recall where the function y = sin(x) crosses

the y-axis.

3. Using your knowledge of the graph of

y = sin(x), does it cross the y-axis nearer a minimum, a

maximum, or halfway between each? Explain?

4. We need to find this same point on our regression. To do this, change your window to see the

minimum value just left of the y-axis. Find the point that was discussed in the previous

question.

5. How far right of the y-axis is this point? How would you describe the phase shift?

6. What specific day of the year does this phase shift represent? What is the significance of this

date?

3

8/2/2019 Developing the Sine Function

4/4

Vertical Shift:

Finally, lets consider the vertical shift. The vertical shift is the shift up or down. So, we need to recall

where the function y = sin(x) crosses the y-axis.

7. Using your knowledge of the graph of

y = sin(x), does it cross the x-axis nearer a minimum, a

maximum, or halfway between each? Explain?

8. We need to find this same point on our regression. To do this, change your window to see the

minimum value just left of the y-axis. Find the point that was discussed in the previous

question. (This is the same answer to #2 to find the Horizontal Shift.)

9. How far above of the x-axis is this point? How would you describe the vertical shift?

10. Where in the equation ( ) dcbxay ++= sin do you see this value? a, b, c, or d?

Alternative way of finding the Vertical shift.

Use the maximum and minimum values to find the median line to find the value ofd.

4