8.2 Exponential Decay

P. 474

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

• State whether f(x) is an exponential growth or decay function

• f(x) = 5(2/3)x

• b=2/3, 0<b<1 it is a decay function.

• f(x) = 8(3/2)x

• b= 3/2, b>1 it is a growth function.

• f(x) = 10(3)-x

• rewrite as f(x)=10(1/3)x so it is decay

Recall from 8.1:

• 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)

Graph:• y = 3(1/4)x

• Plot (0,3) and (1,3/4)

• Draw & label asymptote

• Connect the dots using the asymptote

Domain = all reals Range = reals>0

y=0

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 realsRange : 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.

Example 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 realsRange : 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 aa is the initial amount and rr is the percent decrease expressed as a decimal.

• The quantity 1-r is called the decay factor

Ex: 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!!!

Assignment