7-7B Exponential Decay
Functions
Algebra 1
Exponential Decay
y  a b
x
a is the initial amount
b is the decay factor
y-intercept is (0,a)
Where a>0 & 0 < b< 1
WRITING EXPONENTIAL DECAY MODELS
A quantity is decreasing exponentially if it decreases by the same
percent in each time period.
EXPONENTIAL DECAY MODEL
C is the initial amount.
t is the time period.
y = C (1 – r)t
(1 – r ) is the decay factor, r is the decay rate.
The percent of decrease is 100r.
Ex. 1)Identify the initial amount
a & the decay factor b in each
exponential function.
• f(x) = 10  0.1x
Ex. 2 State whether the equation
represents exponential growth,
exponential decay or neither
y  0.93  2
y  7  0.5
x
x
Ex. 3
• You bought a used boat for $2300. The
value of the boat will be less each year
because of depreciation. The boat
depreciates at the rate of 8% per year.
• A.) Write an exponential decay model to
represent this situation.
Ex. 3 ( Cont.’d)
• You bought a used boat for $2300. The
value of the boat will be less each year
because of depreciation. The boat
depreciates at the rate of 8% per year.
• B.) Estimate the value of the boat in 2
years.
Example 4
• You bought a computer for $1800. The
value of the computer will be less each year
because of depreciation. The computer
depreciates at the rate of 29% per year.
• a.) Write an exponential decay model to
represent this situation.
Example 4 (Cont.’d)
• You bought a computer for $1800. The
value of the computer will be less each year
because of depreciation. The computer
depreciates at the rate of 29% per year.
• b.) Estimate the value of the computer in 3
years.
Assignment
•
GRAPHING EXPONENTIAL DECAY MODELS
CONCEPT
EXPONENTIAL GROWTH AND DECAY MODELS
SUMMARY
EXPONENTIAL GROWTH MODEL
y = C (1 + r)t
EXPONENTIAL DECAY MODEL
y = C (1 – r)t
An exponential model y = a • b t represents exponential
b > 1 and
exponential (1
decay
0 <decay
b < 1.factor,
(1 growth
+ r) is theifgrowth
factor,
– r) isifthe
Ct is
is the
the initial
time period.
amount.
C)
(0, C) rate.
r is the(0,growth
rate.
r is the decay
1+r>1
0<1–r<1
Writing an Exponential Decay Model
From 1982 through 1997, the purchasing power
of a dollar decreased by about 3.5% per year. Using 1982 as the base
for comparison, what was the purchasing power of a dollar in 1997?
COMPOUND INTEREST
SOLUTION
Let y represent the purchasing power and let t = 0 represent the year
1982. The initial amount is $1. Use an exponential decay model.
y = C(1 – r) t
Exponential decay model
= (1)(1 – 0.035) t
Substitute 1 for C, 0.035 for r.
= 0.965 t
Simplify.
Because 1997 is 15 years after 1982, substitute 15 for t.
y = 0.96515
Substitute 15 for t.
0.59
The purchasing power of a dollar in 1997 compared to 1982 was $0.59.
GRAPHING EXPONENTIAL DECAY MODELS
Graphing the Decay of Purchasing Power
Graph the exponential decay model in the previous example.
Use the graph to estimate the value of a dollar in ten years.
SOLUTION
Help
Make a table of values, plot the points in a coordinate
plane, and draw a smooth curve through the points.
t 0
1
2
3
4
5
6
7
8
9
10
y 1.00 0.965 0.931 0.899 0.867 0.837 0.808 0.779 0.752 0.726 0.7
y = 0.965t
0.8
(dollars)
Purchasing Power
1.0
0.6
0.4
0.2
0
1
2
3
Your dollar of today
will be worth about 70
cents in ten years.
4
5
6
7
8
Years From Now
9
10
11
12