Transcript 8.8 Exponential Growth and Decay • Exponential Growth – Modeled with the function: y = a • bx for a > 0 and.

8.8 Exponential Growth and Decay
• Exponential Growth
– Modeled with the function:
y = a • bx for a > 0 and b > 1.
y = a • bx
a = the starting amount (when x = 0)
b = the base, which Is greater than 1, is the growth
factor
x = exponent
Modeling Exponential Growth
• Since 1985, the daily cost of patient care in community
hospitals in the United States has increased about 8.1%
per year. In 1985, such hospital costs were an average of
$460 per day.
a. Write an equation to model the cost of hospital care.
Relate: y = a • bx
Define: Let x = the number of years since 1985
Let y = the cost of community hospital care at
various times
Let a = the initial cost in 1985, $460
Let b = the growth factor, which is
100% + 8.1% = 108.1% = 1.081
Write: y = 460 • 1.081x
Modeling Exponential Growth
b. Use your equation to find the approximate cost per day in
2000.
y = 460 • 1.081x
y = 460 • 1.08115
y ≈ 1480
The average cost per day in 2000 was about $1480.
Compound Interest
• Suppose your parents deposited $1500 in an account
paying 6.5% interest compounded annually (once a year)
when you were born. Find the account balance after 18
years.
Relate: y = a • bx
Define: Let x = the number of interest periods
Let y = the balance
Let a = the initial deposit, $1500
Let b = 100% + 6.5% = 106.5% = 1.065
Write: y = 1500 • 1.065x
= 1500 • 1.06518
≈ 4659.98
The balance after 18 years will be $4659.98.
Annual Interest Rate of 8%
Compounded
Annually
Periods per
Year
1
Interest Rate
per Period
8% every year
Semi-annually
2
Quarterly
4
Monthly
12
4% every 6
months
2% every 3
months
0.6% every
month
Compound Interest
• Suppose the account in the other example paid interest
compounded quarterly instead of annually. Find the
account balance after 18 years.
Relate: y = a • bx
Define: Let x = the number of interest periods
Let y = the balance
Let a = the initial deposit, $1500
Let b = 100% + 6.5%
4
= 1 + 0.01625 = 1.01625
Write: y = 1500 • 1.01625 x
= 1500 • 1.0162572
≈ 4787.75
The balance after 18 years will be $4787.75.