In Lesson 8A, you considered the characteristics of sinusoidal functions and their corresponding graphs. This lesson introduces sinusoidal functions as mathematical models. Beginning with a realistic situation, you will use a sinusoidal function to construct a regression model. Then, you will use the model to answer questions about the situation.

The next example reviews how to graph a scatter plot and corresponding regression model using the graphing calculator. As you read the example, complete the steps on your calculator. Contact your teacher if you have any difficulty following the examples with your calculator.

Note: It is not necessary to graph a scatter plot before finding a regression equation. It is done in many of the examples in this lesson to give you an idea of the general shape of the graph.

When working with sinusoidal regression ensure that your calculator MODE is in radians. See Lesson 8A for instructions on how to set MODE to radians.

Botanical Beach Provincial Park on Vancouver Island is noted for its striking geological features and abundant shore life. When the tide is out, visitors see a wide variety of live creatures in tidal pools. The following data on water depth were collected on one day. Graph the data and determine the sinusoidal regression equation that best fits the data.

Time heart
5.15
11.32
17.67
23.70
Depth (m)
1.19
2.86
1.11
2.71

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

Note: You can find a regression equation without turning on the STAT PLOT. It is used in this example to give you an idea of the general shape of the graph. To find a 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.

Step 8: Choose the regression model; press STAT, right arrow.

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

In the example above, an estimate of the period is used because there is not enough data to generate a sinusoidal regression equation. You will not be required to do this on your assigned questions in this unit.

Recall from Lesson 7D, the calculator model used to solve logarithmic problems graphically has a different form than the equation used to solve algebraically. The same is true of the sinusoidal regression model.

In Lesson 8A, you studied the characteristics of the sinusoidal equation y = a sin b(x – c) + d. When solving with a calculator or computer the form used most often is y = a sin (bx + c) + d. This is the form you will use throughout Lesson 8B.