1. CHAPTER 1: FUNCTIONS, GRAPHS, AND MODELS; LINEAR FUNCTIONS Section 1.6: Fitting Lines to Data Points: Modeling Linear Functions 1

2. SECTION 1.6: MODELING LINEAR FUNCTIONS How do we come up with equations to model a set of points? We find a line of ‘ best fit ’ The table below give the number of full- and part- time employees and clinics of dentists for selected years between 1970 and 1998. 2 Year1970197519801985199019951998 Employees (in thousands)222331415480580644666

3. SECTION 1.6: MODELING LINEAR FUNCTIONS Draw a scatterplot of the data: x is the year since 1970 and y is the number of employees (in thousands) Graph each of the following equations with the data y 1 = -12x + 660 y 2 = 13x + 220 y 3 = 16x + 42 Which is the best fit? 3

4. SECTION 1.6: MODELING LINEAR FUNCTIONS The line of ‘ best fit ’ is found using the procedure of linear regression. Typically, the regression employs the least-squares method. reduces the some of the squares of the distance between the data points and line How do we get this line? Luckily, the calculator does it for us. 4

5. SECTION 1.6: MODELING LINEAR FUNCTIONS Linear Regression on the calculator Enter your data into the STAT List Check the scatterplot – would a line be a reasonable fit? Find the Linear Regression STAT  CALC  4: LinReg(ax+b) To save in your Y= menu: same as above, then VARS  YVARS  Y 1 Graph the scatterplot and Linear Regression together ZOOM  9: ZoomStat Does the line seem to fit? Report your equation (from Y 1 ) in an appropriate fashion. Try this for the employee data. What do you find? 5

6. SECTION 1.6: MODELING LINEAR FUNCTIONS The table below shows the earnings of year-round full-time workers by gender and educational attainment. Create a linear model that expresses females ’ annual earnings as a function of males ’ earnings. Pay attention to units!! 6 Educational Attainment Average Annual Earnings for Males ($ thousand) Average Annual Earnings for Females ($ thousand) < 9 th grade18.74312.392 Some high school18.90812.057 High school grad.30.41418.092 Some college33.61420.241 Associate ’ s degree40.04725.079 Bachelor ’ s degree or more66.81036.755

7. SECTION 1.6: MODELING LINEAR FUNCTIONS A scatter plot is a way to represent a discrete function – when there are a finite number of inputs. series of dots When we ‘ fit ’ a scatter plot with a function, we are using a continuous function to describe the data continuous curve Often we use the continuous function to determine other data that are not given. interpolate – find a value within the given domain extrapolate – find a value outside of the given domain 7

8. SECTION 1.6: MODELING LINEAR FUNCTIONS The average math SAT scores in Beaufort County, SC are given in the table below. Write the equation of the line that is the best fit for these data, aligning the data so that x = 0 in 1990. Compare the outputs from the equation with the original data for each of the years 1994 to 1999 and determine the year in which the model output is closest to the data value. 8 Year199419951996199719981999 Avg. Math SAT Score472464470471473470

9. SECTION 1.6: MODELING LINEAR FUNCTIONS According to your model, what would you predict the average Math SAT score to be in 2000. Do you think the prediction is accurate? Why or why not? 9 Year199419951996199719981999 Avg. Math SAT Score472464470471473470

10. SECTION 1.6: MODELING LINEAR FUNCTIONS Homework: pp. 94-101 1-7 odd, 13-16 all, 17, 19, 21, 27, 29, 33, 37, 41, 45 10