1. 5.2 exponential functions

2. Quiz
Fill in the blanks below:
2x+y = __ * __

3. Exponential function Standard form: f(x) = ax , where a>0, a ≠ 1.
Example: f(x) = 2x, f(x) = (1/3)x
Compare g(x) = x2 and f(x) = 2x

4. properties
Graph f(x) = 2x and g(x) = (1/2)x

5. Properties Characteristics of f(x) Continuous One to one
Domain: (- ∞, ∞)
Range: (0, ∞)
Increasing if a>1(growth)
Decreasing if 0<a<1(Decay)
Horizontal asymptote at y = 0.
Key points on the graph: (1,a), (0,1)

6. Graphs by transformations
Describe how each of the following can be obtained from the graph of f(x) = 2x.
a. f(x) = 2x+3
b. f(x) = 2x – 1
c. f(x) = x
Example: exercise #37, #69

7. Exponential Equations
ab = ac  b = c
Solve for x:
1. 2x-3 = 8
2. (1/4)3 = 8x
3. 274x = 9x+1
Use a graphing calculator to solve 2x-3 > 8 or 2x-3 ≤ 8

8. Natural Base -- e e = (1 + 1/k)k as k approaches positive infinite.
Natural exponential function: f(x) = ex

9. A = P(1 + r/n)nt Compound interest
Suppose that a principal of P dollars is invested at an annual interest rate r, compounded n times per year. Then the amount A accumulated after t years is given by the formula
A = P(1 + r/n)nt
A = Amount accumulated after t years
P = principal
r = annual interest rate
n = compounded times of a year

10. Typical Compounding periods
Compound annually: n = 1
Compound semi- annually: n = 2
Compound quarterly: n = 4
Compound monthly: n = 12
Compound weekly: n = 52
Compound daily: n = 365

11. example
Suppose that $100,000 is invested at 6.5% interest, compounded semi-annually.
1. Find a function for the amount of money after t years
2. Find the amount of money in the account at t = 1,4,10 years.

12. Continuous Compounding
As the number of compounding periods increases without bound, the model becomes
A = Pert

13. example
If you put $7000 in an money market account that pays 4.3% a year compounded continuously, how much will be in the account in 15 years?
You have $1500 to invest. Which is better – 2.25% compounded quarterly for 3 years? Or 1.75% compounded continuously for r years?

14. Homework PG. 339: 3-18(M3), 38, 39-75(M3), 80 KEY: 38, 54, 60, 75
Reading: 5.3 Logarithms and their properties