Piecewise modeling with multiple functions: Federal Debt

**Name:** Teacher** **

**Time
period:** 1989 to 1997

*What function type do you think best approximates your data?*

Linear*Why did you select this function type? Be sure to include difference tables and r*^{2}values in your explanation.

Question 2. Federal debt Year Debt First difference $ millions 1989 379,993 26,613 1990 406,606 37,951 1991 444,557 31,547 1992 476,104 38,253 1993 514,357 43,247 1994 557,604 38,273 1995 595,877 39,062 1996 634,939 16,185 1997 651,124

The r^{2}value of the least-squares line (line of best fit) is 1.00, which means the data are approximated by a linear model very well. Although the first differences are not exactly equal, this is to be expected because we are using real data. The first differences are close though, and the data are best approximated by a linear model.*What is the basic form of the equation (e.g., y = mx + b) for the type of function you selected?*

y = mx + b*What does each variable represent in the equation? (e.g., What do***m**and**b**represent in a linear equation?)

**m**represents the slope of the line.

**b**represents the y-intercept of the line.*For this dataset, what does each variable represent in the real world?*

For this dataset,**m**represents the rate at which the federal debt is increasing or decreasing annually whereas**b**represents the value of the federal debt in 1989, the first year of the period.*What are your best values for each variable?*

My best value for**m**is 35,900 and my best value for**b**is 374,000.*What do the values of these variables mean in the real world?*

The**m**value of 35,900 means that the federal debt is increasing at a rate of $35,900,000 per year, whereas the**b**value of 374,000 means that the value of the federal debt in the first year is $374,000,000. (Note: Federal debt values are provided in thousands of dollars.)*Write your equation here, with your best values inserted.*

y = 35,900x + 374,000*Consult with the students who analysed the data for the time periods immediately before and after your assigned time period. Compare their data and functions with yours. What are the similarities and differences?*

Previous time period: 1975 to 1988

In this time period, the debt was best modelled by a quadratic or an exponential function, which means it changed more rapidly each year than during my time period. However, in both their and my time period, the debt is increasing.

Next time period: 1998 to 2008

In this time period, debt decreased quadratically, whereas in my period, debt increased linearly. In the next time period, the data points do not follow the function model as well as the data in my time period.*What historical events may have contributed to the shape of your graph, and how?*

Answers vary.*Print a copy of your graph (with the function model and best values for your variables showing) and hand it in with your answers to the above questions.*

**Fathom**graph

(Fathom is licensed by the Ministry of Education and used by schools in some provinces. Providing the Fathom format is in no way an endorsement or recommendation of the Fathom software by Statistics Canada.)

*Note: this is the least squares line calculated by Fathom.*

**Excel**graph

To generate the graph below, select**XY (Scatter)**and**Add Trendline (linear)**.

(Providing the Excel format is in no way an endorsement or recommendation of the Excel software by Statistics Canada.)