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Linear Functions and Models

Linear Functions and Models. Lesson 2.1. 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 …

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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

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