Algebra II Items to Support Formative Assessment
Unit 2: Exponential and Logarithmic Functions
Construct and compare linear and exponential models and solve problems.
F.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given
a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
(Cross-cutting)
F.LE.A.2 Task
Use the graph below to construct an exponential model of the data in the scatter plot. Use your model to
complete the chart.
Use the given graph to estimate the input value when y=10. Then use your model to find the exact input
value when y=10. Compare your answers.
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F.LE.A.2 Item 1
A biologist is studying a new strain of the influenza virus. She begins with a sample that has 20 mg of the
flu spores. She tracks it for 3 weeks and notices that the sample triples every 8 hours.
a.
A person will show symptoms of the virus when they have 3000 mg of spores in their system. How
long will it take the 20 mg sample to cause a person to show symptoms?
b.
A person is contagious when they have 800 mg of the virus in their system. How long are they
contagious, but are not showing symptoms?
c.
A new medication is given to a person 2 days after they have been infected with the 20 mg. The
medication stops the growth of new spores, and kills half of the spores that are in your system every 12
hrs. How long after the medication is taken will you no longer be contagious? How long until you no
longer show symptoms? Show all work that leads to your conclusions.
Answers:
a. 36.48 hours after.
b. 9.63 hours when they show no symptoms, but are contagious.
c. After 2 days, the person will have 14580 mg of influenza. It will take 27.372 hours for the number of
spores to drop below the 3000 needed to cause symptoms, and 50.256 hours to not be contagious.
F.LE.A.2 Item 2 Overarching Standard N.Q.A.2
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A furniture store, Store A, is going out of business after 6 weeks and needs to sell everything. The sales
team has decided that each week they will reduce the previous weeks price in half. You have found a
monogrammed bedroom set for $1,675.00 you want to purchase. Create a function that models this
situation.
a. You just started a new part-time job that pays $8.00 an hour. How many hours will you need to work to
be able to purchase the bedroom set before the store goes out of business? Use the model you previously
found and a new linear model to solve for the number of hours.
b. Your parents decide to help you and give you $100.00. You want to purchase the bedroom set soon, as
you are afraid someone else will. How many hours must you work in order to purchase the set after the
second week of the sale?
c. You found another furniture store, Store B, that has the same bedroom set with a different sale. They
will take off $630.00 each week. After the first week, what is the difference in the price of Store A and
Store B? When will Store B’s price be better than Store A and by how much? Show a table of values to
prove your answer.
Answers:
a. You need to work at least 7 hours
b. At least 40 hours
c. Difference: $207.50, After two weeks, Store B’s price will be less by $3.75
F.LE.A.2 Item 3
A geometric sequence has a first term of 4 and a fifth term of 20.25. What is the explicit rule for the
sequence? Write the rule as a function.
Answer:
an=4(1.5)n-1 f(x)= 4(1.5)x
note: In function notation if you use x as the exponent, the initial amount (x=0) will give the first term. If
you use x-1 as the exponent, the initial amount will occur at x = 1, which matches the explicit rule.
F.LE.A.2 Item 4
Use the graph below to construct an exponential model of the data in the scatter plot.
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Answer:
y
1 x
(3)
2
Note: It is important to emphasize that for sequences, the domain is restricted for the function.
Interpret functions that arise in applications in terms of a context.
F.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs
and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of
the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing,
positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*
(Cross-cutting)
F.IF.B.4 Task Overarching Standard N.Q.A.2
A recent outbreak of a highly contagious disease being referred to as PC-20 has the Centers for Disease
Control and Prevention (CDC) working overtime to develop a vaccine. The outbreak originated from 5
Peace Corps members who traveled overseas to a remote location on a volunteer assignment. The CDC
has determined that a human being is between 65% and 75% likely to become infected after sustained
proximity with an infected individual. On average, Americans are in sustained proximity with
approximately 2-5 unique people per day. The virus appears to be contagious for only 24 hours but has a
high mortality rate.
Group 1: The CDC needs to be able to predict how quickly this new disease will spread to the maximum
number of people. Create an appropriate table of values and use it to find a mathematical model that will
assist the CDC.
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Group 2: The CDC needs to be able to predict how quickly this new disease will spread to an average
number of people. Create an appropriate table of values and use it to find a mathematical model that will
assist the CDC.
Group 3: The CDC needs to be able to predict how quickly this new disease will spread to the minimum
number of people. Create an appropriate table of values and use it to find a mathematical model that will
assist the CDC.
All Groups:
Explain how you determined your model and discuss how useful it is in the context of the situation. Who
would want to use your model? Include a discussion of any shortcomings you think the model may have.
a. A state of emergency is declared when 500,000 people are infected. According to your model, when
will this occur if a vaccine is not developed?
b. Explain why you think there could or could not be variability amongst models.
c. Choose one of the following stakeholders in which your model would benefit most. Explain why they
would use the model you chose in the context of their position.
a. CDC scientist charged with assisting in vaccine development
b. Parent of three young children
c.
Government official in charge of supervising budget
Answers
Group 1: f(x) = 2(1.65)x
x
y
0
2
1
3.3
2
5.445
3
8.9843
4
14.824
Group 2: f(x) = 3.5(1.7)x
x
y
0
3.5
1
5.95
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2
10.115
3
17.196
4
29.232
Group 3: f(x) = 5(1.75)x
x
y
0
5
1
8.75
2
15.313
3
26.797
4
46.895
All Groups:
To find the model, take the initial value and the rate as a decimal. The model is exponential, thus in the
form f(x) = a(b)x, where a is the initial value and b is (1±r).
Who would want to use your model?
Group 1: CDC scientist charged with assisting in vaccine development
Group 2: Parent of three young children
Group 3: Government official in charge of supervising budget
a. Group 1: 24.820 days
Group 2: 22.369 days
Group 3: 20.573 days
b. There is variability because different people would want to know different values of the ranges of
percentages and number of people infected.
c. Group 1: CDC scientist charged with assisting in vaccine development because they would need to help
every person who is infected.
Group 2: Parent of three young children because she would want to know a safe middle group of whether
her children are at risk or not.
Group 3: Government official in charge of supervising budget because they would want to spend a little
as possible.
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F.IF.B.4 Item 1 Overarching Standard N.Q.A.2
The table below represents the temperature of a batch of brownies just as they are taken out of the oven.
(t =0 represents the moment the brownies were removed from the oven.) Use the table below to answer
the following.
t (in
minutes)
T (in
degrees F)
1
3.5
5
7.5
9
12.45
14
16.5
18.25
250°
200°
150°
103°
92°
85°
80°
76°
75°
a.
Describe the relationship between the time and temperature of the brownies mathematically. Be
sure to include the correct terminology and give an appropriate model for the situation.
b.
During which time period were the brownies cooling the fastest? Show all work that supports
your conclusion.
c.
What appears to be happening to the temperature of the brownies as the time out of the oven gets
larger? What graphical feature does that relate to?
Answers:
a. As the time increases the temperature decreases by less each time. The temperature seems to be
leveling off at approximately 74°, indicating there is an asymptote there. The graph indicates that the
function is exponential and could be modeled by f(x)=238.588(.783)x+74 or by f(x)=219.157(.933)x if
they do not account for the asymptote when using exponential regression.
b. Over the time period from 1 to 3.5 or from 1 to 5 could be acceptable answers. (-20 or -25
respectively.)
c. The temperature is decreasing by less and less and appears to be leveling off to approximately 74°,
which would be an asymptote.
F.IF.B.4 Item 2
A radioactive isotope decays at a certain rate per hour. The decay is displayed in the table below.
Hours
Amount of the Isotope
0
45 mg
1
34.65 mg
2
26.681 mg
3
20.544 mg
4
15.819 mg
5
12.181 mg
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6
9.379 mg
Given the table of values, find an appropriate model to best represent the data. Round to the nearest
thousandth. Then, graph the function.
a. Will the amount of the isotope ever be 0? Why or why not?
b. What is the rate at which the isotope is decaying?
c. How much of the isotope will be left after 10 hours?
Answer:
f(x) = 45(0.77)x
a. No, there is a horizontal asymptote at y = 0
b. 23%
c. 3.297 mg
F.IF.B.4 Item 3
An initial deposit of $200 is deposited into an account that earns 8.4% interest, compounded
continuously. At the time of withdrawal, the money has grown to $331.07. Determine how long
the money was in the account. Round your answer to the nearest year.
Answer:
6 years
F.IF.B.4 Item 4
Sodium-24 has a half-life of 15 hours. When will 4 mg of Sodium-24 remain if the original
sample contains 64 mg?
Answer:
60 hours
Interpret expressions for functions in terms of the situations they model.
F.LE.B.5 Interpret the parameters in an linear or exponential function in terms of a context. (Crosscutting)
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F.LE.B.5 Task
The table below shows the number of international adoptions by year in the U.S.
Year
Number of International Adoptions
2004
22991
2005
22734
2006
20680
2007
19608
2008
17456
2009
12744
2010
11058
2011
9319
2012
8668
2013
7092
U.S. State Department, "Immigrant Visas Issued to Orphans Coming to the U.S." (www.travel.state.gov/
orphan_numbers.html, accessed Jun. 26, 2014)
Use the modeling cycle to analyze the data. This includes identifying and defining variables, stating any
restrictions to essential features, formulating a model, and responding to the following questions. Validate
your choice of model with in the context of the problem.
How many adoptions are expected in the year 2014?
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F.LE.B.5 Item 1
The population of Howard county doubled from 1975 to 1995. The model for the growth over that time
period is P = 96071e.0365t , where t is the time in years since 1975, and P is the number of people.
a. Explain the meaning of 96071 from the model in context.
b. Use the model to predict how many people there will be in 2000, and 2010.
c. Can the county maintain this growth indefinitely? Explain your reasoning.
Answers:
a. It is the starting population 1975.
b. 239,268 people, 344,670 people
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c. No, eventually the growth should slow. There will not be any more land/space to develop.
F.LE.B.5 Item 2
A container of cooked food is left out on the counter at room temperature. The food contains 16,000
bacteria, which grow at a rate of 35% per hour in this unrefrigerated state. Write a function to model the
situation.
a. How many bacteria will be present in the food after 3 hours?
b. Health regulations state that food is unsafe to eat when the number of bacteria reaches 2,000,000. If
the food described remains unrefrigerated, how many hours may pass until the food is considered unsafe?
Round to the nearest thousandth.
c. When food is refrigerated, the growth of bacteria slows to 5% per hour. If the same container of food
(containing 16,000 bacteria) is promptly refrigerated, how many hours may pass until the food is
considered unsafe? Show the new function.
Answers:
f(t) = 16,000(1.35)t
a. 39,366
b. 16.089 hours
c. f(t) = 16,000(1.05)t, 98.961 hours
F.LE.B.5 Item 3
The price of a car depreciates at an initial rate of 15% as it is driven off the lot, then, at an additional rate
of 2% per month. You have found a $32.625.00 BMW you want to purchase. You plan to keep the car for
two years, then sell it. What will the car be worth when you are ready to sell it?
Answers:
f(t) = 27,731.25(0.98)t
$17,076.36
F.LE.B.5 Item 4
Matthew’s grandparents started a college savings account for him on his fifth birthday. The initial deposit
was $40,000 and the interest rate is 3.8%.
a. How much money will be in the account on Matthew’s tenth birthday?
b. How old will Matthew be when the amount in the account is $60,000?
Answer:
a. $48,199.97
b. This occurs in 11.872 years, so Matthew will be approximately 16 years, 10 months, 14 days.
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