1. Exponential functions have a variable in the Exponent and a numerical base. Ex. Not to be confused with power functions which Have a variable base. Ex.

2. A function that can be expressed in the form above is called an Exponential Function. Exponential Functions with positive values of x are increasing, one-to-one functions. For f(x) = b x the graph has a y-intercept at (0,1) (initial value) and passes through (1,b). The value of b determines the steepness of the curve. The function is neither even nor odd. There is no symmetry. There is no local extrema.

3. The domain: The range: End Behavior: The y-intercept is The horizontal asymptote: There is no x-intercept. There is no x-intercept. There are no vertical asymptotes. There are no vertical asymptotes. This is an increasing, continuous function. This is an increasing, continuous function. Initial value is a. Initial value is a.

4.  How would you graph Domain: Range: Y-intercept: Hor.Asy :  How would you graph

5.  Naming a, b a, (initial value)=1: b, growth rate = 3  Rewritten:  In form

6.  Recall that if reflection of about the y-axis.  Thus, if Initial value: (0,1) Rate of decay: ½ H.A.:

7.  Notice that the reflection is decreasing, or decaying at a rate of 1/3 and the end behavior is:  How would you graph

8. If b>1, then f is an increasing function, and If 0<b<1, then f is a decreasing function, and

9. If a =1, (0,1) is the y intercept If a > 1, then (0,a) is the y intercept a = 4, b = 3, (0, 4) is the initial value Which means when x = 0, y = 4

10. Given: (0,6), (1, 12), (2, 24), (3, 48) Hint: find a first, then substitute one point into to solve for b.

11. Given: (0,6), (1, 12), (2, 24), (3, 48) a = 6 to solve for b: SOLUTION:

12. Exponential graphs, like other functions we have studied, can be stretched/shrunk, reflected and translated. It is important to maintain the same base as you analyze the transformations. Vertical shift up 3: Reflect in x-axis Vertical stretch 3 Vertical shift down 1

13. Domain: Range : HA Domain: Range : H.Asym: Inc/decreasing?

14. Reflect about the x-axis. Horizontal shift right 1. Vertical shift up 1. Vertical shrink ½. Horizontal shift left 2. Vertical shift down 3. Domain: Range : HA Domain: Range : Y-intercept: H.Asym: decreasing increasing

16. Domain: Range: Y-intercept: H.A.: Domain: Range: Y-intercept: H.A.: Domain: Range: Y-intercept: H.A.: Transformations Vertical stretch 3. Vertical shift up 2. Reflect @ x-axis. Vertical shift down 1. Horizontal shift left 2. Vertical shift up 2 Inc/dec?increasing Concavity?up Inc/dec? decreasing Concavity?down increasing Concavity?up

17. Exponential function: y = ab x Growth rate: A = a(1+r) x Decay rate: A = a(1-r) x If b>1 it’s a growth….. What is the rate of growth? If b<1, it’s a decay… What is the rate of decay?

18. Growth rate: A = a(1+r) t Decay rate: A = a(1-r) t What is the rate of growth? 1.05 – 1 =.05 = 5% What is the rate of decay? 1 -.98 =.02 = 2%

19. Growth rate: A = a(1+r) t Decay rate: A = a(1-r) t What is the rate of growth? 1.05 – 1 =.05 = 5% What is the rate of decay? 1 -.98 =.02 = 2% See page 296 #13,14,20, 31

20. Radioactive decay or growth, (half life is an example) Growth model: (b > 1), t may have a coefficient A = a 0 (b) t Decay model: (0<b<1) Half life: (K is the half life of the substance…) A = a 0 (.5) t/k t is the time we are calculating k is the half life of the substance

21. The half life of a substance is 20 days. Express the amount of the substance remaining as a function of time, t, when the initial amount is 30 grams Use the function to determine how many grams remain after 40 days.

22. The half life of a substance is 20 days. Express the amount of the substance remaining as a function of time, t, when the initial amount is 30 grams Use the function to determine how many grams remain after 40 days.

23. When will the substance remaining be 5 grams? Graph the function and check the Table of values Now try page 297 31

24. Do Now: Page 298 # 55

25. “logistic growth” C is max value a is initial value, b is growth Logistic function

26. A problem that seems reasonable: Under favorable conditions, a single cell of the bacterium Escherichia coli divides into two about every 20 minutes. If the same rate of division is maintained for 10 days, how many organisms will be produced from a single cell? Solution: 10 days = 720 20-minute periods There will be 1 ∙ 2^720 ≈ 5.5 ∙ 10^216 bacteria after 10 days.

27. Makes sense… …until you consider that there are probably fewer than 10^80 atoms in the entire universe. Real world Bizarro world

28. Why didn’t they tell us the truth? Most of those classical “exponential growth” problems should have been “logistic growth” problems! ExponentialLogistic

29. The logistics function is like capping off an Exponential function. Where c is the limited growth and a is determined By the initial Value. It is bounded by y=0 and y=c

30. Do page 288 # 41,50

31. see page 297 # 24 Find the initial value first, (0,12) Cross multiply and solve for a

32. see page 297 # 24 Now solve for b:

34. Now try 26 and 28 from page 297

35. You may use the calculator to find the Point of symmetry. See page 288, # 49 to find it. The point of symmetry for a logistics function is at the point: (x,.5c)

37. Common (base 10) and natural logarithms (base e): “a logarithm is an exponent” Exponentials and logarithms are inverses log 10 = 1, log 100 = 2, log 1 = 0 ln e = 1, ln e 2 = 2, ln 1 = 0 log (-5) does not exist! Examples:

38. Common and natural logarithms: “a logarithm is an exponent”

39. Common and natural logarithms: “a logarithm is an exponent”

40. Common and natural logarithms: “a logarithm is an exponent”

41. Practice page 308 12-24 evens New example: Now try 34 and 36 Hw: page 308 1-35 odds

42. Express each side as a base 5 number:

43.  Equations that contain one or more exponential expressions are called exponential equations.  Steps to solving some exponential equations: 1.Express both sides in terms of same base. 2.When bases are the same, exponents are equal. i.e.:

44.  DO NOW:  Get like bases and solve:

45.  Get like bases:

46.  Set exponents equal:  Solve and check!

47.  Isolate the base  & solve

48.  Isolate the base  solve

49.  Isolate the base  & solve

50.  Isolate the base  & solve

51.  Isolate the base  “log both sides” & solve

53. Exponential Equations Sometimes it may be helpful to factor the equation to solve: Sometimes it may be helpful to factor the equation to solve: There is no value of x for which is equal to 0. or

54. Try this…….

55. Exponential Equations or

56. INTEREST FORMULA amount at the end Principal (amount at start) annual interest rate (as a decimal) time (in years)

57. You invest 1200 into a 12 month cd at 2% interest. How much is it worth at the end of the year?

58. You invest 1200 into a 12 month cd at 2% interest. How much is it worth at the end of the year?

59. COMPOUND INTEREST FORMULA amount at the end Principal (amount at start) annual interest rate (as a decimal) time (in years) number of times per year that interest in compounded

60. You invest 1200 into a 12 month cd at 2% compounded quarterly. How much is it worth at the end of the year?

61. You invest 1200 into a 12 month cd at 2% compounded quarterly. How much is it worth at the end of the year?

62. Find the amount of an investment of $500 Compounded quarterly for two years at 8 % annual rate.

63. 50 0.08 4 4 (2). Find the amount that results from $500 invested at 8% compounded quarterly after a period of 2 years.

64. Solve for A 0:

67. What about continously? What does that mean? Things that are continuous: Population, medicine, compound continuous Quarterly = 4 times a year Monthly = 12 Semi-annually = 2 Daily = 360

68. The number e The letter e is the initial of the last name of Leonhard Euler (1701-1783) who introduced the notation. Since has special calculus properties that simplify many calculations, it is the natural base of exponential functions. The value of e is defined as the number that the expression approaches as n approaches infinity. The value of e to 16 decimal places is 2.7182818284590452. The function is called the Natural Exponential Function

69. Try it for n = 1000 Try it for n = 10,000

70. Compound continously:

71. Calculate: How much money will you have if you invest 500 at an annual rate of 8 %, compounded continuously for two years.

72. Calculate: How much money will you have if you invest 500 at an annual rate of 8 %, compounded continuously for two years.

73. Do Now: copy the rules on page 310 in the green box. Logarithmic Properties… Then try page 317 1,3,5 See “Change of base formula” on page 313 Do # 33 on page 317