read pages 531-534

8.6 Write and Graph Exponential Decay Functions

xy a b Exponential function.

0, 0, 1a b b

To be an exponential function, “a” can’t be zero, the base must be positive but can’t be one.

For GROWTH, “a” must be positive and the base must be greater than one

For DECAY, “a” must be positive and the base must be between zero and one.

Exponential Growth or Decay? WHY?

2xy

13

6

x

y

4 8x

y

8xy

6 2x

y

3 7x

y

14

4

x

y

12

8

x

y

DECAY, base is between zero and one

GROWTH, base is greater than one

GROWTH, base is greater than one

NEITHER, “a” is negative

GROWTH, base is greater than one

DECAY, base is between zero and one

NEITHER, “a” is negative

GROWTH, base is greater than one

decay

decay

decaygrowth

growth growth

As x increases by 1, each y value is multiplied by 1/5. This is an exponential function.

1

5b

1a

1

5

x

y

EXPONENTIAL DECAY MODELy = a(l - r)t

a is the initial amount. r is the decay rate.1 - r is the decay factor. t is the time period.

Notice how this differs from the Growth model…………..

Instead of adding the rate you subtract it! Everything else is the same.

Population The population of a city decreased from 1995 to 2003 by 1.5% annually. In 1995 there were about 357,000 people living in the city. Write a function that models the city’s population since 1995. Then find the population in 2003.

EXPONENTIAL DECAY MODELy = a(l - r)t

357,000(1 .015)ty

357,000(0.985)ty

1995: t=0 2003: t=8

8357,000(0.985)y

316,343y

In 2003 the population decreased to ~ 316,343 people.