In Lesson 6A, you considered the characteristics of exponential functions and their corresponding graphs. This lesson introduces exponential functions as mathematical models. Beginning with a realistic situation, you will use an exponential function to construct a regression model. Then, you will decide whether the model is appropriate for the given data. If it is appropriate, 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.

Sarah did not feel well, so she visited her family doctor. The doctor thought she might have strep throat and sent a bacterial culture from Sarah's throat to the laboratory to confirm her suspicions. A lab technician recorded the growth of the bacterial population in the culture at fixed time intervals. The first five observations are given in the table below. Determine the exponential regression model for this data.

Observation
1
2
3
4
5
Number of Bacteria
6 180
9 270
13 905
20 857
31 285


*Note: You can find a regression equation without turning on the STAT PLOT. To find a regression equation only, omit steps 1, 2, 3, and 7

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

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 9: Select the exponential regression; press 0 (zero).

Step 10: Place the exponential regression into the function area; press VARS, right arrow, ENTER, ENTER, ENTER.

Step 11: View the scatter plot and regression model; press GRAPH.
Because the regression curve goes through all data points, this
model is an excellent representation of the information in the table.

The exponential regression model is y = 4120.1(1.5)x.

In Example 1, Step 10 is a shortcut to place the exponential regression into the function area, Y=. An alternate method for doing this step is demonstrated in the video on the next page.