STEM-Centric Lesson
Exponential Functions
Author: Hannah Knisely, Kent Island High School, Queen Anne’s County Public Schools
Background Information
Subject:
Identify the course the unit will be implemented in.
Grade Band:
Identify the appropriate grade band for the lesson.
Duration:
Identify the time frame for the unit.
Overview:
Provide a concise summary of what students will
learn in the lesson. It explains the unit’s focus,
connection to content, and real world connection.
Background Information:
Identify information or resources that will help
teachers understand and facilitate the lesson.
STEM Specialist Connection:
Describe how a STEM Specialist may be used to
enhance the learning experience. STEM Specialist
may be found at http://www.thestemnet.com/
Enduring Understanding:
Identify discrete facts or skills to focus on larger
concepts, principles, or processes. They are
transferable - applicable to new situations within or
beyond the subject.
Algebra II
10-12
One 90 minute class period
In this lesson, students learn the real-world application of exponential models. In
subsequent lessons they will learn how to solve exponential and logarithmic equations
as well as base e, natural logarithms, and exponential decay and growth problems.
Teachers should understand how exponential functions model both growth and decay
in our everyday world. Applications of exponential decay can be found with the
following examples: medications/caffeine leaving the body, radioactive decay, half-life,
carbon dating, depreciation of material objects, etc. Applications of exponential growth
can be found with the following examples: Population growth, bacteria growth,
appreciation of material objects, compound interest, etc.
The STEM Specialist can:
 Help students interpret the results of their dice activity.
 Engage students in a discussion of how exponential growth/decay can be found in
our everyday lives.
 Encourage students to look for patterns in the real-world application problems.
 Discuss careers in which application of exponential models is prevalent (Biology,
Anthropology, Banking, Investments, Financial Advisors, Real-Estate Agents,
Census workers, Food Production, etc.)


Mathematical models are used to develop solutions to real-world problems.
Exponential models carefully define the percent rate of change in real-world
applications.
Page 1 of 24
STEM-Centric Lesson
Exponential Functions
Background Information
Essential Questions:
Identify several open-ended questions to provoke
inquiry about the core ideas for the lesson. They are
grade-level appropriate questions that prompt
intellectual exploration of a topic.
1. How can exponential functions be used to model real-world problems and
solutions?
2. How does a STEM professional use exponential functions?
Students will be able to:
1. Graph exponential functions expressed symbolically and show key features of
Identify the transferable knowledge and skills that
the graph by hand and using technology.
students should understand and be able to do when
the lesson is completed. Outcomes must align with
2. Determine if a graph/equation is representing exponential growth or
but not limited to Maryland State Curriculum and/or
exponential decay.
national standards.
3. Explain the application of exponential models in real-world situations.
Audience:
☒Peers
 Students will accurately graph and explain the results of the
☒Experts /
Product, Process, Action, Performance,
dice
activity.
etc.:
Practitioners
 Students will think through a problem and persevere in solving
Identify what students will produce to
☒Teacher(s)
demonstrate that they have met the challenge,
an exponential equation given bank account information over a
☐School
learned content, and employed 21st century
period of time.
Community
skills. Additionally, identify the audience they will
 Students will analyze a situation (exponential model) and make
present what they have produced to.
☐Online
sense of the situation by solving and explaining the solution.
Community
☐Other______
Student Outcomes:
Standards Addressed in the Unit:
Identify the Maryland State Curriculum Standards
addressed in the unit.
Common Core Algebra II Standards:
Domain: Modeling with Functions
 Cluster Statement: Interpret expressions for functions in terms of the situation they
model.
Standard F.LE.5 Interpret the parameters in a linear or exponential function in
terms of context.

Cluster Statement: Construct and compare linear, quadratic, and exponential models
and solve problems.
Standard F.LE.2 Construct linear and exponential functions, including arithmetic
and geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (including reading these from a table).
Page 2 of 24
STEM-Centric Lesson
Exponential Functions
Background Information

Cluster Statement: Analyze functions using different representations
Standard F.IF.7: Graph functions expressed symbolically and show key features
of the graph, by hand in simple cases and using technology for more complicated
cases.
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and amplitude.

Suggested Materials and Resources:
Identify materials needed to complete the unit. This
includes but is not limited to websites, equipment,
PowerPoints, rubrics, worksheets, and answer keys.
Cluster Statement: Analyze functions using different representations.
Standard F.IF.9: Compare properties of two functions each represented in a
different way (algebraically, graphically, numerically in tables, or by verbal
descriptions). For example, given a graph of one quadratic function and an
algebraic expression for another, say which has the larger maximum.
Equipment:
 computer with internet access
 projector
Websites*
 Magnitude of an Earthquake (http://www.youtube.com/watch/?v=nxqRQgRAe9o)
 The Science of Overpopulation (http://www.youtube.com/watch?v=dD-yN2G5BY0)
 Futurama video clips: (http://threeacts.mrmeyer.com/frysbank/)
This particular website has been edited into video clips for convenience in the
classroom. Those clips can be found as media clips in the PowerPoint and/or file
folder associated with this lesson.
* The sites have been chosen for their content and grade-level appropriateness. Teachers
should preview all websites before introducing the activities to students and adhere to their
school system’s policy for internet use.
Materials:
 Dice (36 per group of 2-4)
 Dice Activity
 Exponential Functions PowerPoint
 Fry's Bank Account Video Questions
 Fry’s Bank Account Video Questions Answer Key
 Fry's Bank Account (First Clip)
 Fry's Bank Account (Second Clip)
 Exponential Functions Homework
 Exponential Functions Homework Answer Key
Page 3 of 24
STEM-Centric Lesson
Exponential Functions
Lesson Overview: Students will discover the use of exponential function in the real-world. They will work
through several examples independently as well as a class. Additionally they will view video clips explaining the real-world
use of exponential modeling.
Duration: 90 Minutes
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
☐Engagement
☒Exploration
☐Explanation
Details
Materials:
 Dice (36 per group of 2-4)
 Dice Activity worksheet (1 per group)
 Computer
 Projector
 Exponential Functions PowerPoint
☐Make sense of problems
and persevere in solving
them.
Preparation:
 Gather enough dice needed to complete the activity for your class.
 Copy the Dice Activity worksheet for each of your groups. The
groups could be pre-determined or you may allow students to
choose; the preference is yours based on your classroom
dynamics.
☒Construct viable arguments
and critique the reasoning
of others.
☐Extension
☐Evaluation
Standards for
Mathematical Practice
Facilitation of Learning Experience:
Allow students time for experimentation/exploration. Students will start
with 36 dice. Each time a 6 is rolled that die should be removed from
the bunch and the remaining dice should be recorded in the table. After
15 trials (or when the dice run out), the students will be able to show
you an exponential decay regression graph. They will then answer the
following questions:
 What do you notice about the graph?
☒Reason abstractly and
quantitatively.
☒Model with mathematics.
☐Use appropriate tools
strategically.
☒Attend to precision.
☒Look for and make use of
structure.
☐Look for and express
regularity in repeated
reasoning.
Page 4 of 24
STEM-Centric Lesson
Exponential Functions
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
☐Engagement
☐Exploration
☐Explanation
☒Extension
☐Evaluation
Details
 What type of function do you think this is?
Transition:
 The questions from the activity will be discussed as a class once
each group is finished with their experiment.
 This discussion should naturally transition into the remaining
aspects of the PowerPoint.
Materials:
 Exponential Functions PowerPoint
 Computer
 Projector
Preparation:
Go through the PowerPoint as the instructor before presenting the
information with your students to ensure your own understanding of the
material.
Facilitation of Learning Experience:
 Discuss, in detail the difference between linear and exponential
functions. The graphs on the PowerPoint will provide clarification.
Emphasize that linear graphs have a constant rate of change and
exponential graphs have a percent rate of change.
 Work through the job offers example together as a class. Discuss
the choices present: Which is better immediately? Which is better in
the long-term?
 Allow students time to complete the table from the savings account
example on their own; then discuss the results as a class. Help
them derive the equation for the model of the bank account. Once
finished with this, allow the students to be the ones to determine
Standards for
Mathematical Practice
☒Make sense of problems
and persevere in solving
them.
☒Reason abstractly and
quantitatively.
☒Construct viable arguments
and critique the reasoning
of others.
☐Model with mathematics.
☐Use appropriate tools
strategically.
☒Attend to precision.
☒Look for and make use of
structure.
☐ Look for and express
regularity in repeated
Page 5 of 24
STEM-Centric Lesson
Exponential Functions
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
Details
how much money is in the account after 20 years. Compare
answers as a class.
Standards for
Mathematical Practice
reasoning.
Transition:
Discuss how exponential models are used in the real-world. Allow
students time to think about how these models are used in the realworld before giving them the answers.
Or
A STEM Specialist can be used to help students understand how
exponential functions are used in the STEM workforce. The Specialist
will engage students in a hands-on learning experiences that
demonstrates how he/she employs exponential functions in his/her
field. To find a STEM Specialist visit theSTEMnet.com.
☐Engagement
☐Exploration
☒Explanation
☐Extension
☐Evaluation
Materials:
 Exponential Functions PowerPoint
 Computer
 Projector
Preparation:
 Familiarize yourself with exponential equations: 𝑓(𝑥) = 𝑎𝑏 𝑥 .
 Know that a is the constant in each exponential equation. This
relates to the y-intercept of the graph (initial value).
 b is the base of the equation and tells us the rate in which the
function increases/decreases.
 The variable is in the exponent allowing us to calculate values per
interval of time (usually time).
☐Make sense of problems
and persevere in solving
them.
☒Reason abstractly and
quantitatively.
☐Construct viable arguments
and critique the reasoning
of others.
☒Model with mathematics.
☐Use appropriate tools
Page 6 of 24
STEM-Centric Lesson
Exponential Functions
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
Details
Facilitation of Learning Experience:
 Go through each slide of the PowerPoint explaining the symbolic
meaning of an exponential function as well as the graphical
meaning.
 Show a graphical representation of an exponential function and
explain how you can tell if the function is an increasing/decreasing
exponential function based on the graph alone.
 Break down the individual pieces of the equation of an exponential
function and stress what each means.
 Discuss with the class the meaning behind the mathematical
language of 𝑓(𝑥) = 𝑎 ∙ 𝑏 𝑥 for 𝑏 > 1 (growth) and 𝑓(𝑥) = 𝑎 ∙ 𝑏 𝑥 for
0 < 𝑏 < 1 (decay).
 Challenge student understanding of the meaning behind the
symbolic representation of an exponential function by asking, “Can
you automatically conclude that an exponential function models
decay if the base of the power is a fraction or decimal?”
Standards for
Mathematical Practice
strategically.
☒Attend to precision.
☒Look for and make use of
structure.
☒ Look for and express
regularity in repeated
reasoning.
Transition:
Discuss the answer to the question posed above. You cannot conclude
that a fraction/decimal implies exponential decay. This is true because
not all fractions/decimals are less than 1. Some fractions/decimals
represent numbers larger than 1 which would imply exponential growth.
☒Engagement
☐Exploration
Materials:
 Computer
 Projector
 Exponential Functions PowerPoint
 Fry's Bank Account Video Questions
☒Make sense of problems
and persevere in solving
them.
☒Reason abstractly and
Page 7 of 24
STEM-Centric Lesson
Exponential Functions
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
☐Explanation
☐Extension
☐Evaluation
Details



Fry's Bank Account (First Clip)
Fry's Bank Account (Second Clip)
Fry’s Bank Account Video Question Answer Key
Preparation:
 Preview each of the following video clips to ensure they are
appropriate for your classroom audience.
 Complete the Fry’s Bank Account Video Questions in advance to
prepare for any questions that may arise from your students.
Facilitation of Learning Experience:
 Pass out Fry’s Bank Account Video Questions to each student (the
supplement provided is cut into two—half sheet per student should
suffice).
 Allow the students to watch the video clip from the TV show
Futurama. If students have not seen the show explain that the
character Fry is from the past. The show takes place in the year
3000. Fry is from the year 2000. This will help with the
understanding of the clip.
 After Clip 1 has been shown have students guess how much is in
Fry’s bank account. They should write down a number they know is
too small and a number they think is too large. The actual number
will be bleeped out in the video clip to allow for students to think
about it.
 Once all the students have answered take a quick poll of the class
to see what kind of numbers they came up with.
 View video clip 2. In this clip they say that Fry’s bank account with a
balance of $0.93 with 2.25% interest over 1000 years has
Standards for
Mathematical Practice
quantitatively.
☒Construct viable arguments
and critique the reasoning
of others.
☒Model with mathematics.
☒Use appropriate tools
strategically.
☒Attend to precision.
☒Look for and make use of
structure.
☐ Look for and express
regularity in repeated
reasoning.
Page 8 of 24
STEM-Centric Lesson
Exponential Functions
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
Details
Standards for
Mathematical Practice
accumulated to $4.3 Billion. This amount can be confirmed using
𝑟 𝑛∙𝑡
the equation 𝐴 = 𝑃 (1 + 𝑛) . Assume the account compounds
interest once, annually. Show the students the mathematics behind
this amount manually and using a calculator (reinforce the
importance of the order of operations).
 Students should then answer the remaining questions:
 It took Fry 1,000 years to get that much money. How long
will it take him to double it?
 How long will it take him to get a trillion dollars?
 Be sure to let the students figure these questions out
independently or with a partner before discussing the answers as a
class.
 After the Futurama video clips there is one more real-world example
in the PowerPoint. This helps students see how slow (sometimes
fast) dilution of medication happens in the blood system.
 This would be an interesting place to discuss how all
medications/supplements affect the human body (a great example
is caffeine). Work this example through with the students as a class
and discuss how to find a growth factor.
 If time allows, show students the video clip discussing the
magnitude of earthquakes and how exponential models help us
mathematically define the epicenter.
 If additional time allows, show students the Science of
Overpopulation clip. It is lengthy put does a nice job outlining how
human population will one day outgrow the world’s ability to
Page 9 of 24
STEM-Centric Lesson
Exponential Functions
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
Details
Standards for
Mathematical Practice
produce food.
Transition:
Re-emphasize to the students how examples of exponential
decay/growth can be found everywhere in the real-world.
☐Engagement
☐Exploration
☐Explanation
☐Extension
☒Evaluation
Materials:
Exponential Functions Homework
Exponential Functions Homework Answer Key
☒Make sense of problems
and persevere in solving
them.
Preparation:
 Take time to complete the exponential functions homework
assignment before handing it out to students. Be prepared to ask
any questions the students may have about the assignment.
 Feel free to alter the questions to the ability level of your students.
☒Reason abstractly and
quantitatively.
Facilitation of Learning Experience:
Allow the students time to discover information/work through problems
on their own.
☒Construct viable arguments
and critique the reasoning
of others.
☒Model with mathematics.
☒Use appropriate tools
strategically.
☒Attend to precision.
☒Look for and make use of
structure.
☒ Look for and express
regularity in repeated
Page 10 of 24
STEM-Centric Lesson
Exponential Functions
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
Details
Standards for
Mathematical Practice
reasoning.
Interventions/Enrichments
Identify interventions and enrichments for
diverse learners.
Supporting Information
Struggling Learners
 Group students based upon ability, learning style, or other appropriate
criteria, so all students can equally contribute to group work.
 If the questions asked in class are too vague, try guiding your
questioning strategies to allow students to look for repeated reasoning.
 Class time should be effectively used—let students know the time frame
allotted for each activity.
 Provide resources to define and/or pronounce difficult vocabulary.
 Provide additional time for work completion.
English Language Learners
 Strategies to help English Language Learners are similar to those listed
above.
 Provide resources to define and/or pronounce difficult vocabulary. A
native language dictionary may also be beneficial.
 Use visuals.
 Read directions and documents aloud to students, when appropriate.
Gifted and Talented
 Ask students to research further a particular situation that appears to
have an exponential relationship.
 The instructor should foster independent thinking.
 Higher level thinking questions should be asked throughout the lesson
with the expectation of responses that are thoughtful and elaborate.
 Encourage students to develop discussion questions for the STEM
Specialist.
Page 11 of 24
STEM-Centric Lesson
Exponential Functions
Supporting Information
 Think about careers that use exponential modeling and explain how that
modeling is used.
Page 12 of 24
Names: ___________________________
___________________________
Dice
Equipment Needed: 36 number cubes
1.
2.
3.
4.
5.
In the table below, the original number of dice (36) has been recorded.
Roll these dice.
Remove any die that show a 6 on the top face.
Count the remaining dice and record that number in the table below.
Using only the dice remaining, repeat steps 2-4 until there are no die remaining or until you
have rolled 15 times, whichever occurs first.
Number of
Rolls
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Number of
Dice Left
36
Using the results from your table sketch a graph of your results here:
What do you notice about the graph?
What type of function do you think this is?
Fry’s Bank Account (From “Futurama”)
Name: ____________________
Clip 1:
1.
How much money does Fry have in his bank account?
a. Write down an answer you know is too high.
b.
Write down an answer you know is too low.
Clip 2:
2.
3.
It took Fry 1,000 years to get that much money. How long will it take him to double it?
How long will it take him to get a trillion dollars?
Fry’s Bank Account (From “Futurama”)
Name: ____________________
Clip 1:
1. How much money does Fry have in his bank account?
a. Write down an answer you know is too high.
b.
Write down an answer you know is too low.
Clip 2:
2.
3.
It took Fry 1,000 years to get that much money. How long will it take him to double it?
How long will it take him to get a trillion dollars?
Fry’s Bank Account (From “Futurama”)
Answer Key
Clip 1:
1. How much money does Fry have in his bank account?
a. Write down an answer you know is too high.
Students guess
b. Write down an answer you know is too low.
Students guess
Clip 2:
2.
It took Fry 1,000 years to get that much money. How long will it take him to double it?
𝑟 𝑛𝑡
𝐴 = 𝑃 (1 + )
𝑛
Remember:
A= amount in the account
P=Principle (amount deposited)
r= interest rate (as a decimal, not a percent)
n= number of times compounded yearly
t= number of years
0.0225 1∙1000
4,300,000,000 ≈ 0.93 (1 +
)
1
So… to double it we are looking at 8,600,000,000
0.0225 𝑡
8,600,000,000 ≈ 0.93 (1 +
)
1
9247310000 = (1.0225)𝑡
log(9247310000) = 𝑡 ∙ log(1.0225)
𝑡=
log(9247310000)
log(1.0225)
𝑡 = 1,031.32 years
3.
How long will it take him to get a trillion dollars?
1,000,000,000,000 ≈ 0.93 (1 +
0.0225 𝑡
)
1
1075270000000 = (1.0225)𝑡
log(1075270000000) = 𝑡 ∙ log(1.0225)
𝑡=
log(1075270000000)
log(1.0225)
𝑡 = 1,245.07 years
Name: _____________________
Exponential Functions Homework
Directions: Read the following situations and answer the questions that follow using the information
learned in class today.
1. A golf ball manufacturer packs 3 golf
balls into a single package. Three of
these packages make a gift box. Three
gift boxes make a value pack. The
display shelf is high enough to stack 3
value packs on top of the other. Three
such columns of value packs make up a
display front. Three display fronts can
be packed in a single shipping box and
shipped to various retail stores. How
many golf balls are in a single shipping
box?
2. The cost of a pair of sneakers increases
about 4.9% every year. About how
much would a $60 pair of sneakers cost
30 years from now?
3. The initial number of bacteria in a
culture is 10,000. The number after 5
days is 320,000.
a. Write an exponential function to
model the population y of
bacteria after x days.
b. How many bacteria are there
after 10 days?
4. Tom opened a savings account that
accrues compound interest at a rate of
2.10% quarterly. Let P be the initial
amount Tom deposited and let t be the
number of years the account has been
open.
a. Write an equation to find A, the
amount of money in the account
after t years. (Hint: we discussed
this in class today)
b. If Tom opened the account with
$450 and made no deposits or
withdrawals, how much is in the
account 10 years from now?
5. A university with a graduating class of
4,000 students in 2013 predicts it will
have a graduating class of 4,862 in 4
years. Write an exponential function to
model the number of students y in the
graduating class t years after 2013.
6. Sketch the graph of each function. Then state the function’s domain and range.
a. 𝑦 = 3(2)𝑥
1 𝑥
b. 𝑦 = 2 ( )
2
c. 𝑦 = 1.5(0.4)𝑥
Domain:
Domain:
Domain:
Range:
Range:
Range:
7. Determine whether each function represents exponential growth or decay:
a. 𝑦 = 3(8)𝑥
b. 𝑦 = 2(2.5)𝑥
d. 𝑦 = 10−𝑥
e. 𝑦 = 4 (10)
9
𝑥
c. 𝑦 = 0.1(3)𝑥
f. 𝑦 = 5 ∙ 4−𝑥
8. Analyze #6b and #7b. How do each of them compare and contrast?
9. Mark is one of 1,200 workers at the UPS
warehouse in Baltimore, which operates
seven days a week. He arrives at the
factory on Monday with a slight fever. While
working with three others on a project, he
becomes so ill that he goes to the nurse
and is sent home. Later that day, the nurse
determines that Mark has a contagious
virus that will appear one day after
exposure in those with whom he has been
in close contact. An epidemic is about to
happen. The warehouse must close if more
than 40 percent of its workers are ill. If the
three workers with whom Mark worked on
Monday come down with the virus on
Tuesday and each day each worker infects
three others, when will the factory need to
close?
Complete the following table and
describe the pattern of the virus.
DAY
Monday
Tuesday
Wednesday
Thursday
Friday
# of
workers
infected
Describe the pattern:
BONUS: If each sick worker must stay home
for three days, when will the factory reopen?
10. In the science-fiction novel The Pride of
Chanur, author C. J. Cherryh imagined
an alien race, the stsho, which has three
sexes instead of two as we humans do.
Thus, each stsho has three parents,
which will be referred to as X, Y, and Z
instead of "mother" and "father." Create
a family tree for a certain stsho, named
Tle-nle, showing three parents (XYZ),
three grandparents (GX,GY,GZ), and
three great-grandparents
(GGX,GGY,GGZ). How many ancestors
does Tle-nle have, going back through
the sixth generation of the family tree?
Name: _____________________
Exponential Functions Homework Answer Key
Directions: Read the following situations and answer the questions that follow using the information
learned in class today.
1. A golf ball manufacturer packs 3 golf
balls into a single package. Three of
these packages make a gift box. Three
gift boxes make a value pack. The
display shelf is high enough to stack 3
value packs on top of the other. Three
such columns of value packs make up a
display front. Three display fronts can
be packed in a single shipping box and
shipped to various retail stores. How
many golf balls are in a single shipping
box?
729 golf balls
2. The cost of a pair of sneakers increases
about 4.9% every year. About how
much would a $60 pair of sneakers cost
30 years from now?
4. Tom opened a savings account that
accrues compound interest at a rate of
2.10% quarterly. Let P be the initial
amount Tom deposited and let t be the
number of years the account has been
open.
a. Write an equation to find A, the
amount of money in the account
after t years. (Hint: we discussed
this in class today)
0.0210 4𝑡
𝐴 = 𝑃 (1 +
)
4
b. If Tom opened the account with
$450 and made no deposits or
withdrawals, how much is in the
account 10 years from now?
$554.85
≈$252.00
3. The initial number of bacteria in a
culture is 10,000. The number after 5
days is 320,000.
a. Write an exponential function to
model the population y of
bacteria after x days.
𝑦 = 10,000𝑒 0.693147𝑥
b. How many bacteria are there
after 10 days?
10,240,000
5. A university with a graduating class of
4,000 students in 2013 predicts it will
have a graduating class of 4,862 in 4
years. Write an exponential function to
model the number of students y in the
graduating class t years after 2013.
𝑦 = 4000(1.05)𝑡
6. Sketch the graph of each function. Then state the function’s domain and range.
a. 𝑦 = 3(2)𝑥
1 𝑥
b. 𝑦 = 2 ( )
2
c. 𝑦 = 1.5(0.4)𝑥
Domain: All Reals
Domain: All Reals
Domain: All Reals
Range: y > 0
Range: y > 0
Range: y > 0
7. Determine whether each function represents exponential growth or decay:
a. 𝑦 = 3(8)𝑥
growth
b. 𝑦 = 2(2.5)𝑥
growth
d. 𝑦 = 10−𝑥
e. 𝑦 = 4 ( )
decay
decay
9
10
𝑥
c. 𝑦 = 0.1(3)𝑥
growth
f. 𝑦 = 5 ∙ 4−𝑥
decay
8. Analyze #6b and #7b. How do each of them compare and contrast?
#6b represents an exponential decay equation. #7b represents an exponential growth
equation. Both have a b-value (y=abx) that is a fraction/decimal but the difference is that in
#6b the fractional value is between 0 and 1. #7b the decimal is a number larger than 1, thus
it is an exponential growth equation.
9. Mark is one of 1,200 workers at the UPS
warehouse in Baltimore, which operates
seven days a week. He arrives at the
factory on Monday with a slight fever. While
working with three others on a project, he
becomes so ill that he goes to the nurse
and is sent home. Later that day, the nurse
determines that Mark has a contagious
virus that will appear one day after
exposure in those with whom he has been
in close contact. An epidemic is about to
happen. The warehouse must close if more
than 40 percent of its workers are ill. If the
three workers with whom Mark worked on
Monday come down with the virus on
Tuesday and each day each worker infects
three others, when will the factory need to
close?
Complete the following table and
describe the pattern of the virus.
DAY
Monday
Tuesday
Wednesday
Thursday
Friday
# of
workers
infected
1
3
9
27
81
Each day that passes, 1 infected person infects 3 new people. This is
an exponential model. So, an exponential function to model this
situation is 𝑓(𝑥) = 3𝑥 where 𝑓(𝑥) is the number of persons with the
illness and x is the number of days since the illness first came to work.
It comes out to be that 243 have the illness on day 5 and 729 have the
illness on day 6. So, the factory would have to close on the Sunday
following the Monday after Mark came to work since 40% of 1,200
workers is 480. That number of people are infected somewhere
between Saturday and Sunday.
But if you look closer at the situation it breaks down as follows:
 on day zero, no one is at home
 on day one, one person is at home
 on day two, 1 + 3 = 4 people are at home
 on day three, 1 + 3 + 9 = 13 people are at home
 on day four, 1 + 3 + 9 + 27 - 1 = 39 people are at home
 on day five, 1 + 3 + 9 + 27 + 81 - 1 - 3 = 117 people are at home
 on day six, 1 + 3 + 9 + 27 + 81 + 243 - 1 - 3 - 9 = 351 people are
at home
There are actually less than 40% of the factory workers who have the
illness on day six and the factory won't close on day six but on day
seven instead.
 on day seven, 1 + 3 + 9 + 27 + 81 + 243 + 729 - 1 - 3 - 9 - 27 =
1053 people are at home and the factory closes.
BONUS: If each sick worker must stay
home for three days, when will the
factory reopen?
Since the workers must wait 3 days to return to
work after the illness breaks out the factory
reopens on day 10. The factory can reopen the
second Saturday after the Monday that Mark
came to work with the illness.
10. In the science-fiction novel The Pride of
Chanur, author C. J. Cherryh imagined
an alien race, the stsho, which has three
sexes instead of two as we humans do.
Thus, each stsho has three parents,
which will be referred to as X, Y, and Z
instead of "mother" and "father." Create
a family tree for a certain stsho, named
Tle-nle, showing three parents (XYZ),
three grandparents (GX,GY,GZ), and
three great-grandparents
(GGX,GGY,GGZ). How many ancestors
does Tle-nle have, going back through
the sixth generation of the family tree?
(See Answer on Next page)
Tle-nle
X
GX GY GZ
Y
GX GY GZ
Z
GX GY GZ
GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ
How many ancestors does Tle-nle have, going back through the sixth generation of the family tree?
𝑓(𝑥 ) = 3𝑥 = 36 = 729 𝑎𝑛𝑐𝑒𝑠𝑡𝑜𝑟𝑠