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Linear Functions and Models

Linear Functions and Models

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Linear Functions and Models

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  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 ofthe data points • Close to beingin a straight line

  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 programsuch 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. Answer: Creating a Function from a Table • Determine slope by using

  10. Creating a Function from a Table • Now we know slope m = 3/2 • Use this and one ofthe points to determiney-intercept, b • Substitute an orderedpair into y = (3/2)x + b

  11. Creating a Function from a Table • Double check results • Substitute a different ordered pair into the formula • Should give a true statement

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

  16. You Try It • Consider table of ordered pairsshowing calories per minuteas a function of body weight • Enter data into data matrix ofcalculator • APPS, Date/Matrix Editor, New,

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

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

  19. Using Regression On Calculator • When all options areset, press ENTER andthe calculator comesup with an equation approximates your data Note both the original x-y values and the function which approximates the data

  20. Using the Function • Resulting function: • Use function to find Caloriesfor 195 lbs. • C(195) = 5.24This 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

  21. Interpolation • Use given data • Determine proportional“distances” x 25 0.8 30 Note : This answer is different from the extrapolation results

  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. 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