1. Linear Functions and Models Lesson 2.1

2. Problems with Data Real data recorded Experiment results Periodic transactions Problems Data not always recorded accurately Actual data may not exactly fit theoretical relationships In any case … Possible to use linear (and other) functions to analyze and model the data

3. Fitting Functions to Data Consider the data given by this example Note the plot of the data points Close to being in a straight line Temperature Viscosity (lbs*sec/in 2 ) 16028 17026 18024 19021 20016 21013 22011 2309

4. Finding a Line to Approximate the Data Draw a line “by eye” Note slope, y-intercept Statistical process (least squares method) Use a computer program such as Excel Use your TI calculator

5. Graphs of Linear Functions For the moment, consider the first option Given the graph with tic marks = 1 Determine Slope Y-intercept A formula for the function X-intercept (zero of the function)

6. Graphs of Linear Functions Slope – use difference quotient Y-intercept – observe Write in form Zero (x-intercept) – what value of x gives a value of 0 for y?

7. Modeling with Linear Functions Linear functions will model data when Physical phenomena and data changes at a constant rate The constant rate is the slope of the function Or the m in y = mx + b The initial value for the data/phenomena is the y-intercept Or the b in y = mx + b

8. Modeling with Linear Functions Ms Snarfblat's SS class is very popular. It started with 7 students and now, 18 months later has grown to 80 students. Assuming constant monthly growth rate, what is a modeling function? Determine the slope of the function Determine the y-intercept Write in the form of y = mx + b

9. Creating a Function from a Table Determine slope by using xy 37 48.5 510 611.5 Answer:

10. Creating a Function from a Table Now we know slope m = 3/2 Use this and one of the points to determine y-intercept, b Substitute an ordered pair into y = (3/2)x + b xy 37 48.5 510 611.5

11. Creating a Function from a Table Double check results Substitute a different ordered pair into the formula Should give a true statement xy 37 48.5 510 611.5

12. Piecewise Function Function has different behavior for different portions of the domain

13. Greatest Integer Function = the greatest integer less than or equal to x Examples Calculator – use the floor( ) function

14. Assignment Lesson 2.1A Page 88 Exercises 1 – 65 EOO

15. 15 Finding a Line to Approximate the Data Draw a line “by eye” Note slope, y-intercept Statistical process (least squares method) Use a computer program such as Excel Use your TI calculator

16. 16 You Try It Consider table of ordered pairs showing calories per minute as a function of body weight Enter data into data matrix of calculator APPS, Date/Matrix Editor, New, WeightCalories 1002.7 1203.2 1504.0 1704.6 2005.4 2205.9

17. 17 Using Regression On Calculator Choose F5 for Calculations Choose calculation type (LinReg for this) Specify columns where x and y values will come from

18. 18 Using Regression On Calculator It is possible to store the Regression EQuation to one of the Y= functions

19. 19 Using Regression On Calculator When all options are set, press ENTER and the calculator comes up with an equation approximates your data Note both the original x-y values and the function which approximates the data

20. 20 Using the Function Resulting function: Use function to find Calories for 195 lbs. C(195) = 5.24 This is called extrapolation Note: It is dangerous to extrapolate beyond the existing data Consider C(1500) or C(-100) in the context of the problem The function gives a value but it is not valid WeightCalories 1002.7 1203.2 1504.0 1704.6 2005.4 2205.9

21. 21 Interpolation Use given data Determine proportional “distances” WeightCalories 1002.7 1203.2 1504.0 1704.6 195?? 2005.4 2205.9 30 0.8 25 x Note : This answer is different from the extrapolation results

22. 22 Interpolation vs. Extrapolation Which is right? Interpolation Between values with ratios Extrapolation Uses modeling functions Remember do NOT go beyond limits of known data

23. 23 Correlation Coefficient A statistical measure of how well a modeling function fits the data -1 ≤ corr ≤ +1 |corr| close to 1  high correlation |corr| close to 0  low correlation Note: high correlation does NOT imply cause and effect relationship

24. Assignment Lesson 2.1B Page 94 Exercises 85 – 93 odd