Successfully reported this slideshow.
Upcoming SlideShare
×

AA Notes 2-7 and 2-8

640 views

Published on

Fitting a Model to Data

Published in: Education, Technology
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

AA Notes 2-7 and 2-8

1. 1. SECTIONS 2-7 AND 2-8 FITTING A MODEL TO DATA
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 23. MATHEMATICAL MODEL
24. 24. MATHEMATICAL MODEL A mathematical representation of a real-world situation
25. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 45. STEPS TO FINDING A MODEL FROM DATA
46. 46. STEPS TO FINDING A MODEL FROM DATA 1. Graph the data
47. 47. STEPS TO FINDING A MODEL FROM DATA 1. Graph the data 2. Describe the graph
48. 48. STEPS TO FINDING A MODEL FROM DATA 1. Graph the data 2. Describe the graph 3. State the equation
49. 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. 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. 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. 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. 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. 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. 55. D = kt 2
56. 56. D = kt 2 .6 = k(1) 2
57. 57. D = kt 2 .6 = k(1) 2 k = .6
58. 58. D = kt 2 .6 = k(1) 2 k = .6 D = .6t 2
59. 59. D = kt 2 .6 = k(1) 2 k = .6 D = .6t 2 D = .6(2) 2
60. 60. D = kt 2 .6 = k(1) 2 k = .6 D = .6t 2 D = .6(2) 2 D = 2.4
61. 61. D = kt 2 .6 = k(1) 2 k = .6 D = .6t 2 D = .6(2) 2 D = 2.4 D = .6t 2
62. 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. 63. CONVERSE OF THE FUNDAMENTAL THEOREM OF VARIATION
64. 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. 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. 66. HOMEWORK
67. 67. HOMEWORK Make sure to read Sections 2-7 and 2-8 p. 113 #1-11, p. 119 #1-11