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AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
AA Notes 2-7 and 2-8
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AA Notes 2-7 and 2-8

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Fitting a Model to Data

Fitting a Model to Data

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  1. SECTIONS 2-7 AND 2-8 FITTING A MODEL TO DATA
  2. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k b. y = x c. y = kx 2 k d. y = 2 x
  3. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k b. y = 20 = 10k x c. y = kx 2 k d. y = 2 x
  4. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k b. y = 20 = 10k x k=2 c. y = kx 2 k d. y = 2 x
  5. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k b. y = 20 = 10k x k=2 y = 2x c. y = kx 2 k d. y = 2 x
  6. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k b. y = 20 = 10k y = 2(30) x k=2 y = 2x c. y = kx 2 k d. y = 2 x
  7. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k b. y = 20 = 10k y = 2(30) x k=2 y = 60 y = 2x c. y = kx 2 k d. y = 2 x
  8. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k b. y = 20 = 10k y = 2(30) x k k=2 y = 60 20 = 10 y = 2x c. y = kx 2 k d. y = 2 x
  9. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k b. y = 20 = 10k y = 2(30) x k k=2 y = 60 20 = 10 y = 2x k = 200 c. y = kx 2 k d. y = 2 x
  10. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 20 = 10k y = 2(30) x x k k=2 y = 60 20 = 10 y = 2x k = 200 c. y = kx 2 k d. y = 2 x
  11. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 10 y = 2x k = 200 c. y = kx 2 k d. y = 2 x
  12. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k d. y = 2 x
  13. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k d. y = 2 20 = k(10) 2 x
  14. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k d. y = 2 20 = k(10) 2 x k= 1 5
  15. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k d. y = 2 20 = k(10) 2 x k= 1 5 y= x1 5 2
  16. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k d. y = 2 20 = k(10) 2 y = (30) 1 2 x 5 k= 1 5 y= x1 5 2
  17. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k d. y = 2 20 = k(10) 2 y = (30) 1 2 x 5 k= 1 5 y = 180 y= x1 5 2
  18. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k d. y = 2 20 = k(10) 2 y = 5 (30) 1 2 k x 20 = 2 k= 51 y = 180 10 y= x1 5 2
  19. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k d. y = 2 20 = k(10) 2 y = 5 (30) 1 2 k x 20 = 2 k= 51 y = 180 10 y= x k = 2000 1 2 5
  20. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k 2000 d. y = 2 y = 2 20 = k(10) 2 y = 5 (30) 1 2 k x x 20 = 2 k= 51 y = 180 10 y= x k = 2000 1 2 5
  21. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k 2000 d. y = 2 y = 2 20 = k(10) 2 y = 5 (30) 1 2 k x x 20 = 2 y = 2000 k= 51 y = 180 10 2 30 y= x k = 2000 1 2 5
  22. WARM-UP The point with coordinates (10, 20) appears on the graph for each variation function below. Determine the value of k in each case, and then determine the value of y when x is 30. a. y = kx k 200 b. y = y= 200 20 = 10k y = 2(30) x x y= k 30 k=2 y = 60 20 = 20 10 y= y = 2x k = 200 3 c. y = kx 2 k 2000 d. y = 2 y = 2 20 = k(10) 2 y = 5 (30) 1 2 k x x 20 = 2 y = 2000 k= 51 y = 180 10 2 30 20 y= y= 5x1 2 k = 2000 9
  23. MATHEMATICAL MODEL
  24. MATHEMATICAL MODEL A mathematical representation of a real-world situation
  25. MATHEMATICAL MODEL A mathematical representation of a real-world situation A good model fits all possibilities as best it can; often is just an approximation
  26. EXAMPLE 1 Matt Mitarnowski and Fuzzy Jeff measured the intensity of light at various distances from a lamp and got the data where d = distance in meters and I = intensity in watts/m2. d I 2 560 2.5 360 3 250 3.5 185 4 140
  27. EXAMPLE 1 Matt Mitarnowski and Fuzzy Jeff measured the intensity of light at various distances from a lamp and got the data where d = distance in meters and I = intensity in watts/m2. d I 600 2 560 450 2.5 360 3 250 300 3.5 185 150 4 140 0 0 1.25 2.50 3.75 5.00
  28. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations.
  29. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k I= d
  30. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k I= 560 = d 2
  31. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k I= 560 = k = 1120 d 2
  32. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 I= 560 = k = 1120 I= d 2 d
  33. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4
  34. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280
  35. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280 Not close
  36. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280 Not close k I= 2 d
  37. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280 Not close k k I= 2 560 = 2 d 2
  38. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280 Not close k k I= 2 560 = 2 k = 2240 d 2
  39. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280 Not close k k 2240 I= 2 560 = 2 k = 2240 I= 2 d 2 d
  40. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280 Not close k k 2240 2240 I= 2 560 = 2 k = 2240 I= 2 I= d 2 d 16
  41. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280 Not close k k 2240 2240 I= 2 560 = 2 k = 2240 I= 2 I= d 2 d 16 I = 140
  42. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280 Not close k k 2240 2240 I= 2 560 = 2 k = 2240 I= 2 I= d 2 d 16 I = 140 Right on!
  43. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280 Not close k k 2240 2240 I= 2 560 = 2 k = 2240 I= 2 I= d 2 d 16 I = 140 Right on! 2240 I= 2 d
  44. EXAMPLE 1 This looks like either a hyperbola or an inverse-square curve. Let’s test them out by checking some points in our equations. k k 1120 1120 I= 560 = k = 1120 I= I= d 2 d 4 I = 280 Not close k k 2240 2240 I= 2 560 = 2 k = 2240 I= 2 I= d 2 d 16 I = 140 Right on! 2240 I= 2 Make sure to check that all points are close d
  45. STEPS TO FINDING A MODEL FROM DATA
  46. STEPS TO FINDING A MODEL FROM DATA 1. Graph the data
  47. STEPS TO FINDING A MODEL FROM DATA 1. Graph the data 2. Describe the graph
  48. STEPS TO FINDING A MODEL FROM DATA 1. Graph the data 2. Describe the graph 3. State the equation
  49. STEPS TO FINDING A MODEL FROM DATA 1. Graph the data 2. Describe the graph 3. State the equation 4. Find k, plug it in
  50. STEPS TO FINDING A MODEL FROM DATA 1. Graph the data 2. Describe the graph 3. State the equation 4. Find k, plug it in 5. Check other points
  51. STEPS TO FINDING A MODEL FROM DATA 1. Graph the data 2. Describe the graph 3. State the equation 4. Find k, plug it in 5. Check other points 6. State the model
  52. EXAMPLE 2 Matt Mitarnowski measured how far a marble rolled down a ramp over a period of time. He obtained the following data where t is time in seconds and D is distance in inches. t D 1 0.6 2 2.4 3 5.2 4 9.8 5 15.2
  53. EXAMPLE 2 Matt Mitarnowski measured how far a marble rolled down a ramp over a period of time. He obtained the following data where t is time in seconds and D is distance in inches. 20 t D 15 1 0.6 2 2.4 10 3 5.2 5 4 9.8 5 15.2 0 0 1.25 2.50 3.75 5.00
  54. EXAMPLE 2 Matt Mitarnowski measured how far a marble rolled down a ramp over a period of time. He obtained the following data where t is time in seconds and D is distance in inches. 20 t D 15 1 0.6 2 2.4 10 3 5.2 5 4 9.8 5 15.2 0 0 1.25 2.50 3.75 5.00 Should be a parabola
  55. D = kt 2
  56. D = kt 2 .6 = k(1) 2
  57. D = kt 2 .6 = k(1) 2 k = .6
  58. D = kt 2 .6 = k(1) 2 k = .6 D = .6t 2
  59. D = kt 2 .6 = k(1) 2 k = .6 D = .6t 2 D = .6(2) 2
  60. D = kt 2 .6 = k(1) 2 k = .6 D = .6t 2 D = .6(2) 2 D = 2.4
  61. D = kt 2 .6 = k(1) 2 k = .6 D = .6t 2 D = .6(2) 2 D = 2.4 D = .6t 2
  62. D = kt 2 .6 = k(1) 2 k = .6 D = .6t 2 D = .6(2) 2 D = 2.4 D = .6t 2 Check the rest of the points to make sure satisfy the model
  63. CONVERSE OF THE FUNDAMENTAL THEOREM OF VARIATION
  64. CONVERSE OF THE FUNDAMENTAL THEOREM OF VARIATION a. If multiplying every x-value of a function by c results in multiplying the corresponding y-value by cn, then y varies directly as the nth power of x, or y = kxn.
  65. CONVERSE OF THE FUNDAMENTAL THEOREM OF VARIATION a. If multiplying every x-value of a function by c results in multiplying the corresponding y-value by cn, then y varies directly as the nth power of x, or y = kxn. b. If multiplying every x-value of a function by c results in dividing the corresponding y-value by cn, then y varies inversely as the n th power of x, or k . y= n x
  66. HOMEWORK
  67. HOMEWORK Make sure to read Sections 2-7 and 2-8 p. 113 #1-11, p. 119 #1-11

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