**Different balls bounce at various heights depending on things like the type of ball, the**

pressure of air in the ball, and the surface on which it is bounced. The rebound percentage

of a ball is found by determining the quotient of the rebound height (that is the height of

each bounce) to the height of the ball before that bounce, converted to a percentage.

pressure of air in the ball, and the surface on which it is bounced. The rebound percentage

of a ball is found by determining the quotient of the rebound height (that is the height of

each bounce) to the height of the ball before that bounce, converted to a percentage.

Find the average rebound percentage for your ball. Show your work.

3. Tennis balls are sealed in a pressurized container to maintain the rebound percentage

of the balls. A tennis ball has a rebound percentage of 55% when it is taken out of the

pressurized can. Suppose a tennis ball is dropped from a height of 2 meters onto a tennis

court. Use the rebound rate given to predict the height of the ball’s first seven bounces.

6. Enter the bounce height data into a graphing calculator. Make a scatterplot and then

sketch the graph below.

11. What is the total vertical distance that the ball from Question 10 has traveled after six

bounces?

4. Make a scatterplot of the data you generated in the table and compare the scatterplot to

the function rule you found for Question 1. How does adding $50 per month to Derrick’s

savings change the way in which his money grows?

6. REFLECTION: How would you write a recursive routine to model this situation? A function

rule? Explain your reasoning for each type of rule and compare your responses.

a=b(0.10)+b+$50

Y= a(n-1)*1.1+50

**GOAL!**

**SAS: 3 and 4 Recursion and Exponential Functions**

**By Rachel and Nicholle**

**1. Collect data on a bouncing ball that show the maximum height of at least five bounces of**

the ball. Then make a scatterplot of the maximum height as a function of the bounce

number. (Let Bounce 0 be the initial drop height of the ball.)

the ball. Then make a scatterplot of the maximum height as a function of the bounce

number. (Let Bounce 0 be the initial drop height of the ball.)

60

30

15

9

4

1

60

30

15

9

4

1

30/60

15/30

9/15

4/9

1/4

50%

50%

60%

40%

25%

.55 x 2

.55 x 1.1

.55 x .605

.55 x .33275

.55 x .183

.55 x .1007

.55 x .0559

1.1

.605

.3327

.183

.1007

.0554

.0304

4. Write a recursive rule for the height of the ball for each successive bounce.

a =2

1

a= (a ) x .55

n

n7

5. Describe, in words, how the height of each bounce is calculated from the height of the

previous bounce.

Take the last height and multiply it by the rebound percentage (55%)

7. What kind of function might model the tennis ball bounce situation?

9. What is the height of the fifth bounce of a new tennis ball if the initial drop height is

10 meters above the ground? Use a function rule to find your answer.

Exponential

8. Look back at the table you generated in Question 3. Write a function rule for bounce

height in terms of bounce number. Graph the function rule with the scatterplot on your

graphing calculator to see if the function rule models the data.

y=2(.55 )

x

y=10(.55 )

5

.503 meters

=

10. Suppose a new tennis ball is dropped from a height of 20 feet. How many times does it

bounce before it has a bounce height of less than 4 inches (the diameter of the ball)?

Explain your solution.

Y=20(.55 )

x

.3333=20(.55 )

x

y=6.86

If you add the distance from each of the ball's first 6 bounces it has traveled almost 67 feet

12. REFLECTION: How can you decide if a data set can be modeled by an exponential

function? How are recursive rules different from function rules for modeling

exponential data? How are they the same?

13. EXTENSION: As our population uses more antibiotics for minor infections, bacteria

adapt and become resistant to the medications that are available. Methicillin-resistant

Staphylococcus aureus (MRSA) is strain of Staphylococcus bacteria that is resistant to

antibiotics. MRSA causes skin and respiratory infections and can be fatal. The Center for

Disease Control (CDC) reported that in the United States 94,360 MRSA infections

occurred in 2005, with 18,650 of these cases resulting in deaths. To put this number in

perspective, CDC reported 16,316 deaths in 2005 related to AIDS.

A research lab is observing the growth of a new strain of MRSA in an agar dish. The

initial area occupied by the bacteria is 2 square millimeters. In previous experiments

with MRSA, scientists observed the bacteria to increase by 20% each week.

• Fill in the table, showing the increase in area of the bacteria over eight weeks.

• Write a recursive rule for the area (a) of the bacteria after n weeks.

• Then make a scatterplot of the data.

Derrick is trying to save money for the down payment on a used car. His parents have said

that, in an effort to help him put aside money, they will pay him 10% interest on the money

Derrick accumulates each month. At the moment, he has saved $200.

1. Suppose Derrick does not add any money to the savings. Write a recursive rule and an

explicit function rule that model the money Derrick will accumulate with only the

addition of the interest his parents pay.

2. How long will it take Derrick to save at least $2,000 for the down payment if the only

additions to his savings account are his parents’ interest payments?

3. In an effort to speed up the time needed to save $2,000, Derrick decides to take on some

jobs in his community. Suppose he commits to adding $50 per month to his savings,

starting with the initial deposit from his parents. Fill in the table, showing the amount of

money Derrick will have over several months.

5. How long will it take Derrick to save $2,000 for the down payment if he continues to add

$50 every month? Explain how you arrived at your answer.

7. EXTENSION: Suppose Derrick changes the amount of money he adds to his savings each

month to $100. How does this affect the time it takes to save $2,000? How much does he

have to add to the savings each month to have enough money for the down payment on

his car in six months? Explain your responses.

Recursive rule for modeling exponential data: A formula that requires the computation of all previous terms in order to find the value of a

Function rules: Where you express y in terms of x. Ab ordered pair can be written.

On a graphing calculator you can add the data to the graph according to the order it is in.

The difference is...

Function rules use exponents and recursive rule use percentages.

n

2(.20)+2

2.4(.20)+2.4

2.6(.20)+2.6

2.9(.20)+2.9

3.48(.20)+3.48

4.176(.20)+4.17

5.01(.20)+5.0112

6.0(.20)+6.013

2.4

2.6

2.9

3.48

4.17

5.0112

6.013

7.216

n(.2)+n

a =(1.10)dn-1 V=(1.10)208

n

approximately 25 months

200+(.10 x 200)

200(.10)+50

270(.10)+50

347(.10)+50

431(.10)+50

864(.10)+50

740(.10)+50

627(.10)+50

524(.10)+50

270

347

431

524

627

740

864

1000

864

270

1 2 3 4 5 6 7 8

347

431

524

627

740

It will take Derrick 13 months. I continued the graph from number 3 to get my answer.

If you replace the $50 with 100 it will help reach his goal quicker. He would have to add $220 each month to reach his goal of $2,000 in 6 months.

14.)

Dog Population

Y=a -1(1.1) each year the dog population will increase by about 10%

n

• Research a set of population data. Use data from a country’s census report or other

state/national population data or animal population data.

• Cite the source of the data.

• Decide if the data, or a portion of the data, can be modeled by an exponential

function.

• Support your decision with a mathematical argument.

• Find an appropriate model for the data.

• Based on your model, make a prediction about the population.

A) Recursive rule deals with percentages while this data has decimals.

B) Y=2(1.2)^x

C) Y=2(1.2)^20 = 76.67mm