College Algebra 9th txtbk

MATCHED PROBLEM 3
SECTION 5–1 Exponential Functions 333
y
f 1 f 2
y
5
5
y 4
x
5
5
x
5
5
5
5
f 3
g(x) 4 e x/2
(a)
(b)
Z Figure 6
Let g(x) 2e x2 5. Use transformations to explain how the graph of g is related to the
graph of f 1 (x) e x . Describe the increasing/decreasing behavior, find any asymptotes, and
sketch the graph of g.
Z Compound Interest
The fee paid to use someone else’s money is called interest. It is usually computed as a
percentage, called the interest rate, of the original amount (or principal) over a given
period of time. At the end of the payment period, the interest paid is usually added to the
principal amount, so the interest in the next period is earned on both the original amount,
as well as the interest previously earned. Interest paid on interest previously earned and
reinvested in this manner is called compound interest.
Suppose you deposit $1,000 in a bank that pays 8% interest compounded semiannually.
How much will be in your account at the end of 2 years? “Compounded semiannually”
means that the interest is paid to your account at the end of each 6-month period, and the
interest will in turn earn more interest. To calculate the interest rate per period, we take
the annual rate r, 8% (or 0.08), and divide by the number m of compounding periods per
year, in this case 2. If A 1 represents the amount of money in the account after one compounding
period (6 months), then
Factor out $1,000.
We will next use A 2 , A 3 , and A 4 to represent the amounts at the end of the second, third,
and fourth periods. (Note that the amount we’re looking for is A 4 .) A 2 is calculated by multiplying
the amount at the beginning of the second compounding period (A 1 ) by 1.04.
A 2 A 1 (1 0.04)
[$1,000(1 0.04)](1 0.04)
$1,000(1 0.04) 2
A 3 A 2 (1 0.04)
A 1 $1,000 $1,000 a 0.08
2 b
[$1,000(1 0.04) 2 ](1 0.04)
$1,000(1 0.04) 3
Principal 4% of principal
$1,000(1 0.04)
A 4 A 3 (1 0.04)
[$1,000(1 0.04) 3 ](1 0.04)
$1,000(1 0.04) 4
$1,169.86
Substitute our expression for A 1 .
Multiply.
P a1 r 2
m b
Substitute our expression for A 2 .
Multiply.
P a1 r 3
m b
Substitute our expression for A 3 .
Multiply.
P a1 r 4
m b