Alg2 Lesson 1.5 Modeling w/ Exp

Lesson 1.5: Modeling
Nonlinear Data with

Exponential Functions

Obj: I can formulate exponential models and use
them to make predictions.

EQ: How do I use Desmos to find an exponential
regression and use it make predictions?

Roles:
Facilitator
Scribe
Resourcer
Includer

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Communicates conflicts or questions to the teacher.

Word Cloud

How do you plan to contribute today?

Part 1: Modeling an
Exponential Situation

Go to Pre-AP Classroom find Handout 1.5a

Pre–AP Classroom: https://myap.collegeboard.org/

1st Period: AZPD3J

9th Period: 4Q4ZGY

10th Period: EL4L9Y

Modeling the World Population

1.

Create a scatterplot of the data and select the type of function
that you think could be used to most appropriately model this
data. Explain your choice.

2.

Determine the linear regression equation for your data.
{In Desmos, you will type: y1~mx1+b}

3.

What do the values of the parameters mean in the context of
the problem?

4.

Now that you have the linear model, plot the residuals. What,
if anything, do the residuals indicate about the
appropriateness of a linear model for this data? Explain.

5.

Next, determine an exponential regression equation for the
data. {In Desmos, you will type: y1~a(c)^x1}

6.

What do the values of the parameters mean in the context of
the problem?

7.

Now that you have an exponential model, plot the residuals.
What, if anything, do the residuals indicate about the
appropriateness of an exponential model for this data? Explain.

8.

Use both of the models to estimate the world population in
2030. Which estimate seems more plausible to you and why?

A Note About “Log Mode”

Multiple Choice

Which model seems to fit better?

• Linear

• Exponential

• Quadratic

Multiple Choice

What is the growth factor of this model? (1+r)

• 1.015
• 1.05
• 0.015
• 1.5

The rate of increase of this model is 1.5%. This means it it increasing by 1.5%. It is 101.5% of the original amount.

Multiple Choice

What is the exponential regression formula for this data?

• P(t) = 1.317(1.015)^t

• P(t) = 1.75(1.015)^t

• P(t) = 1.522(0.985)^t

• P(t) = 1.651(1.15)^t

Multiple Choice

According to the exponential model, what will the world population be in 2030?

• 8.5 billion people
• 10.3 billion people
• 9.12 billion people
• 6.02 billion people

Part 2: Identifying the Best

Model

Go to Pre-AP Classroom find Handout 1.5b

Pre–AP Classroom: https://myap.collegeboard.org/

1st Period: AZPD3J

9th Period: 4Q4ZGY

10th Period: EL4L9Y

A Cooling Water Experiment

Have you ever wondered how crime scene investigators can estimate how long a body has
been dead? They rely on physical principles, such as that the temperature of liquids, like the
ones in your body, cool predictably. Laura was curious about this principle and wondered if
she could determine how quickly a cup of boiling water cools. She heated water on a stove,
poured the water into a cup, stirred the water, and took the temperature of the water every
minute for 30 minutes. She noted that the temperature of her kitchen was 21.7°C. The data
that she collected and the resulting scatterplot can be found at
preap.org/Desmos-CoolingWater

1.

Analyze the scatterplot. What kind of function do you think would be most appropriate
to model the data? Why?

2.

Determine an appropriate regression equation for the data.

3.

Using the model you determined using Desmos, predict the temperature of the water
after 45, 60, and 300 minutes.

Multiple Choice

What kind of graph is this?

• Linear

• Exponential Growth

• Exponential Decay

• Quadratic

Multiple Choice

What is the exponential regression
equation?

• y = 83.747(1.019)^x
• y = 83.747(0.981)^x
• y = 100(0.95)^x
• y = 50(0.5)^x

Multiple Choice

Using the model y = 83.747(0.981)^x, what is the temperature after 45 minutes?

• 35.324 °C
• 25.750 °C
• 40.031 °C

• 30.500 °C

Multiple Choice

Using the model y = 83.747(0.981)^x, what is the temperature after 60 minutes?

• 24.200 °C
• 29.750 °C

• 26.492 °C
• 30.500 °C
• 24.000 °C

Multiple Choice

Using the model y = 83.747(0.981)^x, what is the temperature after 300 minutes?

• -0.5 °C
• 10 °C
• 0.997 °C

• 0.265 °C
• 50 °C

Check for Reasonableness

The exponential model says 0.265 °C would
be the temperature after 300 minutes. Is this
reasonable? Why or why not? Write your
explanation in your journal.

Room Temperature = 21.7 °C

On Desmos, add another line with:

y2 ~ac^x1, this creates a
transformation set of y-values.

This new y value is the difference
between water temperature and
room temperature.

Add 21.7 to the y-values here
to get the predicted
temperature.

Multiple Choice

What is the new regression model to represent the temperature?

• y = 63(0.973)^x

• y = 63(0.973)^x + 21.7

• y = 21.7(0.973)^x

• y = 63(0.973)^x - 21.7

Multiple Choice

Using the new regression equation, y = 63(0.973)^x + 21.7, what is the closest estimate to the temperature of the water after 45 minutes?

• 39.7°C

• 60.5°C

• 35.2°C

Multiple Choice

Using the new regression equation, y = 63(0.973)^x + 21.7, what is the closest estimate to the temperature of the water after 60 minutes?

• 33.5°C

• 31.2°C

• 20.5°C

Multiple Choice

Using the new regression equation, y = 63(0.973)^x + 21.7, what is the closest estimate to the temperature of the water after 300 minutes?

• 30.5°C
• 25.0°C
• 21.7°C

Exponential Formulas

Growth Factor

A = New Amount

P = Principle Amount

r = rate of increase/decrease

n = # times compounded per period

t = time periods

e ~ 2.718…

Ln = natural log operator

Compound Interest

Given points
(x1, y1) and (x2, y2)

Go to Pre-AP Classroom find Handout 1.5c

Pre–AP Classroom: https://myap.collegeboard.org/

1st Period: AZPD3J

9th Period: 4Q4ZGY

10th Period: EL4L9Y

Part A: Exponential Functions and Percent Change

1. Sandy collected 10 pounds of apples at an apple orchard.
She is determined to eat 20% of the apples remaining every
day after the day she went to the orchard. Write an
exponential function to model this scenario and use it to
estimate how many pounds of apples she has left after 1
week.

Part A: Exponential Functions and Percent Change

2. Ginny is running a simulation to track the spread of the
common cold. Her simulation uses an exponential function to
model the number of people with the common cold. The
simulation uses the function p(t) = 3(1.25)^t, where p(t) is the
number of people with the common cold and t is the number of days. What is the initial number of people with the common cold in Ginny’s simulation, what is the growth factor of the number of people with the common cold, and what is the percent change in the number of people with the common cold?

Part A: Exponential Functions and Percent Change

3. Rohiza sold her stamp collection and every year thereafter
donates 30% of the remainder of this money to charity. If her
stamp collection sold for $6,000, how much of the initial amount
will she have left after t years?

Part A: Exponential Functions and Percent Change

4. Suppose that one of Peter’s social media posts receives 45
likes on the first day of a particular month and that the total
number of likes increases by about 15% every day. Write an
exponential function model for this scenario and use it to
estimate the number of likes Peter would have on the 25th day
of the month.

Part A: Exponential Functions and Percent Change

5. Daunte’s favorite video game store is running a clearance
sale. Every week until everything is sold, the store’s owner is
going to reduce the price of the remaining merchandise by 25%.
Duante figures that he should wait 4 weeks to go to the store
because then the merchandise will be 100% off! Construct a
mathematical argument involving an exponential function to
explain whether Daunte is correct or incorrect.

Part B: Exponential Function from Two Data Points

1. Dylan releases new music on one of his social media
platforms. One particular song of his had 500 listens on the
day he released it and 1,725 listens 5 days after he released
it. Write an exponential function model for this scenario and
use it to estimate the number of listens he might expect the
song to get 21 days after he released it.

Part B: Exponential Function from Two Data Points

2. Each of Meghna’s cats has fleas. Her veterinarian
recommended a treatment plan that includes combing out her cats’
fur every day. Meghna kept track of the number of fleas that she
removed from her cats while combing them. On Day 0, the day
she started the treatment, she removed a total of 87 fleas. On Day
6, she removed a total of 14 fleas. If Meghna keeps up the
treatment and the number of fleas can be modeled by an
exponential function, approximate the number of fleas she should
expect to remove from the cats on Day 14.Write the exponential
function and show all your work.

Part B: Exponential Function from Two Data Points

3. Naomi is a social scientist who is studying how a rumor
travels through a community. Naomi tracked that 7 people
had heard the information 2 days after it was introduced to
the community. Then she tracked that 11 people had heard
the rumor 5 days after the information was introduced to the
community. Write an exponential function that Naomi could
use to model the scenario, and use it to predict the number of
people who have heard the rumor 20 days after the rumor
was introduced to the community.

Part B: Exponential Function from Two Data Points

4. In 1979, the richest 1% of people in the U.S. owned 23% of
the nation’s wealth. By 2019, the richest 1% of the people in
the U.S. owned nearly 35% of the nation’s wealth. Determine
an exponential function model for this scenario and use it to
estimate the percent of the nation’s wealth that the richest 1%
of people in the U.S. owned in 1995. What does the growth
factor mean in the context of this scenario?

Part B: Exponential Function from Two Data Points

5. Ishi is performing an experiment to investigate how quickly a certain
strain of bacteria reproduces in a controlled environment. She places
1,000 bacteria in a petri dish on Day 0. On Day 4 of the experiment, she
observes 2,000 bacteria in the petri dish. She reasons that the bacteria
population is doubling every 4 days. She predicts that if the bacteria
population continues to grow at this rate, the petri dish should contain
4,000 bacteria on Day 8. She reasons that since 3,000 is halfway
between 2,000 and 4,000 and Day 6 is halfway between Day 4 and Day
8, she should expect about 3,000 bacteria to be present on Day 6. Use
an exponential function to construct an argument to confirm or refute
Ishi’s reasoning.

Part C: Compound Interest

1. Pierre invests $1,000 in an account that earns 15% interest
compounded quarterly. Suppose that he makes no deposits
and no withdrawals. Write an exponential function that you
could use to determine the account balance after t years,
then use it to determine the account balance after 10 years.

Part C: Compound Interest

2. Maria has a choice between two different savings bonds,
each priced at $100. A saving bond is a long-term investment
that involves loaning money to the U.S. Treasury in exchange
for a guaranteed return on the-loan. Bond A returns 2%
annual interest, while Bond B returns 1% interest
compounded semiannually (twice per year). After 20 years,
which savings bond would be worth more? Write two different
exponential functions to determine the solution.

Part C: Compound Interest

3. Gokul needs to borrow $10,000 to purchase a car. One
bank is offering loans at an interest rate of 2%, compounded
twice per year, and another bank is offering loans at an
interest rate of 4% interest, compounded once per year. For
both banks, the loan length is 5 years. Gokul isn’t sure if 2%
interest compounded twice per year is the same as 4%
interest compounded once per year. Write two exponential
functions and use them to help Gokul determine which loan
he should choose.

Part D: Exponential Regression

1a. Create a scatterplot of the data. Do you think an exponential function would be an
appropriate fit for the data? Why or why not?

1b. Determine an exponential regression equation for the data. Does it seem to
appropriately model the data? Check the residual plot to see if the model is
inappropriate.

1c. What is the decay factor for this scenario? Explain what it means in terms of the
annual percent change in multiple births in the United States.

1d. Use your exponential regression equation to estimate the number of multiple births in
the United States in 2025, assuming the decline in multiple births continues as predicted
by the model.

1. In recent years, the number of
triplets and other higher-order births in
the U.S. has been declining. The
numbers of multiple births, excluding
twins, for 2010 through 2019 are
shown in the table.

Part D: Exponential Regression

2a. Create a scatterplot of the data. Do you think an exponential function would be an
appropriate fit for the data? Explain.

2b. Determine an exponential regression equation for the data. Does it seem to
appropriately model the data? Check the residual plot to see if the model is
inappropriate.

2c. What is the growth factor for this scenario? Explain what it means in terms of the
annual percent change in the number of Starbucks stores.

2d. Use your exponential regression equation to estimate the number of Starbucks stores
in 2021, assuming the growth continues as predicted by the model.

2. Starbucks is a popular coffeehouse that
originated in Seattle, Washington. Since
the mid-1980s, the number of Starbucks
stores has been increasing. The following
table shows selected years from 1987
through 2015 and the numbers of
Starbucks stores in those years.

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the
wheel as a class and spend a minute or so
discussing our answers.

● Go to your calendar paper.

● Select a skill to work on independently or with a partner.

● Work on Unit 1 Deltamath.

Self-Acquisition Time

Open Ended

Answer the essential question: How do I use Desmos to find an exponential
regression and use it make predictions?