Have you ever wondered how many people might inhabit the US in 20

years?For example, if the US population grows by 1.5% each year, how many people

will there be in 20 years?

In this lesson you will learn how to create and graph

exponential relationships by using exponential functions

Let’s Review

Example: You start with 25 dots, and the number of dots increase by 40% in

every step.

40% growth 40% growth

Let’s Review

40% growth 40% growth

y = a(1+r)x

y = 25(1+.4)x

y = 25(1.4)x

Let’s Review

Exponential growth

Exponential decay

A Common Mistake

If the values are growing, the growth factor is greater than 1

y = 25(1.4)x

If it is decaying, the decay factor is less than 1 but more than 0

y = 25(.4)x

Core Lesson

We will investigate the following:The population of the United States has

grown by approximately 1.5% each year for the past 100 years. If the population of the US was 92.2 million 100 years ago, create

and graph the function relating time and the population of the US.

Core Lesson

p = a(1+r)t

p = (92.2)*(1+r)t

p = (92.2)*(1+.015)t

p = (92.2)*1.015t

92.2 (million) start pop.

1.5% growth every year

Core Lesson

time (years)

pop

ula

tion

(millio

ns)

Input (t) Output (p)

0 92.2

10 107

20 124

30 144

40 167

50 194

75 281

100 408

p = (92.2)*1.015t

050

100150200250300350400450

0 50 100

In this lesson you have learned how to create and graph

exponential relationships by using exponential functions

Guided Practice

We will investigate the following:During the 19th century, American settlers

hunted bison almost to extinction. If 15% of the population was killed each year, and they

started with an initial population of 1,000,000, create and graph the relationship

between time and number of bison remaining.

Guided Practice p = a(1+r)t

p = (1000)*(1+r)t

p = (1000)*(1-.15)t

p = (1000)*0.85t

1000 (thousands) start pop.

15% decay every year

Guided Practice

time (years)

pop

ula

tion

(thou

san

ds)

Input (t) Output (p)

0 1000

5 443

10 197

15 87

20 39

25 17

30 7.6

50 .29

p = (1000)*0.85t

0

200

400

600

800

1000

1200

0 20 40 60

Extension Activities

1. Explore what effect changing the growth/decay factor has on a populations growth/decay.

2. Your bank account may have a “compounded interest”. Investigate compound interest and relate it to exponential functions.

3. Investigate the population of the world over the past 4000 years. Can the population be modeled with an exponential function. Justify your answer.

Quick Quiz

1. A population of bacteria will grow by about 20% each hour. If a colony of bacteria start with 2000, create and graph the relationship between time and colony size.

2. A certain new car’s value is said to “depreciate” (lose value exponentially) at a rate of 12% each year. If the car was originally priced at $28,000, create and graph the relationship between time and the car’s value.