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Exponential Functions
Exponential Function
f(x) = ax
for any positive number a other than one.
Examples
• What are the domain and range of
y = 2(3x) – 4?
• What are the roots of 0 =5 – 2.5x?
Properties of Powers (Review)
• When multiplying like bases, add exponents.
ax ● ay = ax+y
• When dividing like bases, subtract exponents.
x
a
x y
a
y
a
• When raising a power to a power, multiply
exponents.
(ax)y=axy
Properties of Powers (Review)
• When you have a monomial or a fraction
raised to a power (with no add. or sub.), raise
everything to that power.
x
( ab )  a b
x
x
x
or
x
a
a
   x
b
b
Half-Life & Exponential Growth/Decay
• The half-life of a substance is the time it takes
for half of a substance to exist.
▫ Mirrors the behavior of Exponential Growth &
Decay functions.
 Exponential Growth: y = kax, if a > 1
 k is the initial amount present
 a is the rate at which the amount is growing
 Exponential Decay: y = kax, 0 < a < 1
 k is the initial amount present
 a is the rate at which the amount is growing
Example
• Suppose the half-life of a certain radioactive
substance is 20 days and that there are 5 grams
present initially. When will there be only 1 gram
of the substance remaining?
After 20 days:
IN GENERAL:
1 5
5  
2 2
After 40 days:
1
y  5 
2
t
Models the mass
of the substance
after t days.
y1  1
2
5
 1  1 
1
5    5  
4
 2  2 
2
20
Therefore, let
1
y 2  5 
2
t
20
graph, and find intersection.
t ≈ 46.44 days
Exponential Growth/Decay
Example:
A population initially contains 56.5 grams of a
substance. If it is increasing at a rate of 15% per
week, approximately how many weeks will it
take for the population to reach 281.4 grams?
Exponential Growth
Example:
How long will it take a population to triple if it is
increasing at a rate of 2.75%?
The Number e
• Many real-life phenomena are best modeled
using the number e
▫ e ≈ 2.718281828
• e can be approximated by: f  x    1  1 
x

As x  , f  x   e
x
• Interest compounding continuously:
I = Pert, where P = initial investment,
r = interest rate (decimal)
t = time in years
Example Compounding Interest
• A deposit of $2500 is made in an account that
pays an annual interest rate of 5%. Find the
balance in the account at the end of 5 years if the
interest is compounded
a.) quarterly
b.) monthly
c.) continuously
Suggested HW
• Sec. 1.3 (#5, 7, 11, 19, 21-31 odd)
• 1.3 Web Assign Due Monday night
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