*
Representing Exponential
Growth
*2-1. In the book Of Mice and Men by John
Steinbeck, two good friends named Bert and
Ernie dream of raisin rabbits and living off the
land. What if their dreams came true?
*Suppose Bert and Ernie started with two
rabbits that had two babies after one month,
and suppose that every month thereafter,
each pair of rabbits had two babies.
*
*Our task is to determine how
many rabbits Bert and Ernie
would have after one year.
Spend a few minutes with your
team to discuss the problem and
strategize.
*
* How can you determine the number the
number of rabbits that will exist at the end of
one year?
Organize your information in a table
showing the total number of rabbits
for the first several months.
*
Number
of months
0
1
2
3
4
5
6
7
8
Number
of
Bunnies
2
4
8
16
32
64
128
256
512
*
* How can you determine the number the
number of rabbits that will exist at the end of
one year?
*
*
Display a graph to represent how
the total number of rabbits is
growing each month.
How many rabbits will Bert and
Ernie have after three months?
*
number of bunnies
90
80
70
60
50
40
30
20
10
-1
number of months
1
2
3
4
5
6
7
8
9
number of bunnies
90
80
70
60
50
40
30
20
10
-1
number of months
1
2
3
4
5
6
7
8
9
number of bunnies
number of months
2
4
6
8
10
12
14
* How can you determine the number the number of
rabbits that will exist at the end of one year?
•
write a rule for the relationship
between the total number of
rabbits and the number of
months that have passed since
Bert and Ernie obtained the first
pair of rabbits.
*
Number
of months
0
1
2
3
4
5
6
7
8
Number
of
Bunnies
2
4
8
16
32
64
128
256
512
* Initial amount
* When x=0
* Starting amount.
Multiple by 2
Multiple by 2
Multiple by 2
𝑓 𝑥 =2∙2
*
𝑥
* 2-3. Joey and Chandler want to raise as many
rabbits as possible, so they have a few options
to consider. They could start with a larger
numbers of rabbits, or they could raise a breed
of rabbits that reproduces faster. How would
each of these options change the pattern of
growth you observed in the previous problem?
Which situation would yield the largest
population after one year?
*
*a. To help answer these questions ,
model each case below with a table.
Then use the patterns in each table
to write a rule for each case.
* Case 2: Start with 10 rabbits;
each pair has 2 babies per
month.
*
Number
of months
0
1
2
3
4
5
6
7
Number
of
Bunnies
10
20
40
80
160
* Initial amount
* When x=0
* Starting amount.
Multiple by 2
Multiple by 2
Multiple by 2
𝑓 𝑥 = 10 ∙ 2
8
*
𝑥
* a. To help answer these questions , model each
case below with a table. Then use the patterns in
each table to write a rule for each case.
*Case 3: Start with 2 rabbits; each
pair has 4 babies per month.
*
Number
of months
0
1
2
3
4
5
6
7
Number
of
Bunnies
2
6
18
54
162
* Initial amount
* When x=0
* Starting amount.
Multiple by 3
Multiple by 3
Multiple by 3
𝑓 𝑥 =2∙3
8
*
𝑥
* a. To help answer these questions , model each case
below with a table. Then use the patterns in each
table to write a rule for each case.
* Case 4: Start with 2 rabbits; each
pair has 6 babies per month.
*
Number
of months
0
1
2
3
4
5
6
7
Number
of
Bunnies
2
8
32
128
512
* Initial amount
* When x=0
* Starting amount.
Multiple by 4
Multiple by 4
Multiple by 4
𝑓 𝑥 =2∙4
8
*
𝑥
f(x)=ab
Initial amount
Starting amount
Y intercept
f(0)
x
Growth factor
Rate of change
*
Which case would give Chandler
and Joey the most rabbits after
one year?
*
*Read and
*Homework:
take notes
on:
Methods and
Meanings on
page 56
R&P
Page 57 #13-19
*
Due at the
beginning of
next class
period.
2-3a. To help answer these questions , model
each case below with a table. Then use the
patterns in each table to write a rule for each
case.
Case 2: Start with 10 rabbits; each pair
has 2 babies per month.
Case 3: Start with 2 rabbits; each pair has
4 babies per month.
Case 4: Start with 2 rabbits; each pair has
6 babies per month.
*
* c. Now make up your own case, stating the
initial number of rabbits and the number of
babies each pair has per month. Organize your
information in a table and write a rule from
the pattern you observe in your table. Show
how your table is connected to its equation
using color-coding, arrows, and any other tools
that help you show the connections.
*
2-4 A NEW FAMILY?
Is the data in the “Multiplying like bunnies”
linear, or is it an example of some other
relationship?
a. Look back at the x-y tables you created in
problem 2-3. What do they all have in common?
* 2-4 A NEW FAMILY?
* b. Graph all four of the equations from
problems 2-1 and 2-3 on your graphing
calculator. Adjust the viewing window so that
all four graphs show up clearly. Then, on
paper, sketch the graphs and label each graph
with its equation. How would you describe the
graphs?
* C. Now decide whether the data in the rabbit
problem is linear. Justify your conclusion.
* To represent the growth in the number of
rabbits in problems 2-1 and 2-3, you discovered
a new family of functions that are not linear.
Functions in this new family are called
exponential functions.
* Write a Learning Log entry to record what you
have learned so far about exponential
functions.
*