Day Date Topic Homework - Hostler...

COMMON CORE MATH 1-B

Unit 5 – Exponential Functions

Day Date Topic Homework 1

1/21 Wed Intro to Exponential Models

2 1/22 Thur Growing Sequences

3 1/23 Fri Seq. of Tables and Graphs Who Wants to be Rich worksheet

4 1/26 Mon Now-Next Forms of Exponents The Million Dollar Mission

5 1/27 Tues Explicit Form of Exp. Graphs Killer Plants

6 1/28 Wed

Quiz

Explicit Equations y= & f(x)

Monsters and Amoebas

7 1/29 Thur Compound Interest Compound Interest

8 1/30 Fri Now-Next & Explicit Population Growth

9 2/2 Mon Linear vs. Exponential Growth

10 2/3 Tues Quiz - Translations Translation worksheet

11 2/4 Wed Intro. To Exponential Decay Medication Filtering

12 2/5 Thur Declining Sequences Declining Sequences

13 2/6 Fri Other Models of Exp. Decay Bouncing Ball worksheet

14 2/9 Mon Half Life

15 2/10 Tues Depreciation

16 2/11 Wed Review

17 2/12 Thur Test

Homework: Pay It Forward, Again Name __________________________

At the beginning of this unit we examined the Pay It Forward class project that Trevor McKinney came up with. Let us

revisit this situation and take a deeper look at what transpired.

1. Make a table that shows the number of people who will receive good deeds at each of the next seven stages of

the Pay It Forward process.

Then plot the data on a graph. Make sure you have accurate axes labels and scales.

2. How does the number of good deeds at each stage grow?

3. What is the common ratio?

4. How is that pattern change shown in the plot of the data?

5. How many stages of the Pay It Forward process will be needed before a total of at least 20,000 good deeds will

be done?

6. Write a NOW-NEXT rule to illustrate the Pay It Forward process.

7. Write an NOW-NEXT rule that would show the number of good deeds at a stage number if each person in the

process does good deeds for two others.

8. How would the NOW-NEXT rule change if each person in the process does good deed for four other people?

Stage of

Process

1 2 3 4 5 6 7 8 9 10

Number

of Good

Deeds

3 9 27

Homework: Charity Donations

Mari’s wealthy Great-aunt Sue wants to donate money to Mari’s school for

new computers. She suggests three possible pans for her donations.

Plan 1: Great-aunt Sue’s first plan is give money in the following way: 1, 2, 4,

8, . . . . She will continue the pattern in this table until day 12. Complete the

table to show how much money the school would receive each day.

Plan 2: Great-aunt Sue’s second plan is to give funds in the following way: 1, 3, 9, 27, . . . . She

will continue the pattern in this table until day 10. Complete the table to show how much

money the school would receive each day.

Plan 3: Great-aunt Sue’s third plan is to give money in the following way: 1, 4, 16, 64, . . . She

will continue the pattern in this table until day 7. Complete the table to show how much

money the school would receive each day.

Graph each plan on the same graph to the right.

1. How much does each plan give the school on day

6?

2. What is the common ratio (growth rate) for each

plan?

a. Plan 1 __________

b. Plan 2 __________

c. Plan 3 __________

3. Which plan should the school choose? Why?

4. Which plan will give the school the greatest total amount of money?

Day 1 2 3 4 5 6 7 8 9 10 11 12

Donation $1 $2 $4 $8

Day 1 2 3 4 5 6 7 8 9 10

Donation $1 $3 $9 $27

Day 1 2 3 4 5 6 7

Donation $1 $4 $16 $64

Jason is planning to swim in a charity swim-a-thon. Several

relatives have agreed to sponsor him in this charity event. Each of

their donations is explained below.

Grandfather: I will give you $1 if you swim 1 lap, $3 if you swim 2 laps, $5 if you swim 3 laps, $7

if you swim 4 laps, and so on.

Father: I will give you $1 if you swim 1 lap, $3 if you swim 2 laps, $9 if you swim 3 lops, $27 if

you swim 4 laps, and so on.

Aunt June: I will give you $2 if you swim 1 lap, $3.50 if you swim 2 laps, $5 if you swim 3 laps,

$6.50 if you swim 4 laps, and so on.

Uncle Bob: I will give you $1 if you swim 1 lap, $2 if you swim 2 laps, $4 if you swim 3 laps, $8 if

you swim 4 laps, and so on.

5. Decide whether each donation sequence is exponential, linear, or neither.

a. Grandfather’s Plan _______________________________________

b. Father’s Plan ____________________________________________

c. Aunt June’s Plan _________________________________________

d. Uncle Bob’s Plan _________________________________________

6. Complete the table for each sequence below.

Grandfather’s

Plan

Father’s

Plan

Aunt June’s

Plan

Uncle Bob’s

Plan

# of Laps 1 2 3 4 5 6 7 8 9 10


# of Laps 1 2 3 4 5 6 7 8 9 10

Donation $1 $3 $9 $27

# of Laps 1 2 3 4 5 6 7 8 9 10

Donation $2 $3.50 $5 $6.50

# of Laps 1 2 3 4 5 6 7 8 9 10


The Million Dollar Mission

You’re sitting in math class, minding your own business, when in walks a Bill Gates kind of guy - the real success story of

your school. He's made it big, and now he has a job offer for you.

He doesn't give too many details, mumbles something about the possibility of danger. He's going to need you for 30 days, and you'll have to miss school. (Won't that just be too awful?) And you've got to make sure your passport is current. (Get real, Bill, this isn’t Paris). But do you ever sit up at the next thing he says:

You'll have your choice of two payment options:

1. One cent on the first day, two cents on the second day, and double your salary every day thereafter for the

thirty days; or

2. Exactly $1,000,000. (That's one million dollars!)

You jump up out of your seat at that. You've got your man, Bill, right here. You'll take that million. You are there. And off you go on this dangerous million-dollar mission.

So how smart was this guy? Did you make the best choice? Before we decide for sure, let's investigate the first

payment option. Complete the table for the first week's work.

First Week – First Option

Day Pay for Total Pay

1 .01 .01

2 .02 .03

3

4

5

6

7

So, after a whole week you would have only made .

That's pretty awful, all right. There's no way to make a million in a month at this rate. Right? Let's check out the second week. Complete the second table.

Second Week – First Option

Day No. Pay for that Day Total Pay (In Dollars)

8

9

10

11

12

13

14

Well, you would make a little more the second week; at least you would have made . But there's still a big difference between this salary and $1,000,000. What about the third week?

Third Week – First Option


15

16

17

18

19

20

21

We're getting into some serious money here now, but still nowhere even close to a million. And there's only 10 days left. So it looks like the million dollars is the best deal. Of course, we suspected that all along.

Fourth Week – First Option


22

23

24

25

26

27

28

Hold it! Look what has happened. What's going on here? This can't be right. This is amazing. Look how fast this pay is growing. Let's keep going. I can't wait to see what the total will be.

Last 2 Days – First Option


29

30

In 30 days, it increases from 1 penny to over dollars. That is absolutely amazing.

Questions to consider:

If your boss was so impressed with your reasoning skills that he kept you on for 10 more days and paid you using Payment Option 1. However, since your help is so costly, he is now only willing to give you 50% more each day after the 30th day.

14. How can you determine how much money he would receive on Day 35?

15. How can you determine how much money he would receive on Day 40?

16. If you know how much money he receives on a certain day, how can you determine how much money he will receive 2 days later? . . . 10 days later?

17. Write a sentence that describes how much money the guy receives each day after day 30.

18. What number is being used to advance the pattern? Is this a common difference or a common ratio?

19. Use the words NOW and NEXT to write a rule to express the pattern.

20. Use the pattern you discovered to write an explicit equation for the rice acquired each day. (number of day, money received)?

21. Graph the first ten days of salary option 1 on the graph to the right. Be sure to label your axes and title your graph. Is this a graph of a linear function or an exponential function?

Independent Practice: Killer Plants

Ghost Lake is a popular site for fishermen, campers, and boaters. In recent years, a certain water plant has

been growing on the late at an alarming rate. The surface area of Ghost Lake is 25,000,000 square feet. At

present, 1,000 square feet are covered by the plant. The Department of Natural Resources estimates that the

area is doubling every month.

1. Complete the table below.

2. Use the data to graph the situation. Be sure to label your axes and title your graph.

3. Write 2 equations (NOW-NEXT and y =) to represent the growth pattern of the plant on Ghost Lake.

4. Explain what information the variables and numbers in your equations represent.

5. How much of the lake’s surface will be covered with the water plant by the end of a year?

6. How much of the lake’s surface was covered by the water plant 6 months ago?

Number of Months 0 1 2 3 4 5

Area Covered in Square Feet 1,000

7. In how many months will the plant completely cover the surface of the lake?

Loon Lake has a “killer plant” problem similar to Ghost Lake. Currently, 5,000 square feet of the lake is

covered with the plant. The area covered is growing by a factor of 1.5 each year.

8. Complete the table to show the area covered by the plant for the next 5 years.

9. Graph the data. Be sure to label your axes and title your graph.

10. Write 2 equations (NOW-NEXT and y =) to represent the growth pattern of the plant on Ghost Lake.

11. Explain what information the variables and numbers in your equations represent.

12. How much of the lake’s surface will be covered with the plant by the end of 7 years?

13. The surface area of the lake is approximately 5 acres. How long will it take before the lake is

completely covered if one acre is 43,560 square feet?

Number of Years 0 1 2 3 4 5

Area Covered in Square Feet 5,000

CCM1B Name: ________________________________________

HOMEWORK

More Alien Monster and Amoeba Encounters

For each alien encounter below, write the explicit equation in function notation and then solve.

1. An alien amoeba colony is growing exponentially and had a population of 130 when it was first observed.

An hour later, the population was 260. What was the population 10 hours after it was first observed?

2. The population of the alien city, found on the dark side of the moon, has grown at a rate of 3.2% each year

for the last 10 years. If the population 10 years ago was 25,000, what is the population today? When do

you think they will tell the human population of it’s existence?

3. In 2010, the population of a monster city, called Halloween Town, was 50 monsters. Since then the

population has increased at a constant rate of 25% each year. Assuming this rate of increase stays

constant, what will the monster population of Halloween Town be in 4 years? In 20 years?

4. A population of alien bacteria grows by 35% every hour. If the population begins with 100 alien

specimens, how many are there after 6 hours? How many will there be in 18 hours?

5. The population in the town of Alien Acres is presently 42,500. The town has been growing at a steady rate

of 2.7%. Find the number of years ago that the population was 30,000.

Independent Practice with Compound Interest

Write a NOW-NEXT and explicit equation for each problem situation in order to find the solution.

1) An investment of $75,000 increases at a rate of 12.5% per year. Find the value of the investment after

30 years. How much more would you have if the interest is compounded quarterly?

2) Suppose you invest $5000 at an annual interest of 7%, compounded semi-annually. How much will

you have in the account after 10 years? Determine how much more you would have if the interest

were compounded monthly.

3) Lisa invested $1000 into an account that pays 4% interest compounded monthly. If this account is for

her newborn, how much will the account be worth on his 21st birthday, which is exactly 21 years from

now?

4) Mr. Jackson wants to open up a savings account. He has looked at two different banks. Bank 1 is

offering a rate of 5.5% compounded quarterly. Bank 2 is offering an account that has a rate of 8%, but

is only compounded semi-annually. Mr. Jackson puts $6,000 in an account and wants to take it out for

his retirement in 10 years. Which bank will give him the most money back?

5) Mason deposited $2,000 into a savings account that pay an annual interest rate of 9% compounded

annually. Determine the amount of money in the savings account after 1 year, 5 years, 10 years and

20 years. Using the calculated values, construct a graph.

HOMEWORK: Population Growth and Other Word Problems

The Elk Population

1) The table shows that the elk population in a state forest is

growing exponentially.

a) What is the growth factor? __________

b) How did you find this?

____________________________________________

c) By what percent is the population growing?

_______________________________

d) Write a NOW-NEXT equation you could use to predict the elk

population.

e) Write an explicit equation in function notation to predict the elk

population p for any year n after the elk were first counted.

f) Suppose this growth pattern continues. How many elk will these be after 10 years? How

many elk will there be after 15 years?

g) In how many years will the elk population exceed one million?

Movie Ticket Costs

2) Suppose a movie ticket costs about $7, and inflation causes ticket prices to increase by 4.5% a

year for the next several years.

a) Write an explicit formula in function notation.

b) At this rate, how much will tickets cost 5 years from now?

c) How much will a ticket cost 10 years from now?

d) After which year will a ticket cost $25?

School Population

3) Currently, 1,000 students attend East Garner IB Magnet Middle School. The school can

accommodate 1,300 students. The school board estimates that the student population will

grow by 5% per year for the next several years.

a) Write an explicit formula for population growth.

b) In how many years will the population outgrow the present building?

c) Suppose the school limits its growth to 50 students per year. Write a NOW-Next formula.

d) How many years will it take for the population to outgrow the school?

Time (Year) Population

0 30

1 57

2 108

3 206

4 391

5 743

Linear Functions versus Exponential Functions

Exponential functions, like linear functions, can be expressed by rules relating x and y values and by

rules relating NOW and NEXT y values when an x value increases in steps of 1. Compare the patterns

of (x, y) values produced by these functions: y = 2(3x) and y = 2 + 3x by completing these tasks.

1. Complete a table for each function

x 0 1 2 3 4 5

y = 2(3x)

x 0 1 2 3 4 5

y = 2 + 3x

2. Graph both functions on the same graph.

3. For each function write another rule using NOW

and NEXT that could be used to produce the same

pattern of (x, y) values. Include the start numbers.

y = 2(3x)

_________________________ START = ________

y = 2 + 3x

_________________________ START = ________

4. Identify each function as either

arithmetic or geometric, linear or exponential.

Function Arithmetic or Geometric? Linear or Exponential?

y = 2(3x)

y = 2 + 3x

5. Describe the similarities and differences in the relationships of both functions in terms of their

function graphs, tables, and rules.

6. When will the exponential function “overtake the linear function”? Will this happen all of the

time or just some of the time? Explain your thoughts.

7. What do the numbers a and b in a linear function y = a + bx tell about patterns in the graph of

the function?

8. What do the numbers a and b in a linear function y = a + bx tell about patterns in a table of (x, y)

values for the function?

9. What do the numbers a and b in a exponential function y = a(bx) tell about patterns in the

graph of the function?

10. What do the numbers a and b in a exponential function y = a(bx) tell about patterns in a table

of (x, y) values for the function?

Similarities Differences

Graphs Tables Rules

Homework: Translations of Exponential Functions

A) Graph #1 is the parent function. Graphs #2-4 are transformed from the parent function. In order

to figure out the equation of the parent function, make a table.

B) Equation of parent

function:

___________________________________

____

C) What is the y-intercept for graph #1? _______________________

D) What is the horizontal asymptote? __________________________

E) How did the y-intercept move for graph #2?_________________________________

F) How did the horizontal asymptote move for graph #2? ________________________

G) How did the y-intercept move for graph #3?_________________________________

H) How did the horizontal asymptote move for graph #3? ________________________

I) How did the y-intercept move for graph #4?_________________________________

J) How did the horizontal asymptote move for graph #4? ________________________

K) Write each equation for the four functions here:

1. __________________ 2. __________________ 3. _____________________ 4._____________________

X-Values -5 -4 -3 -2 -1 0 1 2 3 4 5

Y-Values

1. 2.

3. 4.

Graph the following functions on the same coordinate plane. The parent graph is y = 3x and you will

explore the effect of y = 3x ± h

. Draw the horizontal asymptote of each graph. Mark the y-intercept of

the parent function and notice how far above the asymptote it is. Find a point the same distance from

the asymptote on graph 2 and 3 to decide how the parent function has changed.

1) yx

= 3

x f(x)

-1

0

1

2

4)

2) yx

=+

32

x f(x)

-3

-2

-1

0

3) yx

=−

31

x f(x)

0

1

2

3

4) How does h affect the graph of the parent function?

Medication

Assume that your kidneys can filter out 25% of a drug in your blood every 4 hours. You take one 1000-mg dose

of the drug. Fill in the table showing the amount of the drug in your blood as a function of time. The first two

data points are already completed. Round each value to the nearest milligram.

Time since taking the

Medicine (hours)

Amount of Medicine

in your Blood (mg)

0

1000

4

750

8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68

(c) 2010 National Council of Teachers of Mathematics http://illuminations.nctm.org

2. Graph the data below.

3. What is the common ratio?

4. Use your common ratio from #3 to write a NOW-NEXT function for the situation.

5. How many milligrams of the drug are in your blood after 2 days?

6. Will you ever completely remove the medicine from your system?

7. A blood test is able to detect the presence of this medicine if there is at least 0.1 mg in your blood.

How many days will it take before the test will come back negative? Explain your answer.

Other Medication Filtering Problems

3. Assume that your kidneys can filter out 10% of a medication in your blood every 6 hours. You take one

200-milligram dose of the medicine. Fill in the table showing the amount of the medicine in your blood

as a function of time. The first two data points are already completed. Round each value to the nearest

milligram. Graph the data on the coordinate plane below. Make sure to label your axes.

TIME SINCE

TAKING

THE MEDICINE

(HR)

AMOUNT OF

Medicine

IN YOUR

BLOOD (MG)

0 200

6 180

12

18

24

30

36

42

48

54

60

66

A. What is the common ratio?

B. Use the common ratio you calculated in Part A to write a NOW-NEXT equation.

C. How many milligrams of the medicine are in your blood after 2 days?

D. A blood test is able to detect the presence of the medicine if there is at least 0.1 mg in your blood. How

many days will it take before the test will come back negative? Explain your answer.

Time(hours)

4. Calculate the amount of medicine remaining in the blood if you take an initial dose of 1000 mg, but

instead of taking just one dose of the drug, now take a new dose of 100 mg every four hours. Assume

the kidneys can still filter out 25% of the drug in your blood every four hours. Make a complete a table

and graph of this situation (be sure to label the axes). Use your data table and graph to justify their

responses.

A. How do the results differ from the situation explored during the main lesson?

B. As you noted in part A, this problem is a little different, but you can write a NOW-NEXT equation for

it. Give it a try. How does your equation compare with your classmates’? Do you get the same results

when you use each of the equation?

C. How many milligrams of the medication are in your blood after 2 days?

D. A blood test is able to detect the presence of the medicine if there is at least 0.1 mg in your blood. How

many days will it take before the test will come back negative? Explain your answer.

TIME SINCE

TAKING

THE MEDICINE

(HR)

AMOUNT OF

Medicine

IN YOUR

BLOOD (MG)

0 1000

4 850

8

12

16

20

24

28

32

36

40

44

48

Time(hours)

Homework: Declining Sequences

For #1-11, Decide if the following sequences are arithmetic, geometric, or neither. If they are

arithmetic, state the value of d(common difference). If they are geometric, state r(common ratio).

1. 6, 12, 18, 24, ... _______________________________________________________

2. 6, 11, 17, ... __________________________________________________________

3. 2, 14, 98, 686, ... ______________________________________________________

4. 160, 80, 40, 20, ... _____________________________________________________

5. -40, -25, -10, 5, .... _____________________________________________________

6. 7, -21, 63, -189, ... _____________________________________________________

7. 2/3, (4/3)2, (8/3) …___________________________________________________

8. 1/3, 4/3, 7/3, 10/3,…___________________________________________________

9. 10, 10/8, 10/64, …_____________________________________________________

10. 10, 80, 640, 5120, …____________________________________________________

11. 1/3, 8/3, 64/3, 512/3, …_________________________________________________

12. Which of the geometric sequences are growing?____________________________

13. Which of the geometric sequences are declining?____________________________

14. You throw a Bouncy Ball on the cement as

hard as you can and watch it bounce until it

stops. You notice the first bounce reaches a

height of 200ft, but the second bounce

reaches only half of that height.

a) How high will the 7th bounce reach?

b) What type of sequence is illustrated by

this problem?

c) Does the sequence have a common ratio, or a common difference?

What is it?

Independent Practice with Bouncing Balls Exponential Decay Problems 1) When dropped on to a hard surface, a brand new softball should rebound to about 2/5 the

height from which it is dropped.

a. If the softball is dropped 25 feet from a window onto concrete, what pattern of rebound

heights can be expected?

b. Make a table and plot of predicted rebound data for 5 bounces.

Bounce Number 0 1 2 3 4 5

Rebound Height (in

feet)

25

c. What NOW-NEXT rule and “y = “ rules give ways of predicting

rebound height after any bounce?

2) Here are some data from bounce tests of a softball dropped from a

height of 10 feet.

a. What do these data tell you about the quality of the tested softball?

b. What are the first six bounce heights would you expect from this ball if it were dropped

from 20 feet instead of 10 feet?

3) If a basketball is properly inflated, it should rebound to about ½ the height from which it is

dropped.

a. Make a table and plot showing the pattern to be expected in the first 5 bounces after a

ball is dropped from a height of 10 feet.

b. At which bounce will the ball first rebound less than 1 foot?

Show how the answer to this question can be found in the table

and on the graph.


Rebound Height (in

feet)

10 3.8 1.3 0.6 0.2 0.05


Rebound Height (in

feet)

10

c. Write a rule using NOW-NEXT and a rule beginning “y = ” that can be used to calculate

the rebound height after many bounces.

d. How will the data table, plot, and rules change for predicting

rebound height if the ball is dropped from a height of 20 feet?

NOW-NEXT Rule:________________________

Y = ___________________________________

e. How will the data table and rules change for predicting rebound height if the ball is

somewhat over-inflated and rebounds to 3/5 of the height from which it is dropped?

NOW-NEXT Rule:________________________

Y = ___________________________

Adapted from Core-Plus Mathematics, Glencoe McGraw-Hill, 2008.


Rebound Height (in

feet)

20


Rebound Height (in

feet)

20

Homework: Half-Life Problems

Most things are composed of stable atoms. However,

the atoms in radioactive substances are unstable and

the break down in a process called radioactive decay.

The rate of decay varies from substance to substance.

The term half-life refers to the time it takes for half of

the atoms in a radioactive substance to decay. For

example, the half-life of carbon-11 is 20 minutes. This

means that 2,000 carbon-11 atoms will be reduced to 1,000 carbon-11 atoms in 20 minutes.

Half-lives vary from a fraction of a second to billions of years. For example, the half-life of

polonium-214 is 0.00016 seconds. The half-life of rubidium-87 is 49 billion years.

In the problems below, write an exponential decay function in order to find the solution to each

problem. (Use function notation)

1) Hg-197 is used in kidney scans and it has a half-life of 1 day. Write the exponential decay

function for a 12-mg sample. Find the amount remaining after 6 days.

2) Sr-85 is used in bone scans and it has a half-life of 20 days. Write the exponential decay

function for an 8-mg sample. Find the amount remaining after 100 days.

3) I-123 is used in thyroid scans and has a half-life of 5 hours. Write the exponential decay

function for a 45-mg sample. Find the amount remaining after 35 hours.

4) An exponentially decaying radioactive ore originally weighs 28 grams and is reduced to 14 grams

in 1,000 years. How much will be left in 3,000 years? Write an exponential decay function in

order to find the solution.

5) Some radioactive ore which weighed 24 grams 200 years ago has been reduced to 12 grams

today. How much will be left 400 years from now? Write an exponential decay function in

order to find the solution.

Homework: Depreciation Problems

1) A computer valued at $6500 depreciates at the rate of 20% per year.

Write a function that models the value of the computer,then find the

value of the computer after three years.

2) A new truck that sells for $29,000 depreciates 12% each year. Write

a function that models the value of the truck. Find the value of the truck after 7 years.

3) A new car that sells for $18,000 depreciates 25% each year. Write a function that models the

value of the car. Find the value of the car after 4 years.

4) You purchased a car for $19,500. The car will depreciate at a rate of 12% each year. Write a

formula to represent the value of the car after x number of years. Find the value of the car

after 4 years.

5) Each table below shows the expected decrease in a car’s value over the next five years. Both of

the cars’ values are decreasing exponentially. Write a function to model each car’s depreciation.

Determine which car will be worth more after 10 years.

Common ratio?

Depreciating by what percent? Explicit Equation: _______________________

Value after 10 Years:_____________________

Common ratio?

Explicit Equation: ______________________

Depreciating by what percent?

Value after 10 Years:___________________

Therefore, car # _______ will be worth more money after 10 years.

Year 0 1 2

Value of Car 1 $ 30,000 $ 24,000 $ 19,200

Year 0 1 2

Value of Car 2 $ 15,000 $ 14,250 $ 13,537.50

Unit 5B Test Review: Exponential Growth & Decay Name: ___________________________________

1. For #1-5, Describe the transformations used to obtain the graph of g from the graph of f.

1. f(x) = 2x g(x) = 2x - 2 _____________________________________________

2. f(x) = 3x g(x) = 3

x +7 - 6 _____________________________________________

3. f(x) = 4x g(x) = 4x – 5 + 3 _____________________________________________

4. Draw the parent function’s graph � = ��,

then graph the transformation � = �� − �.

Describe the transformation that was made.

_____________________________________

2. State the y-intercept of each function (as an ordered pair) and determine if it is growing or declining.

8. y = 0.3 • 2x 9. y = 6x 10. y= 3•(1/4)x

__________________ __________________ __________________

Is each function growing or declining?

__________________ __________________ __________________

3. Complete the table by answering the following questions about each situation.

4. Given the initial term and either common difference or common ratio, write the first 3 terms of the

sequence.

A) a1 = 4, r = 2/3 _________________________________________________

B) a1 = 7, d = -3 _________________________________________________

C) a1 = 12, r = 1.4 _________________________________________________

SITUATION: Is the sequence

arithmetic, geometric,

or neither?

Does this

represent growth

or decline?

What is the Common

Ratio or Common

Difference?

A) A child’s height increases by 3% each year.

B) A baby gains 3 pounds each month.

C) The amount of sleep you have got each night in the

month of January.

D) The town’s population is decreasing by 5% each

year.

E) 8 students are dropping out of the college each

year.

5. For a sequence, write arithmetic and the common difference or geometric and the approximate common

ratio. If a sequence is neither arithmetic nor geometric, write neither. Also include the equations asked for.

1) 5, 7.5, 11.25, 16.88, 25. 31, ..._____________________ common ___________________ = _______

Explicit Equation:_________________________ NOW-NEXT Equation:__________________________

2) 0.3, 8.3, 16.3, 24.3, ... _____________________ common ___________________ = _______


3) -2, 12, -72, -2592, ... _____________________ common ___________________ = _______


6. An exponentially decaying radioactive ore originally weighs 30 grams and has a half life of 100 years.

How much of the ore will be left in 400 years? Write an exponential decay function in order to find the

solution

Function: _____________________________________ Amount remaining: ___________________

7. Hg-197 is used in kidney scans and it has a half-life of 2 days. Write the exponential decay function for a

7-mg sample. Find the amount remaining after 6 days.

Function: _____________________________________ Amount remaining: ___________________

8. Gilberto a new smart car at a cost of $18,000. The car’s value decreases exponentially at the same rate

each year and one year later the cars value was $16,560.

A) What is the common ratio? ________________

B) By what percentage is the car’s value depreciating each year? _______________

C) Write an equation to model the decay value of this car, where y is the value of the car; x is the number of

years since new purchase. _____________________________________________

9. You have an initial investment of $10,000 to be invested at a 4.2% interest rate compounded annually.

A) Write the NOW-NEXT formula (include START) and explicit formula for the problem.

NOW-NEXT: _____________________________________________ START=_______________

EXPLICIT: _______________________________________________

B) How much will the investment be worth in 5 years? _______________________________________

C) How many years will it take for the investment to be worth $18,000? _________________________

10. The population of a town grows exponentially each year. The population 1 year ago was 8,000. Today

the population is 9280.

A) What is the common ratio? _________________________

B) By what percentage is the population growing each year? ____________________

C) What will the population be in 4 years? __________________________________

D) How many years will it take the population to reach 30,000? ___________________________________

Growth/Decay Word Problem Review

Use your knowledge of exponential functions to answer the following questions. Show your work, especially

equations written to assist in finding solutions.

1. At the end of three years, which investment will give you the most money?

a. $4,000 at 12% compounded annually

b. $5,500 at 8% compounded annually

c. $6,000 at 4% compounded annually

2. The population of a bacteria culture doubles every hour. An experiment begins with 5 bacteria.

Determine the number of bacteria after

a. 3 hours c. 10 hours

b. 6 hours d. 1 day

3. The half-life of a radioactive material is about 2 years. How much of a 5-kg sample of this material

would remain after

a. 4 years b. 6 years c. 2 years d. 10 years

4. The population of Littleton is currently 23,000. Assume that Littleton’s exponential growth rate is 2%

per year.

a. Copy and complete the table by predicting the population for the next six years.

Time (years) 0 1 2 3 4 5 6

Population 23,000

b. Write the explicit equation to model the equation. _______________________

c. Use your equation to predict the population in 10 years. _________________

d. Use your graph or graph to estimate how long it will take the population to reach 30,000.

_________________________

e. Predict the population of Littleton after 10 years if the growth rate is 3%. ____________

5. A population, P, is increasing exponentially. At time t = 0, the population is 35,000. In 1 year, the

population is 42,000.

a. Find the common ratio. ________________________

b. Write the explicit equation in function notation that models the population, P, after t years.

_________________________________

c. After which year will the population reach 100,000?

6. A bacteria culture starts with 300 bacteria and grows to a population of 1,200 after 1 hour.

a. Find the common ratio. _______________________

b. Write the explicit equation for the population (p) after t hours. ______________

c. Determine the number of bacteria after 8 hours. _________________________

7. The half-life of caffeine in a child’s system when a child eats or drinks something with caffeine in it is 5

hours. If a child ate a chocolate bar with 20 mg of caffeine in it 15 hours ago, how much caffeine is still

in the child’s body? ____________________

8. The half-life of tritium is 4 years. How many years will it take a sample of 15 grams of tritium to decay

to 1.875 grams? _______________________________________

9. A radioactive form of uranium has a half-life of 2000 years.

a. Write the explicit equation for the remaining mass of 1 gram sample after t years.

_______________________________________________________

b. Determine the remaining mass of this sample after 8000 years.

10. The half-life of carbon-14 is about 5000 years. What percent of the original carbon-14 would you

expect to find in a sample after 2500 years? __________________________

11. An old stamp is currently worth $60. The stamp’s value will grow exponentially 15% per year.

a. What will the value of the stamp be in 8 years? ___________________________

b. When will the value of the stamp be worth 3 times the initial value? __________

12. A photocopier, which originally costs $500,000, depreciates exponentially by 10% each year.

a. What will the photocopier’s value be worth in 5 years? ___________________

b. After which year will the photocopier’s value be $175,000? _______________

13. After an accident at a nuclear power plant, which caused a radiation leak, the radiation level at the

accident was 950 roentgens. One hour later, the radiation level was 798 roentgens. Radiation levels

decay exponentially. Find the common ratio. r = _______________________

14. Annie bought a new car for $35,000 and sold it 1 year later for $28, 000. Assume that the value of the

vehicle depreciates exponentially each year. Calculate the common ratio. r = _________

15. Mark invests $500 in a savings plan that pays interest, which is compounded annually. At the end of 1

year, his initial investment is worth $570. What interest rate did the plan pay? ______

16. An exponential function is expressed in the form y = a(b)x. How can you tell whether the relation

represents growth or decay?

____________________________________________________________________________________

________________________________________________________________________

17. The population of a small town increases exponentially each year. In 1999, the population was 16,000

and in 2000 it was 21,120.

What is the common ratio? ___________________

What will the population be in 2010?______________________

1. Which is the best investment if the money in each case is invested for three years?

a. $5,000 at 8% compounded monthly

b. $5,000 at 8.2% compounded annually

c. $5,000 at 8.1% compounded semiannually

2. The population of a bacteria culture doubles after 1.5 hours. An experiment begins with 620 bacteria.

Make a table and equation in order to determine the number of bacteria after

a. 3 hours c. 10 hours e. 3 days

b. 6 hours d. 1 day f. 1 week

3. The half-life of a radioactive material is about 2 years. How much of a 5-kg sample of this material

would remain after

a. 4 years b. 3 years c. 5.5 years d. 18 months

4. The population of Littleton is currently 23,000. Assume that Littleton’s exponential growth rate is 2%

per year.

a. Copy and complete the table by predicting the population for the next six years.

Time (years) 0 1 2 3 4 5 6

Population 23,000

b. Graph the data.

c. Create the equation to model the equation.

d. Use your equation to predict the population in 10 years.

e. Use your graph to estimate how long it will take the population will reach 30,000.

f. Predict the population of Littleton after 10 years if the growth rate is 3%.

5. A population, P, is increasing exponentially. At time t = 0, the population is 35,000. In 10 years, the

population is 44,400.

a. Find a in P = k(a)t.

b. Use the value of a that you calculated, write an equation that models the population, P, after t

years.

c. Using your equation, find when the population reaches 100,000.

6. A bacteria culture starts with 3,000 bacteria and grows to a population of 12,000 after 3 hours.

a. Find the doubling period.

b. Find the population after t hours.

c. Determine the number of bacteria after 8 hours.

d. Determine the number of bacteria after 1 hour.

7. The half-life of caffeine in a child’s system when a child eats or drinks something with caffeine in it is

2.5 hour. How much caffeine would remain in a child’s body if the child ate a chocolate bar with 20 mg

of caffeine 8 hours before?

8. Twelve grams of tritium decays to 9.25 grams in 2.5 years. Use a method to estimate the half-life of

tritium.

9. A radioactive form of uranium has a half-life of 2.5 x 105 years.

a. Find the remaining mass of 1 gram sample after t years.

b. Determine the remaining mass of this sample after 5000 years.

10. The half-life of carbon-14 is about 5370 years. What percent of the original carbon-14 would you

expect to find in a sample after 2500 years?

11. An old stamp is currently worth $60. The stamp’s value will grow exponentially 15% per year.

a. What will the value of the stamp be in 8 years?

b. When will the value of the stamp be worth 3 times the initial value?

12. A photocopier, which originally costs $500,000, depreciates exponentially by 10% each year.

a. What will the photocopier’s value be worth in 5 years?

b. When will the photocopier’s value be $175,000?

13. After an accident at a nuclear power plant, which caused a radiation leak, the radiation level at the

accident was 950 roentgens. Five hours later, the radiation level was 800 roentgens. Radiation levels

decay exponentially. Find the rate of decay.

14. Annie bought a new car for $35,000 and sold it 5 years later for $18, 475. Assume that the value of the

vehicle depreciates exponentially. Calculate the rate of depreciation per year.

15. Mark invests $500 in a savings plan that pays interest, which is compounded monthly. At the end of

10 years, his initial investment is worth $909.70. What interest rate did the plan pay?

16. An exponential function is expressed in the form y = a(b)x. How can you tell whether the relation

represents growth or decay?

17. The population of a small town increases exponentially. In 1999, the population was 16,000 and in

2002 it was 60,000. What will the population be in 2010?

18. In 1996, Ontario’s population was about 10.7 million. Ontario’s population will be about 13.7 million in

2016.

a. Calculate the annual growth rate of Ontario’s population.

b. What would Ontario’s population have been in 1980?

c. What have you assumed for part a and part b?

19. During an archaeological dig, Selma found a tool that resembled a small hatchet with a wooded handle.

a. Carbon-14 has a half-life of about 5370 years. Explain what this means.

b. Create an equation that relates the percent of carbon-14 remaining to the tool’s age. Explain

what each part of the equation represents.

c. Explain how you can tell from the equation that the amount of carbon-14 is decreasing.

d. Explain how you can tell from the equation that the amount of carbon-14 is decreasing

exponentially.

1. You are collecting pennies each month and you decide to add 1.5% more pennies each month. You start

with 5 pennies in your piggy bank.

a) What is the initial value? ______________

b) What is the common ratio? ________________

c) Write an explicit equation in function notation form that models the amount of pennies saved.

_________________________________________

d) How many months will it take to get $25.00? ____________________________

2. The population of a small town is predicted using the exponential equation � = ��(�. �)�.

a) By what percent is the population growing? ______________________________________

b) What will the population be in 3 years? ___________________________________________

c) After which year from the initial population will the population reach 35,000? ____________

3. You have an initial investment of $6,000 to be invested at a 2.7% interest rate compounded annually.

a) Write the NOW-NEXT formula (include START) and explicit formula for the problem.

NOW-NEXT: _____________________________________________ START=_______________

EXPLICIT: _______________________________________________

b) How many years will it take for the investment to be worth $10,000? _________________________

4. Convert the NOW-NEXT form of the linear equation to an explicit equation.

Next = Now+(2.6), Start = 13

_____________________________________________________

Day Date Topic Homework - Hostler...

Documents

Transcript of Day Date Topic Homework - Hostler...