1. Exponential Functions Functions that have the exponent as the variable.

2. Exponential Functions Functions that have the exponent as the variable. - “a” is our base raised to some exponent “ x “ that varies - if a > 1, the graph shows exponential growth

3. Exponential Functions Functions that have the exponent as the variable. - “a” is our base raised to some exponent “ x “ that varies - if a > 1, the graph shows exponential growth - these functions explode as “x” gets larger

4. Exponential Functions Functions that have the exponent as the variable. - “a” is our base raised to some exponent “ x “ that varies - if a > 1, the graph shows exponential growth - these functions explode as “x” gets larger

5. Exponential Functions Functions that have the exponent as the variable. - “a” is our base raised to some exponent “ x “ that varies - if 0 < a < 1, the graph shows exponential decay - the graph approaches zero as “x” gets larger

6. Exponential Functions Functions that have the exponent as the variable. - “a” is our base raised to some exponent “ x “ that varies - if 0 < a < 1, the graph shows exponential decay - the graph approaches zero as “x” gets larger

7. Exponential Functions To graph these functions we only need to complete an x / y table…

8. Exponential Functions To graph these functions we only need to complete an x / y table… EXAMPLE : Graph xy 38 2 1 0 -2 -3

9. Exponential Functions To graph these functions we only need to complete an x / y table… EXAMPLE : Graph xy 38 24 1 0 -2 -3

10. Exponential Functions To graph these functions we only need to complete an x / y table… EXAMPLE : Graph xy 38 24 12 0 -2 -3

11. Exponential Functions To graph these functions we only need to complete an x / y table… EXAMPLE : Graph xy 38 24 12 01.5 -2.25 -3.125

12. Exponential Functions To graph these functions we only need to complete an x / y table… EXAMPLE # 2 : Graph xy 30.02 2 1 0 -2 -3

13. Exponential Functions To graph these functions we only need to complete an x / y table… EXAMPLE # 2 : Graph xy 30.02 20.06 1 0 -2 -3

14. Exponential Functions To graph these functions we only need to complete an x / y table… EXAMPLE # 2 : Graph xy 30.02 20.06 10.25 0 -2 -3

15. Exponential Functions To graph these functions we only need to complete an x / y table… EXAMPLE # 2 : Graph xy 30.02 20.06 10.25 01 4 -216 -364

16. Exponential Functions Applications : Compound Interest Interest is sometimes paid from the day you make the investment to the day you liquidate that investment. When interest is compounded more than once per year, your interest earned in the first period(s) makes interest in subsequent periods.

17. Exponential Functions Applications : Compound Interest Interest is sometimes paid from the day you make the investment to the day you liquidate that investment. When interest is compounded more than once per year, your interest earned in the first period(s) makes interest in subsequent periods. Compound Interest equation

18. Exponential Functions Applications : Compound Interest Interest is sometimes paid from the day you make the investment to the day you liquidate that investment. When interest is compounded more than once per year, your interest earned in the first period(s) makes interest in subsequent periods. EXAMPLE # 1 : If $7,500 is invested at 12% interest compounded yearly, how much would be in the account after 5 years ?

19. Exponential Functions Applications : Compound Interest Interest is sometimes paid from the day you make the investment to the day you liquidate that investment. When interest is compounded more than once per year, your interest earned in the first period(s) makes interest in subsequent periods. EXAMPLE # 1 : If $7,500 is invested at 12% interest compounded yearly, how much would be in the account after 5 years ?

20. Exponential Functions EXAMPLE # 2 : If $9,000 is invested at 8% annual interest compounded monthly, how much would be in the account after 6 years ?

21. Exponential Functions EXAMPLE # 2 : If $9,000 is invested at 8% annual interest compounded monthly, how much would be in the account after 6 years ? A = 9000, r = 0.0067 ( 0.08/12 …divide your interest by 12 ) t = 72 ( 12 months x 6 years )

22. Exponential Functions EXAMPLE # 2 : If $9,000 is invested at 8% annual interest compounded monthly, how much would be in the account after 6 years ? A = 9000, r = 0.0067 ( 0.08/12 …divide your interest by 12 ) t = 72 ( 12 months x 6 years )

23. Exponential Functions EXAMPLE # 3 : How much money needs to be invested at 9% annual interest compounded monthly to get $15,000 in 5 years ?

24. Exponential Functions EXAMPLE # 3 : How much money needs to be invested at 9% annual interest compounded monthly to get $15,000 in 5 years ? A = 15,000 r = 0.0075 ( 0.09 / 12 ) T = 60 ( 5 x 12 )

25. Exponential Functions EXAMPLE # 3 : How much money needs to be invested at 9% annual interest compounded monthly to get $15,000 in 5 years ? A = 15,000 r = 0.0075 ( 0.09 / 12 ) T = 60 ( 5 x 12 )

26. Exponential Functions The “half – life “ of a radioactive element is the time it takes a given quantity of a substance to decay to one half of its original mass. Where :c = original mass t = time h = half life

27. Exponential Functions The “half – life “ of a radioactive element is the time it takes a given quantity of a substance to decay to one half of its original mass. Where :c = original mass t = time h = half life Example : Plutonium has a half-life of 24,360 years. How much of a 2 kg sample would be left after 50,000 years ?

28. Exponential Functions The “half – life “ of a radioactive element is the time it takes a given quantity of a substance to decay to one half of its original mass. Where :c = original mass t = time h = half life Example : Plutonium has a half-life of 24,360 years. How much of a 2 kg sample would be left after 50,000 years ?