7.2 Exponential
Decay
Algebra II
Exponential Decay
• Has the same form as growth
functions f(x) = abx
• Where a > 0
• BUT:
• 0 < b < 1 (a fraction between 0 & 1)
Recognizing growth and
decay functions
• Ex. 1) State whether f(x) is an
exponential growth or decay
function
• a.) f(x) = 5(2/3)x
• b=2/3, 0<b<1 it is a decay function.
• b.) f(x) = 8(3/2)x
• b= 3/2, b>1 it is a growth function.
• c.)f(x) = 10(3)-x
• rewrite as f(x)=10(1/3)x so it is decay
Recall from 7.1:
• The graph of y= abx
• Passes thru the point (0,a) (the y
intercept is a)
• The x-axis is the asymptote of the
graph
• a tells you up or down
• D is all reals (the Domain)
• R is y>0 if a>0 and y<0 if a<0
• (the Range)
Ex. 2) Graph:
• y = 3(1/4)x
• Plot (0,3) and
(1,3/4)
• Draw & label
asymptote
• Connect the dots
using the
asymptote
y=0
Domain = all reals
Range = reals>0
Ex. 3) Graph
• y = -5(2/3)x
• Plot (0,-5) and
(1,-10/3)
• Draw & label
asymptote
• Connect the dots
using the
asymptote
y=0
Domain : all reals
Range : y < 0
Now remember: To graph a
general Exponential
Function:
•
•
•
•
y = a bx-h + k
Sketch y = a bx
h= ??? k= ???
Move your 2 points h units left or
right …and k units up or down
• Then sketch the graph with the 2
new points & horizontal asymptote
of y=k.
Ex. 4) graph y=-3(1/2)x+2+1
• Lightly sketch
y=-3·(1/2)x
• Passes thru (0,-3)
& (1,-3/2)
• h=-2, k=1
• Move your 2
points to the left
2 and up 1
• AND your
asymptote k units
(1 unit up in this
case)
y=1
Domain : all reals
Range : y<1
Using Exponential Decay
Models
• When a real life quantity decreases by
fixed percent each year (or other time
period), the amount y of the quantity
after t years can be modeled by:
• y = a(1-r)t
• Where a is the initial amount and r is
the percent decrease expressed as a
decimal.
• The quantity 1-r is called
the decay factor
Ex. 5: Buying a car!
• You buy a new car for $24,000.
• The value y of this car decreases
by 16% each year.
• Write an exponential decay model
for the value of the car.
• Use the model to estimate the
value after 2 years.
• Graph the model.
• Use the graph to estimate when
the car will have a value of
$12,000.
• Let t be the number of years since
you bought the car.
• The model is: y = a(1-r)t
•
= 24,000(1-.16)t
•
= 24,000(.84)t
•
Note: .84 is the decay factor
• When t = 2 the value is
y=24,000(.84)2 ≈ $16,934
Now Graph
The car will have a
value of $12,000 in 4 years!!!
since
Assignment
•