1. 7.2 Exponential Decay
Algebra II

2. Exponential Decay Has the same form as growth functions f(x) = abx
Where a > 0
BUT:
0 < b < 1 (a fraction between 0 & 1)

3. 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

4. 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)

5. 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

6. 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

7. 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.

8. 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)

9. y=1
Domain : all reals
Range : y<1

10. 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

11. 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.

12. 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

13. Now Graph
The car will have a
value of $12,000 in 4 years!!!
since

14. Assignment