2. The essence of mathematics is not to make simple things complicated, but to make complicated things simple. Stan Gudder John Evans Professor of Mathematics University of Denver

3. Chapter 3 Exponential and Logarithmic Functions

4. Day I Exponential and Logarithmic Equations (3.1)

5. Exponential functions can be used to model the amount of defoliation caused by the gypsy moth.

6. GOAL I. To recognize and evaluate exponential functions with the base a

7. I. Exponential Functions

8. Transcendental Function A function which is not an algebraic function. In other words, a function which “transcends,” or cannot be expressed in terms of algebra.

9. Transcendental Functions: Exponential Logarithmic Trigonometric Inverse Trigonometric

10. Definition of an Exponential Function

11. The exponential function f with the base a is denoted by f (x) = a x where a > 0, a  1, and x is any real number.

12. For instance, f (x) = 3 x and g (x) = 0.5 x are exponential functions.

13. The value of f (x) = 3 x when x = 2 is f (2) = 3 2 = 9 The value of f (x) = 3 x when x = –2 is f (–2) = 3 –2 = 1919

14. The value of g (x) = 0.5 x when x = 4 is g(4) = 0.5 4 =0.0625

15. Your Turn

16. The value of g (x) = 0.5 x when x = -4 is g (-4) = 0.5 -4 =16 Don’t use a calculator!

17. Example 1. Evaluating Exponential Expressions

18. Using a calculator, evaluate the expression to 1/1000. 156[(1/2) -3.2 ]= 1433.576.5 x  y 3.2 + / - = * 156.5  (-)3.2 * 156 OR

19. Your Turn

20. Using a calculator, evaluate each expression to 1/1000. 1.5000(2 -1.5 ) = 2.8 2  = 3. 3 4395 = 1767.767 472,369.379 16.380

21. GOAL II. To graph exponential functions

22. II. Graphs of Exponential Functions

23. Standard 12.0.2 Students understand exponential functions (graphs).

24. Example 2. Graphs of y = a x

25. xf(x)f(x)(x, f(x)) -2 0 1 2 Example: Graph f(x) = 2 x Sketch the graph of f (x) = 3 x. 1/91/9 (-2, 1 / 9 ) 3 -2 3 -1 1/31/3 (-1, 1 / 3 ) 3030 1(0, 1) 3131 3(1, 3) 3232 9 (2, 9)

26. Sketch the graph of f (x) = 3 x. x xf(x)f(x) -2 1/91/9 1/31/3 01 13 29 6 2–2 2 4 Example: Graph f(x) = 2 x 8 y

27. Your Turn

28. xf(x)f(x)(x, f(x)) -2 0 1 2 Example: Graph f(x) = 2 x On same graph, sketch f (x) = 4 x. 1 / 16 (-2, 1 / 16 ) 4 -2 4 -1 1/41/4 (-1, 1 / 4 ) 4040 1(0, 1) 4141 4(1, 4) 4242 16 (2, 16)

29. Add the graph of f (x) = 4 x. x xf(x)f(x) -2 1 / 16 1/41/4 01 14 216 6 2–2 2 4 Example: Graph f(x) = 2 x 8 y

30. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The graph of f(x) = a x, a > 1 y x (0, 1) Domain: (– ,  ) Range: (0,  ) Horizontal Asymptote y = 0 Graph of Exponential Function (a > 1) 4 4

31. Example 3. Graphs of y = a -x

32. xf(x)f(x)(x, f(x)) -2 0 1 2 Example: Graph f(x) = 2 x Sketch the graph of f (x) = 3 -x. 9(-2, 9) 3 -(-2) 3 -(-1) 3 (-1, 3) 3030 1(0, 1) 3 -11/31/3 (1, 1 / 3 ) 3 -21/91/9 (2, 1 / 9 )

33. x 6 2 –2 4 Sketch the graph of f (x) = 3 -x. xf(x)f(x) -29 3 01 1 1/31/3 2 1/91/9 8 y

34. Your Turn

35. x 6 2 –2 4 Which is the graph of f (x) = 4 -x ? 8 y a b a or b

36. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The graph of f(x) = a x, 0 < a < 1 y x ( 0, 1) Domain: (– ,  ) Range: (0,  ) Horizontal Asymptote y = 0 Graph of Exponential Function (0 < a < 1) 4 4

37. Transformations of y = b  a x-c or y =  a x-c + b

38. The graph of f (x) = 4 x. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

39. negative If a is negative the graph is reflected over the x-axis.

40. The graph of f (x) = 4 x and g (x) = -4 x x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

41. What is the domain and range of f (x)? D = (- ,  ) R = (0,  )

42. What is the domain and range of g (x)? D = (- ,  ) R = (- , 0)

43. negative If x is negative the graph is reflected over the y-axis.

44. The graph of f (x) = 4 x and g (x) = 4 -x x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

45. The graph of f (x) = 4 x-0. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

46. When you subtract a positive number c from x, you are translating horizontally the graph of the function c units to the right. If c = 1, then x – 1, so move 1 unit right

47. The graph of f (x) = 4 x-0 and g (x) = 4 x-1 g (x) = 4 x-1. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

48. When c is negative, you are translating horizontally the graph of the function |c| units to the left. If c = -1, then x - (-1) = x + 1, so move 1 unit left

49. The graph of f (x) = 4 x-0 and g (x) = 4 x+1 g (x) = 4 x+1. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

50. When you add a positive number b to a function, you are translating vertically the graph of the function b units upwards. If b = 1, move up 1 unit.

51. The graph of f (x) = 4 x and g (x) = 4 x + 1. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

52. When b is negative, you are translating vertically the graph of the function |b| units downwards. If b = -1, move down 1 unit.

53. The graph of f (x) = 4 x and g (x) = 4 x - 1. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

54. Example 4. Transformations of Graphs of Exponential Functions

55. Describe the movement of the transformation of f (x) = 5 x. 1.-5 x+2 2.5 x – 3 3.5 -x + 2

56. 1.-5 x+2 2.5 x – 3 3.5 -x + 2 2 left and reflected over x-axis 3 down 2 up and reflected over y-axis

57. Your Turn

58. Match the exponential function with its graph. 1.6 x-2 2.6 x + 1 3.6 -x

59. a 2

60. b 3

61. c 1

62. What has six wheels and flies?

63. A garbage truck! What has six wheels and flies?

64. GOAL III. To recognize and evaluate exponential functions with base e

65. III. The Natural Base e

66. e  2.718281828... The natural exponential function is y = e x

67. Example 5. Evaluating the Natural Exponential Function

68. Use a calculator to evaluate each expression to the nearest 1/10000. 1.e -3.7 2.e 6.2 0.0247 492.7490

69. Your Turn

70. Use a calculator to evaluate each expression to the nearest 1/10. 1.2e -0.30 4. 2e 0.15 2.2e -0.15 5. 2e 0.30 3.2e 0

71. Example 6. Graphing Natural Exponential Functions

72. Use the data above, graph the function y = 2e 0.15x

73. xy=2e 0.15x y -2 2e -0.30 1.5 -12e -0.15 1.7 02e 0 2 12e 0.15 2.3 22e 0.30 2.7

75. Your Turn

76. Graph the function y = ½ e -0.6x

77. x y= ½ e -0.6x y -2 ½ e 1.2 1.7 -1½ e 0.6 0.9 0½ e 0 0.5 1½ e -0.6 0.3 2½ e -1.2 0.2

79. GOAL IV. To graph exponential functions to model and solve real-life applications

80. IV. Applications

81. Compounding n times per year Interest Formula A = P 1 + rnrnnt

82. A = amount earned P = principal = amount originally invested r = rate as a decimal n = # of compoundings per year t = number of years

83. Compounding continuously Interest Formula A = Pe rt

84. A = amount earned P = principal = amount originally invested r = rate as a decimal t = number of years

85. Example 7. Compound Interest

86. Complete the table to determine the balance (A) for $1000 invested at 6% for 10 years.

87. n P(1+(r/n)) nt A 1 2 4 12 365 1000(1.06) 10 1790.85 1000(1.03) 20 1806.11 1000(1.015) 40 1814.02 1000(1.005) 120 1819.40 1000(1.----) 3650 1822.03

88. How about continuously? A = 1000e (.06●10) A = Pe rt = $1822.12

89. Your Turn

90. Complete the table to determine the balance (A) for $1000 invested at 3% for 10 years.

91. n P(1+(r/n)) nt A 1 12 365 Con 1000(1.03) 10 1343.92 1000(1.0025) 120 1349.35 1000(1.----) 3650 1349.84 1000e 0.3 1349.86

92. Example 8. Radioactive Decay

93. Let Q (in grams) represent a mass of carbon-14 ( 14 C), whose half-life is 5730 years. The quantity of carbon-14 present after t years is Q = 10 t/5730 1212

94. Determine the initial quantity (when t = 0). Q = 10 1212 0/5730 = 10 1212 0 = 10(1) = 10 g

95. Determine the quantity present after 2000 years. Q = 10 1212 2000/5730 = 7.85 g

96. Your Turn

97. Exponential functions can be used to model the amount of defoliation caused by the gypsy moth.

98. To estimate the amount of defoliation caused by the gypsy moth during a given year, a forester counts the number x of egg masses on 1 / 40 of an acre (circle of radius 18.6 feet) in the fall.

99. To find the percent of defoliation y the next spring, he uses the equation y = which he derived from the actual data. 300 3 + 17e -0.065x

100. Estimate (to the nearest percent) the defoliation if 36 egg masses are counted on the target acre. y = 300 3 + 17e -0.065(36) = 64.7%

101. What is a trees least favorite month?

102. Sep–timber!