Growth Decay

8.2 Exponential Decay

Goal 1: I will graph exponential decay functions.

Goal 2: I will use exponential decay functions to model real-life examples.

Where is this used?

Value of used cars

radioactive decay

Caffeine in body

Vocabularyexponential decay function:involves the equation

where b is the base, and a 00 b 1and a is the initial value of y when x = 0, and b is that growth factor.

y = abx

Exponential Factors

• If the factor b is greater than 1, then we call the relationship exponential growth.

• If the factor b is less than 1, we call the relationship exponential decay.

practice

Goal 1: Graph exponential decay functions

State whether the function is an exponential growth or exponential decay function.

a. f (x) 52

3

x

b. f (x) 83

2

x

growthdecay

c. f (x) 10 3 x

101

3

x

decayd. f (x) 4

3

8

x

decaye. f (x) 3

4

3

x

growth

Graph the function

x f (x) 3g

1

4

x y

1f ( 1) 3g

1

4

1

120 f (0) 3g

1

4

0

31 f (1) 3g

1

4

1 3

4

2 f (2) 3g

1

4

2 3

16

Graph the function

domain: all real #s

range: y > 0

Exponential Decay ModelsWhen a real-life quantity (e.g. value of car) decreases byfixed percent each time period, the amount y of the quantity after t time periods is:

y C r t ( )1

Where

C = initial amount

r = growth rate (percent written as a decimal)

t = time where t 0

(1 - r) = decay factor where 1 - r < 1

Value of CarYou buy a new car for $23,000. The value decreases by 15% each year. Write an exponential decay model for the car’s value. Use the model to estimate the value after 3 years.

a 23,000 r 15% 0.15

y 23000(1 0.15)t

y 23000(0.85)t

f (3) 23000(0.85)3 $14,125

y C r t ( )1

practice

An adult takes 400 mg of ibuprofen. Each hour h, the amount i of ibuprofen in the person’s system decreases by about 29%.

Write an exponential decay model that describes the situation.

How much is left in the system after 4 hours?

f (4) 101.6 mg

y C(1 r)t

y 400(1 0.29)t

y 400(0.71)t

Independent Practice

Suppose that 100 pounds of plutonium (Pu) is deposited at a nuclear waste site. How much of it will still be radioactive in 100,000 years? Solve graphically and algebraically.