1. Exponential Functions Exponential functions Geometric Sequences

2. 8.1 Exponential functions and their graphs Recognize and evaluate exponential functions with base a Graph exponential functions Recognize and evaluate exponential functions with base e Use exponential functions to model and solve real-life problems

3. Exponential functions

4. Graph: To graph exponential functions, make a table, plot the points and connect them with a smooth curve. x f(x) g(x) 0 1 2 Domain: Range: y -intercept:

5. Translations y = b x - original graph y = ab x-h + k k - positive - moves graph up k - negative - moves graph down h - positive - to the right h - negative - to the left a < 0 - reflected over x - axis a > 1 - stretched 0< a < 1 - compressed

6. Graph: green Domain: Range: y -intercept Domain: Range: y -intercept

7. A transformation of the graph: x y Domain:Range:

8. 8.2 Solving exponential equations

9. To solve exponential equations, get the bases equal. Solve for x: One to one property! Bases must be the same

10. BONUS!!

11. Compound Interest Formulas After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formula: 1) For n compounding per year: Find the account balance after 20 years if $100 is placed in an account that pays 1.2% interest compounded twice a month.

12. If $350,000 is invested at a rate of 5½% per year, find the amount of the investment at the end of 10 years for the following compounding methods: a) Quarterly b) Monthly

13. Continuously Compound Interest A = Pe rt Joan was born and her parents deposited $2000 into a college savings account paying 4% interest compounded continuously. What would be the balance after 15 years.

14. exponential decay: P = initial amount (1 - r)/(1 + r) is the decay/growth factor, r is the decay rate…0 < r < 1 t is the time period A = final amount You bought a used car for $18,000. The value of the car will be less each year because of depreciation. The car depreciates (loses value) at the rate of 12% per year. Write an exponential decay model to represent the situation then use that model to estimate the value of the car in 8 years. exponential growth:

15. Graph and find the average rate of change in value from year 0 to year 4 t y years value 2 2000

16. 8.3 Logarithmic Functions Write exponential functions as logarithms Write logarithmic functions as exponential functions

17. A logarithm function is another way to write an exponential function y is the logarithm, a is the base, x is the number We can now convert from one form to the other.

18. Rewrite as a exponential equation:

19. Rewrite as a logarithm:

20. To find the exact value of a logarithm (or evaluate), we can change the equation to an exponential one. Evaluate: log 3 81 log 1/2 256

21. log 13 169 Evaluate:

22. 11.3/11.4 Sum of Geometric Sequences

23. Nth term of geometric sequence: (this is used to find any of the items or terms) a n = a 1 r n-1 a 1 = 1st term r = common ratio (divide any term by the prior term) n = how many terms Write an equation for the nth term of a geometric sequence -.25, 2, -16, 128,...

24. Find the equation for the geometric sequence: If a 3 = 16, and r = 4

25. The geometric means are terms between non- consecutive terms of a geometric sequence. Find the 4 geometric means between.5, __, __, __, __, 512

26. Partial Sum of a Geometric Series Find the sum of the geometric series a 1 = 2, n = 10, r = 3

27. Find the sum of the geometric series a 1 = 2000, a n = 125, r = 1/2 Find a 1 in a geometric series for which S n = -26240, n = 8, r = -3

28. Or use the sum formula: Find the sum: