Representing Exponential Growth

*Sec 2.1.1: How does the pattern grow?

*Multiplying Like Bunnies

*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.

*Continued…

*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.

*Further Guidance

*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.

*Table

Number of

months

Number of

Bunnies0 21 42 83 164 325 646 1287 2568 512

*Further Guidance

*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?

-1 1 2 3 4 5 6 7 8 9

102030405060708090

number of months

number of bunnies

-1 1 2 3 4 5 6 7 8 9

102030405060708090

number of months

number of bunnies

2 4 6 8 10 12 14number of months

number of bunnies

*Further Guidance

*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.

*Table and Rule

Number of

months

Number of

Bunnies0 21 42 83 164 325 646 1287 2568 512

*Initial amount*When x=0*Starting amount.

Multiple by 2Multiple by 2Multiple by 2

𝑓 (𝑥 )=2∙2𝑥

*Friends

*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?

*Friends continued…

*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.

*Table and Rule

Number of

months

Number of

Bunnies0 101 202 403 804 1605678

*Initial amount*When x=0*Starting amount.

Multiple by 2Multiple by 2Multiple by 2

𝑓 (𝑥 )=10 ∙2𝑥

*Friends continued…

*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.

*Table and Rule

Number of

months

Number of

Bunnies0 21 62 183 544 1625678

*Initial amount*When x=0*Starting amount.

Multiple by 3Multiple by 3Multiple by 3

𝑓 (𝑥 )=2∙3𝑥

*Friends continued…

*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.

*Table and Rule

Number of

months

Number of

Bunnies0 21 82 323 1284 5125678

*Initial amount*When x=0*Starting amount.

Multiple by 4Multiple by 4Multiple by 4

𝑓 (𝑥 )=2∙ 4𝑥

*Generalize

f(x)=ab x

Initial amountStarting amountY interceptf(0)

Growth factorRate of change

InputEnding amount

*Friends continued…

Which case would give Chandler and Joey the most rabbits after one year?

*On your Own

*Read and take notes on:

Methods and Meanings on page 56

*Homework:R & PPage 57 #13-19Due at the beginning of next class period.

*Friends continued…

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.

*Friends continued…

*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.

*LEARNING LOG!

*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.