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2. LINEAR FUNCTIONS AND CHANGE FUNCTIONS & FUNCTION NOTATION Chapter 1 Section 1 2

3. A function is a rule which takes certain numbers as inputs and assigns to each input number exactly one output number. The output is a function of the input. The inputs and outputs are also called variables. 3Page 2

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5. “Oecanthus Fultoni” 5Page N/A

6. “The Snowy Tree Cricket” 6Page N/A

7. “Nature’s Thermometer" 7Page N/A

8. By counting the number of times a snowy tree cricket chirps in 15 seconds... 8Page 2 (Example 1)

9. By counting the number of times a snowy tree cricket chirps in 15 seconds & adding 40... 9Page 2

10. By counting the number of times a snowy tree cricket chirps in 15 seconds & adding 40... We can estimate the temperature (in degrees Fahrenheit)!!! 10Page 2

11. For instance, if we count 20 chirps in 15 seconds, then a good estimate of the temperature is? 11Page 2

12. For instance, if we count 20 chirps in 15 seconds, then a good estimate of the temperature is? 20 + 40 = 60°F!!!! 12Page 2

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14. 14Page 3 T R

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17. By doing more substitutions into the formula, we can create: 17Page 3

18. R, chirp rate (chirps/minute) T, predicted temperature (°F) 2045 4050 6055 8060 10065 12070 14075 16080 18Page 3

19. From this table, we can create: 19Page 3

20. 20Page 3

21. When we use a function to describe an actual situation, the function is referred to as a mathematical model. is a mathematical model of the relationship between the temperature and the cricket's chirp rate. 21Page 3

22. R, chirp rate (chirps/minute) T, predicted temperature (°F) 2045 4050 6055 8060 10065 12070 14075 16080 What is the chirp rate when the temperature is 40 degrees? 22Page 4

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29. What if the temperature is 30 degrees? What is R? 29Page 4

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32. What is the moral here? 32Page 4

33. Whether the model's predictions are accurate for chirp rates down to zero and temperatures as low as 40°F is a question that mathematics alone cannot answer; an understanding of the biology of crickets is needed. However, we can safely say that the model does not apply for temperatures below 40°F, because the chirp rate would then be negative. For the range of chirp rates and temperatures in Table 1.1, the model is remarkably accurate.1.1 33Page 4

34. R, chirp rate (chirps/minute) T, predicted temperature (°F) 2045 4050 6055 8060 10065 12070 14075 16080 34Page 4

35. Is T a function of R, or vice-versa? 35Page 4

36. T is a function of R. 36Page 4

37. Will making the cricket chirp faster (or slower) result in a temperature change upward (or downward)?!? 37Page 4

38. Will making the cricket chirp faster (or slower) result in a temperature change upward (or downward)?!? No 38Page 4

39. Saying that the temperature (T) depends on the chirp rate (R) means: Knowing the chirp rate (R) is sufficient to tell us the temperature (T). 39Page 4

40. Saying that the temperature (T) depends on the chirp rate (R) means: Knowing the chirp rate (R) is sufficient to tell us the temperature (T). Again, a change in the chirp rate (R) doesn't cause a change in the temperature (T). 40Page 4

41. A function is a rule which takes certain numbers as inputs and assigns to each input number exactly one output number. The output is a function of the input. The inputs and outputs are also called variables. 41Page 2

42. Function Notation Q is a function of quantity, t Or: Q is a function of t We abbreviate: Q = “f of t” or Q = f(t). 42Page 4

43. Q = f(t) This means: applying the rule f to the input value, t, gives the output value, f(t). Here: Q = dependent variable (unknown, depends on t) t = independent variable (known) 43Page 4

44. Q = f(t). In other words: Output = f(Input) Or: Dependent = f(Independent) 44Page 4

45. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f. (b) Explain in words what the statement f(10,000) = 40 tells us about painting houses. 45Page 4 (Example 2)

46. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f. If n = 1, A = ? ft 2 46Page 4

47. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f. If n = 1, A = 250 ft 2 47Page 4

48. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f. If n = 2, A = ? ft 2 48Page 4

49. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f. If n = 2, A = 500 ft 2 49Page 4

50. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f. If n = 3, A = ? ft 2 50Page 4

51. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f. If n = 3, A = 750 ft 2 51Page 4

52. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f. In general, we have: A=(250)(n) 52Page 4

53. In general, we have: A=(250)(n) Now solve for n: 53Page 4

54. In general, we have: A=(250)(n) Now solve for n: n=A/250 54Page 4

55. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f : n=A/250. 55Page 4

56. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f : n=A/250. So f(A)=A/250 56Page 4

57. The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet. Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft 2. We write n = f(A). (a) Find a formula for f : n=A/250. So f(A)=A/250 (b) Explain in words what the statement f(10,000) = 40 tells us about painting houses. 57Page 4

58. (a) Find a formula for f : n=A/250. So f(A)=A/250 (b) Explain in words what the statement f(10,000) = 40 tells us about painting houses. Since f(A)=A/250, what does f(10,000) mean? 58Page 4

59. (a) Find a formula for f : n=A/250. So f(A)=A/250 (b) Explain in words what the statement f(10,000) = 40 tells us about painting houses. Since f(A)=A/250, what does f(10,000) mean? A=10,000 59Page 4

60. (a) Find a formula for f : n=A/250. So f(A)=A/250 (b) Explain in words what the statement f(10,000) = 40 tells us about painting houses. Since f(A)=A/250, what does f(10,000) mean? A=10,000 What can we deduce next? 60Page 4

61. (a) Find a formula for f : n=A/250. So f(A)=A/250 (b) Explain in words what the statement f(10,000) = 40 tells us about painting houses. Since f(A)=A/250, what does f(10,000) mean? A=10,000 What can we deduce next? Since f(A)=A/250, then f(10,000)=10,000/250 61Page 4

62. ( a) Find a formula for f : n=A/250. So f(A)=A/250 (b) Explain in words what the statement f(10,000) = 40 tells us about painting houses. Since f(A)=A/250, then f(10,000)=10,000/250 Therefore, f(10,000)=40 In English this means? 62Page 4

63. ( a) Find a formula for f : n=A/250. So f(A)=A/250 (b) Explain in words what the statement f(10,000) = 40 tells us about painting houses. f(10,000)=40 In English this means: An area of A=10,000 sq. ft. requires n= 40 gallons of paint. 63Page 4

64. Functions Don't have to be Defined by Formulas: People sometimes think that functions are always represented by formulas. However, the next example shows a function which is not given by a formula. 64Page 5

65. The average monthly rainfall, R, at Chicago's O'Hare airport is given in Table 1.2, where time, t, is in months and t = 1 is January, t = 2 is February, and so on.1.2 The rainfall is a function of the month, so we write R = f(t). However there is no equation that gives R when t is known. Evaluate f(1) and f(11). Explain what your answers mean. t123456789101112 R1.8 2.73.13.53.73.53.43.22.52.42.1 65Page 5 (Example 4)

66. Evaluate f(1) and f(11). Explain what your answers mean. Remember: R = f(t). t123456789101112 R1.8 2.73.13.53.73.53.43.22.52.42.1 f(1) = ? 66Page 5

67. Evaluate f(1) and f(11). Explain what your answers mean. Remember: R = f(t). t123456789101112 R1.8 2.73.13.53.73.53.43.22.52.42.1 f(1) = 1.8 67Page 5

68. Evaluate f(1) and f(11). Explain what your answers mean. Remember: R = f(t). t123456789101112 R1.8 2.73.13.53.73.53.43.22.52.42.1 f(11) = ? 68Page 5

69. Evaluate f(1) and f(11). Explain what your answers mean. Remember: R = f(t). t123456789101112 R1.8 2.73.13.53.73.53.43.22.52.42.1 f(11) = 2.4 69Page 5

70. t123456789101112 R1.8 2.73.13.53.73.53.43.22.52.42.1 70Page 5 (Not shown)

71. As was stated, functions don’t have to be defined by formulas. You can do a linear regression analysis and get: rainfall(R) = 2.6424242 + 0.0255245*month (t) 71Page 5 (Not shown)

72. As was stated, functions don’t have to be defined by formulas. You can do a linear regression analysis and get: rainfall(R) = 2.6424242 + 0.0255245*month (t) BUT... 72Page 5 (Not shown)

73. As was stated, functions don’t have to be defined by formulas. You can do a linear regression analysis and get: rainfall(R) = 2.6424242 + 0.0255245*month (t) BUT... What if t = 12? 73Page 5 (Not shown)

74. You can do a linear regression analysis and get: rainfall(R) = 2.6424242 + 0.0255245*month (t) BUT… What if t=12? rainfall(R) = 2.6424242 + 0.0255245*month (12) 74Page 5 (Not shown)

75. You can do a linear regression analysis and get: rainfall(R) = 2.6424242 + 0.0255245*month (t) BUT… What if t=12? rainfall(R) = 2.6424242 + 0.0255245*(12) R = 2.6424242 +.306294 R = 2.9487182 75Page 5 (Not sh0wn)

76. t123456789101112 R1.8 2.73.13.53.73.53.43.22.52.42.1 76Page 5 (Not shown)

77. t123456789101112 R1.8 2.73.13.53.73.53.43.22.52.42.1 Let’s try to fit a quadratic curve: 77Page 5 (Not shown)

78. rainfall(R) = 3.3403409 + 0.0255245*month (t) - 0.0585664*(month (t)-6.5)^2 R = 3.3403409 + 0.0255245*t - 0.0585664*(t-6.5) 2 78Page 5 (Not shown)

79. Again, for t=12: R = 3.3403409 + 0.0255245*t - 0.0585664*(t-6.5) 2 R = 3.3403409 +.306294 - 0.0585664*(5.5) 2 R = 3.3403409 +.306294 - 0.0585664*30.25 R = 3.3403409 +.306294 – 1.7716336 R = 1.8750013 Closer, but still not 2.1!!! 79Page 5 (Not shown)

80. rainfall(R) = 3.3403409 + 0.0255245*month (t) - 0.0585664*(month (t)-6.5)^2 For t=12, R = 1.8750013 80Page 5 (Not shown)

81. When is a Relationship not a Function? It is possible for two quantities to be related and yet for neither quantity to be a function of the other. 81Page 5

82. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 A national park contains foxes that prey on rabbits. Table 1.3 gives the two populations, F and R, over a 12-month period, where t = 0 means January 1, t = 1 means February 1, and so on. 1.3 (a) Is F a function of t? Is R a function of t? (b) Is F a function of R? Is R a function of F? 82Page 5 (Example 5)

83. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (a) Is F a function of t? Is R a function of t? (b) Is F a function of R? Is R a function of F? Please create 4 graphs with pen/paper: 1) Plot F on the y axis & t on the x axis. 2) Plot R on the y axis & t on the x axis. 3) Plot F on the y axis & R on the x axis. 4) Plot R on the y axis & F on the x axis. 83Page 5 (Example 5)

84. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (a) Is F a function of t? 84Page 5

85. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (a) Is F a function of t? Yes: for each value of t, there is exactly one value of F. 85Page 5

86. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (a) Is R a function of t? 86Page 5

87. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (a) Is R a function of t? Yes: for each value of t, there is exactly one value of R. 87Page 5

88. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (b) Is F a function of R? Is R a function of F? 88Page 5

89. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (b) Is F a function of R? 89Page 5

90. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (a) Is F a function of R? 90Page 5

91. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (a) Is F a function of R? No, F is not a function of R. Suppose R = 567. This happens both at t = 2 & at t = 4. Since there are R values which correspond to more than one F value, F is not a function of R. 91Page 5

92. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (a) Is R a function of F? 92Page 5

93. t01234567891011 R1000750567500567750100012501433150014331250 F1501431251007557505775100125143 (a) Is R a function of F? No, R is not a function of F. Suppose F = 57. This happens both at t = 5 & at t = 7. Since there are F values which correspond to more than one R value, R is not a function of F. 93Page 5

94. How to Tell if a Graph Represents a Function: Vertical Line Test Graphically, this means: Look at the graph of y against x. For a function, each x- value corresponds to exactly one y-value. This means that the graph intersects any vertical line at most once. If a vertical line cuts the graph twice, the graph would contain two points with different y-values but the same x- value; this would violate the definition of a function. Thus, we have the following criterion: 94Page 6

95. How to Tell if a Graph Represents a Function: Vertical Line Test Vertical Line Test: If there is a vertical line which intersects a graph in more than one point, then the graph does not represent a function. 95Page 6

96. This concludes Section 1.1 96