Transcript 5.2 exponential functions

5.2 EXPONENTIAL FUNCTIONS
QUIZ

Fill in the blanks below:
2x+y = __ * __
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

PROPERTIES

Graph f(x) = 2x and g(x) = (1/2)x
PROPERTIES

1.
2.
3.
4.
5.
6.
7.
8.
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)
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) = 3 + 2-x
Example: exercise #37, #69
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
NATURAL BASE -- E
e = (1 + 1/k)k as k approaches positive infinite.
 e ≈ 2.71828
 Natural exponential function: f(x) = ex

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
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

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.
CONTINUOUS COMPOUNDING

As the number of compounding periods increases
without bound, the model becomes
A = Pert
EXAMPLE
1.
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?
2.
You have $1500 to invest. Which is better –
2.25% compounded quarterly for 3 years? Or
1.75% compounded continuously for r years?
HOMEWORK

PG. 339: 3-18(M3), 38, 39-75(M3), 80

KEY: 38, 54, 60, 75

Reading: 5.3 Logarithms and their properties