Math Subject: Exponential Functions .pptx

Exponential Functions

1. Exponential Functions

2. Objectives Define exponential functions Evaluate exponential functions Use compound interest formulas

3. Properties of Exponents Recall that for a variable and integers and :  

4. Simplifying Exponents Example: Simplify!  

5. Simplifying Exponents Example: Simplify! =  

6. Exponential Functions is a function involving exponential expression showing a relationship between the independent variable and dependent variable or . It is in form , where , and , and the exponent must be a variable. Examples of which are and  

7. Exponential Functions   (when is zero)    

8. Exponential Functions   (when is zero)     (what you multiply by)   x Y 5 1 10 2 20 3 40 4 80

9. Exponential Functions   (when is zero)     (what you multiply by)  

10. Exponential Functions   (when is zero)     (what you multiply by)  

11. Exponential Functions   (when is zero)     (what you multiply by)   x Y 1 2 3 4

12. Exponential Functions   (when is zero)     (what you multiply by)   x Y 3 1 6 2 12 3 24 4 48

13. Exponential Functions   (when is zero)     (what you multiply by)   x Y 3 1 6 2 12 3 24 4 48 Exponential Growth

14. Exponential Functions   (when is zero)     (what you multiply by)  

15. Exponential Functions   (when is zero)     (what you multiply by)  

16. Exponential Functions   (when is zero)     (what you multiply by)   x Y 1 2 3 4

17. Exponential Functions   (when is zero)     (what you multiply by)   x Y 144 1 72 2 36 3 18 4 9

18. Exponential Functions   (when is zero)     (what you multiply by)   x Y 144 1 72 2 36 3 18 4 9 Exponential Decay

19. Exponential Functions The graph of an exponential function is called an exponential curve. Exponential Growth Exponential Decay Factor (b) is greater than 1.   Factor (b) is between 0 and 1.  

20. Exponential Functions Determine whether each function shows exponential growth or exponential decay . 2. 3. 4.  

21. Exponential Functions Determine whether each function shows exponential growth or exponential decay. 2. The factor (b) is greater than 1. Exponential Growth 3. 4.  

22. Exponential Functions Determine whether each function shows exponential growth or exponential decay. 2. The factor (b) is greater than 1. The factor (b) is between 0 and 1 Exponential Growth Exponential Decay 3. 4.  

23. Exponential Functions Determine whether each function shows exponential growth or exponential decay . 2. The factor (b) is greater than 1. The factor (b) is between 0 and 1 Exponential Growth Exponential Decay 3. 4. The factor (b) is between 0 and 1. Exponential Decay  

24. Exponential Functions Determine whether each function shows exponential growth or exponential decay. 2. The factor (b) is greater than 1. The factor (b) is between 0 and 1 Exponential Growth Exponential Decay 3. 4. The factor (b) is between 0 and 1. The factor (b) is greater than 1. Exponential Decay Exponential Growth  

25. The Equality Property of Exponential Functions We know that in exponential functions, the exponent is a variable. When we wish to solve for that variable, we have two approaches we can take. One approach is to use a logarithm. The second is to make use of the Equality Property for Exponential Functions.

26. The Equality Property of Exponential Functions Suppose is a positive number other than 1. Then if and only if .  

27. The Equality Property of Exponential Functions Example 1: (Since the bases are the same, we simply set the exponents equal).  

28. The Equality Property of Exponential Functions Let’s try!  

29. The Equality Property of Exponential Functions Let’s try! or  

30. The Equality Property of Exponential Functions Example 2: (When the bases are not the same)   Rewrite the bases so that they are the same.

31. The Equality Property of Exponential Functions Example 2: (When the bases are not the same)   The bases are now the same.

32. The Equality Property of Exponential Functions Let’s Try!  

33. The Equality Property of Exponential Functions Let’s Try!  

34. Exponential Growth or Decay A function that models exponential growth grows by a rate proportional to the amount present. For any real number and any positive real numbers and such that , an exponential growth function has the form where is the initial or starting value of the function is the growth factor or growth multiplier per unit  

35. Writing Exponential Functions Given two data points, how do we write an exponential model? If one of the data points has the form (the ), then is the initial value. Substitute into the equation , and solve for with the second set of values. Otherwise, substitute both points into two equations with the form and solve the system. Using and found in the steps 1 or 2, write the exponential function in the form .  

36. Writing Exponential Functions Example: In 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had growth to 180 deer. The population was growing exponentially. Write an exponential function representing the population of deer over time .  

37. Writing Exponential Functions Example: In 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had growth to 180 deer. The population was growing exponentially. Write an exponential function representing the population of deer over time . If we let be the number of years after 2006, we can write the information in the problem as two ordered pairs: and . We also have an initial value, so , and we can use the process in step 1.  

38. Writing Exponential Functions Set up the initial equation ) and substitute and the second set of values into it. Thus, the function becomes  

39. Writing Exponential Functions Example: Find an exponential function that passes through the points (-2,6) and (2,1). Since we don’t have an initial value, we will need to set up and solve a system. It will usually be simplest to use the first equation with the first set of values for , and then substitute that into the second equation with the second set of values to solve for  

40. Writing Exponential Functions Example: Find an exponential function that passes through the points (-2,6) and (2,1). Thus, the function is  

41. Compound Interest The formula for compound interest (interest paid on both principal and interest) is an important application of exponential functions. Recall that the formula for simple interest, , where is principal (amount deposited), is annual rate of interest, and is time in years.  

42. Compound Interest Now, suppose we deposit at 10% annual interest. At the end of the first year, we have so our account now has 1000 + .1(1000) = At the end of the second year, we have so our account now has 1100 + .1(1100) = .  

43. Compound Interest Another way to write 1000 + .1(1000) is 1000(1 + .1) After the second year, this gives us 1000 (1 + .1) + .1(1000(1 + .1)) = 1000 (1 + .1)(1 + .1) = 1000 (1 + .  

44. Compound Interest If we continue, we end up with This leads us to the general formula. Year Account 1 $1100 1000(1 + .1) 2 $1210 1000(1 + .1 3 $1331 1000(1 + .1 4 $1464.10 1000(1 + .1 1000(1 + .1 Year Account 1 $1100 1000(1 + .1) 2 $1210 3 $1331 4 $1464.10

45. Compound Interest Formulas For interest compounded times per year: For interest compounded continuously: where is the irrational constant 2.718281 …  

46. Compound Interest Formulas Example: If $2500 is deposited in an account paying 6% per year compounded twice per year, how much is the account worth after 10 years with no withdrawals?  

47. Compound Interest Formulas Example: 2. What amount deposited today at 4.8% compounded quarterly will give $15,000 in 8 years?  

48. Compound Interest Formulas Example: 2. What amount deposited today at 4.8% compounded quarterly will give $15,000 in 8 years?  

49. Thank you!

50. Individual Task Using the Equality Property of Exponential Functions, determine the value of . B. Solve. 3. Which is a better deal, depositing $7000 at 6.25% compounded every month for 5 years or 5.75% compounded continuously for 6 years?