The following table shows the hours before sunrise for Calgary, Alberta, at the beginning of each month in 2011. Find the sinusoidal regression function that best models the data and use it to complete the following questions.

  1. On what date did the sun rise the earliest?
  2. How many days of the year 2011 did the sun rise before 7:30 a.m.?

Day Number
2011
Hours Before Sunrise
Day Number
2011
Hours Before Sunrise
1
Jan. 1
8.77
183
July 1
4.20
32
Feb. 1
8.30
214
Aug. 1
4.63
60
Mar. 1
7.30
245
Sept. 1
5.54
91
Apr. 1
6.13
275
Oct. 1
6.66
121
May 1
5.08
306
Nov. 1
7.78
153
June 1
4.35
335
Dec. 1
8.54

Step 1: Turn on the STAT PLOT; press 2nd, Y=, ENTER, ENTER.

Note: To find the sinusoidal regression equation only, omit steps 1, 2, 3, and 7.

Step 2: Clear the functions; press Y= then, CLEAR for each function that must be deleted.

Step 3: Select values for the viewing window; press WINDOW. Use the keypad to type the values from the table. Use the up and down arrow keys to scroll through the list.

Step 4: Go to the lists; press STAT, ENTER.

Step 5: Clear the lists; press the up arrow until the column heading is highlighted; then, CLEAR, ENTER. Repeat Step 5 for all columns that have unneeded data. Use the left and right arrow keys to move between columns.

Step 6: Enter the data.

Step 7: View the data; press GRAPH.

In Step 10, an estimate of the period is used to improve the accuracy of the sinusoidal regression equation. You will not be required to do this on your assigned questions in this unit.

Step 11: View the scatter plot and regression model; press GRAPH.

The sinusoidal regression equation is y = 2.3297 sin(0.0165x + 1.8062) + 6.5145.

You must have this sinusoidal regression equation in the function area to complete the problem.


  1. To find the earliest sunrise means to find the minimum value of the graph.

Step 1: Clear all unnecessary functions; press Y= then, CLEAR for each function that needs deleting. Do not clear the sinusoidal regression.

Step 2: Go to the CALCULATE menu; press 2nd, TRACE.

Step 3: Find the minimum; press 3, left arrow to scroll to the left of the minimum point, ENTER, right arrow to scroll to the right of the minimum point, ENTER, ENTER.

The earliest sunrise time is 4.185 hours on the 176th day. This can be restated as 4:11 a.m. on June 25th.

  1. To find how many days of the year 2011 the sun rose before 7:30 a.m. means to find how many x-values fall below y = 7.5. (30 minutes = 0.5 hours)

Step 1: Input the given y-value; press Y=, down arrow to scroll to Y2= 7.5.

Step 2: Graph the functions; press GRAPH.

Step 3: Find the first intersection point; press 2nd, TRACE, 5, ENTER, ENTER, ENTER.

Step 4: Find the second intersection point; press 2nd, TRACE, 5, left arrow to scroll to the second intersection point, ENTER, ENTER, ENTER.

The number of days with sunrise before 7:30 a.m. (7.5 hours) is 298 – 54 = 244 days.