Objectives: Today we will …
1.Write and solve exponential growth functions.
2.Graph exponential growth functions.
Vocabulary: exponential growth
Exponential Growth Functions8.5
Would You Rather … ?!?!
• After arguing with your family that you should get a higher allowance your
family offers you two allowance options. Either, they will give you $20 each
week or they will give you one penny on the first day and double your
allowance every day for 31 days. What option would you pick?
Back-up your answer with math!
The Solution …
1 $.01
2 $.02
3 $.04
4 $.08
5 $.16
6 $.32
7 $.64
8 $1.28
9 $2.56
10 $5.12
11 $10.24
12 $20.48
13 $40.96
14 $81.92
15 $163.84
16 $327.68
17 $655.36
18 $1310.72
19 $2621.44
20 $5242.88
21 $10,485.76
22 $20,971.52
23 $41,943.04
24 $83,886.08
25 $167,772.16
26 $335,544.32
27 $671,088.64
28 $1,342,177.2
8
29 $2,684,354.56
30 $5,368,709.12
31 $10,737,418.24
Real World Exponential Growth Example
• http://www.mathwarehouse.com/exponential-growth/exponential-models-
in-real-world.php
Exponential Growth Functions8.5
EXPONENTIAL GROWTH MODEL
A quantity is growing exponentially if it increases by the same percent in each time period.
C is the initial amount. t is the time period.
(1 + r) is the growth factor, r is the growth rate.
Exponential growth always has a growth rate greater than or equal to one. (1 + r) ≥ 1
y = C (1 + r)tSometimes use P instead of C Note: measure of
rate and time MUST be in the same time unit
Example 1 Compound Interest
You deposit $1500 in an account that pays 2.3% interest compounded yearly,
1)What was the initial principal (C) invested?
2)What is the growth rate (r)? The growth factor?
3)Using the equation y = C(1+r)t, write the equation that models this situation. Then figure out how much money would you have after 2 years if you didn’t deposit any more money?
C or P = $1500
Growth rate (r) is 0.023. The growth factor is 1.023.
y = $1569.79
Example 2 Compound Interest
A savings certificate of $1000 pay 6.5% annual interest compounded yearly. First, write the
equation that models this situation. Then figure out what is the balance when the certificate
matures after 5 years?
≈ $1370.09
1.What is the percent increase each year?
2.Write a model for the number of rabbits in any given year.
3.Find the number of rabbits after 5 years.
Example 3 Exponential Growth Model
A population of 20 rabbits is released into a wild-life region. The population triples each year for 5 years.
200%
≈ 4860 rabbits
y =20(1+2.0
0)t
Exponential Growth Model
Graph the growth of the rabbits.
Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points.
t
y 486060 180 540 162020
51 2 3 40
0
1000
2000
3000
4000
5000
6000
1 72 3 4 5 6Time (years)
Po
pu
lati
on
P = 20 ( 3 ) t Here, the large
growth factor of 3 corresponds to a rapid increase
Here, the large growth factor of 3 corresponds to a rapid increase
Example 4
1.Write a model for the weight during the first 6 week.
2.Find the weight at the end of six weeks.
Example 5 Exponential Growth Model
A newly hatched channel catfish typically weighs about .3 grams. During the first 6 weeks of life, its growth is approximately exponential, increasing by about 10% a day.
y =.3(1+.10)t
≈ 16.4 grams
Example 6 Exponential Growth Model
Graph
Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points.
t
y
0
y =3(1.10)t
1.Write a model for the number of bacteria at any hour.
2.Find the number of bacteria after 8 hours.
Example 7 Exponential Growth Model
An experiment started with 100 bacteria. They double in number every hour.
y =100(1+1.
00)t≈ 25,600
bacteria
• pgs. 480-481 #1, 4, 5, 14, 15, 21, 22, 24
Homework