7.7 EXPONENTIAL GROWTH AND DECAY:
Exponential Growth: An equation that
increases.
Exponential Decay: An equation that
decreases.
Growth Factor: 1 plus the percent rate of
change which is expressed as a decimal.
Decay Factor: 1 minus the percent rate of
change expressed as a decimal.
GOAL:
Definition:
An EXPONENTIAL FUNCTION is a function
of the form:
𝑦 =𝑎∙𝑏
Constant
Base
𝑥
Exponent
Where a ≠ 0, b > o, b ≠ 1,
and x is a real number.
GRAPHING: To provide the graph of the
equation we can go back to basics and
create a table.
Ex:
What is the graph of y = 3∙2x?
GRAPHING:
X
y = 3∙2x
y
-2
3∙2(-2)
-1
3∙2(-1) =
𝟑
𝟐𝟐
𝟑
𝟐𝟏
0
3∙2(0) = 3∙1
𝟑
𝟒
𝟑
𝟐
3
1
3∙2(1) = 3∙2
6
2
3∙2(2) = 3∙4
12
=
GRAPHING:
X
y
0
𝟑
𝟒
𝟑
𝟐
3
1
6
2
12
-2
-1
This graph grows fast = Exponential Growth
YOU TRY IT:
Ex:
What is the graph of y = 3∙
𝟏 x
?
𝟐
GRAPHING:
X
-2
-1
0
1
2
y = 3∙
𝟏 x
𝟐
y
3∙
𝟏 (-2)
2
=3∙(2)
𝟐
12
3∙
𝟏 (-1)
1
=3∙(2)
𝟐
6
3∙
𝟏 (0)
𝟐
3
3∙
𝟏 (1)
𝟏
=3∙
𝟐
𝟐
3∙
𝟏 (2)
𝟏
=3∙
𝟐
𝟒
= 3∙1
𝟑
𝟐
𝟑
𝟒
GRAPHING:
X
y
-2
12
-1
6
0
3
1
2
𝟑
𝟐
𝟑
𝟒
This graph goes down = Exponential Decay
YOU TRY IT:
Ex:
What are the differences and
similarities between:
y = 3∙2x
and
y = 3∙
𝟏 x
?
𝟐
y = 3∙2x
 Base = 2  Exponential growth
 y- intercept (x=0) = 3
y = 3∙
𝟏 x
𝟐
𝟏
𝟐
 Base =  Exponential Decay
 y- intercept (x=0) = 3
MODELING: We use the concept of
exponential growth in the real world:
Ex:
Since 2005, the amount of money
spent at restaurants in the U.S. has
increased 7% each year. In 2005, about
36 billion was spend at restaurants. If
the trend continues, about how much
will be spent in 2015?
EVALUATING: To provide the solution
we must know the following
formula:
y=
x
a∙b
y = total
a = initial amount
b = growth factor (1 + rate)
x = time in years.
SOLUTION:
Since 2005, … has increased 7% each year. In
2005, about 36 billion was spend at restaurants….
about how much will be spent in 2015?
Y= total:
unknown
Initial:
$36 billion
Growth:
1 + 0.07
Time (x): 10 years
(2005-2015)
x
a∙b
y=
10
y = 36∙(1.07)
y = 36∙(1.967)
y = 70.8 b.
BANKING:
We also use the concept of
exponential growth in banking:
A = P(1+
𝒓 nt
)
𝒏
A = total balance
P = Principal (initial) amount
r = interest rate in decimal form
n = # of times compound interest
t = time in years.
MODELING GROWTH:
Ex:
You are given $6,000 at the beginning
of your freshman year. You go to a bank
and they offer you 7% interest. How
much money will you have after
graduation if the money is:
a) Compounded annually
b) Compounded quarterly
c) Compounded monthly
COMPOUNDED ANNUALLY:
A = P(1+
A=?
P = $6000
r = 0.07
n=1
t = 4 yrs
𝒓 nt
)
𝒏
A = 6000(1+
𝟎.𝟎𝟕 1(4)
)
𝟏
4
6000(1.07)
A=
A = 6000(1.3107)
A = $7864.77
COMPOUNDED QUARTERLY:
A = P(1+
𝒓 nt
)
𝒏
𝟎.𝟎𝟕 4(4)
A=?
A = 6000(1+
)
𝟒
P = $6000
16
A = 6000(1.0175)
r = 0.07
n = 4 times A = 6000(1.3199)
t = 4 yrs
A = $7919.58
COMPOUNDED MONTHLY:
A = P(1+
𝒓 nt
)
𝒏
𝟎.𝟎𝟕 12(4)
A=?
A = 6000(1+
)
𝟏𝟐
P = $6000
48
A = 6000(1.0058)
r = 0.07
n = 12 timesA = 6000(1.3221)
t = 4 yrs
A = $7932.32
MODELING DECAY:
Ex:
Doctors can use radioactive iodine to
treat some forms of cancer. The half-life
of iodine-131 is 8 days. A patient
receives a treatment of 12 millicuries (a
unit of radioactivity) of iodine-131.
How much iodine-131 remains in the
patient after 16 days?:
To provide the solution we g
back to the following formula:
DECAY:
y=
x
a∙b
y = total
a = initial amount
b = decay factor (1 - rate)
x = time in years.
SOLUTION: The half-life of iodine-131 is 8 days. A
patient receives a treatment of 12 millicuries
(a unit of radioactivity) of iodine-131. How much
iodine-131 remains in the patient 16 days later?:
Y= total:
unknown
Initial:
12
Growth:
1- 1/2
Time (x): 16/8 = 2
x
a∙b
y=
2
y = 12∙(1/2)
y = 12∙(.25)
y=3
VIDEOS:
Exponential
Functions
Growth
https://www.khanacademy.org/math/trigonometry/expon
ential_and_logarithmic_func/exp_growth_decay/v/expone
ntial-growth-functions
Graphing
https://www.khanacademy.org/math/trigonometry/expon
ential_and_logarithmic_func/exp_growth_decay/v/graphi
ng-exponential-functions
VIDEOS:
Exponential
Functions
Decay
https://www.khanacademy.org/math/trigonometry/expon
ential_and_logarithmic_func/exp_growth_decay/v/wordproblem-solving--exponential-growth-and-decay
CLASSWORK:
Page 450-452:
Problems: As many as needed
to master the
concept.