1. MAT 150 Unit 2-4 Part 2: Power Models

2. Objectives  Model data using power functions  Use first and second differences and visual comparison to determine if a linear or quadratic function is the better fit to the set of data.  Determine whether a quadratic or power function gives better fit to a given set of data

3. Comparison of Linear and Quadratic Models Use first and second differences and visual comparison to determine if a linear or quadratic function is the better fit to the set of data. Recall that when the changes in inputs are constant and the (first) differences of the outputs are constant or nearly constant, a linear model will give a good fit for the data. Same can be said when comparing the differences of the first differences, which are called the second differences.

4. Consider the data in Table 3.14, which gives the measured height y of a toy rocket x seconds after it has been shot into the air from the ground. Is the function exactly, approximately or not quadratic? Time x (seconds) Height (meters) 1 st Differences 2 nd Differences 168.6 2117.6 3147 4156.8 5147 6117.6 768.6

5. Example The figure and table show that the percent of the U.S. adult population with diabetes (diagnosed and undiagnosed) is projected to grow rapidly in the future.

6. Example (cont) a. Find the power model that fits the data, with x equal to the number of years after 2000. b. Use the model to predict the percent of U.S. adults with diabetes in 2015. Solution

7. Example (cont) c. In what year does this model predict the percent to be 29.6? Solution c.

8. Example Using the data in the table below, find both a linear function and a power function to model the data. Which is better fit for the data? XY 14 29 311 421 532 645

9. Example: The following table gives the monthly insurance rates for $100,000 life insurance policy for smokers 35-50 years of age. Age (yr)Monthly Insurance Rate($) Age (yr)M51onthly Insurance Rate ($) 3517.324323.71 3617.674425.11 3718.024526.60 3818.464628.00 3919.074729.40 4019.954830.80 4121.004932.55 4222.225034.47 a.Create a scatter plot for the data using your calculator.

10. The following table gives the monthly insurance rates for $100,000 life insurance policy for smokers 35-50 years of age. b. Find a quadratic function to model the data. c. Find a power function to model the data. d. Compare the two models by graphing each model on the same axes with the data points. Which model appears to be the better fit?