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
Donation $1 $3 $5 $7
# 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
Donation $1 $2 $4 $8
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
Day No. Pay for that Day Total Pay (In Dollars)
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
Day No. Pay for that Day Total Pay (In Dollars)
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
Day No. Pay for that Day Total Pay (In Dollars)
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.
Bounce Number 0 1 2 3 4 5
Rebound Height (in
feet)
10 3.8 1.3 0.6 0.2 0.05
Bounce Number 0 1 2 3 4 5
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.
Bounce Number 0 1 2 3 4 5
Rebound Height (in
feet)
20
Bounce Number 0 1 2 3 4 5
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 ___________________ = _______
Explicit Equation:_________________________ NOW-NEXT Equation:__________________________
3) -2, 12, -72, -2592, ... _____________________ common ___________________ = _______
Explicit Equation:_________________________ NOW-NEXT Equation:__________________________
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
_____________________________________________________
Top Related