Alg2 Lesson 1.4: Growth Factor & Δ%

Lesson 1.4: Connecting Growth

Factor to Percent Change

Obj: I can formulate exponential equations.

EQ: How do I formulate exponential equations and
use them to solve given a verbal description?

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.

​• Encourages everyone to ask questions.

Poll

Which role will you take on today?

• Includer

• Resourcer

• Scribe

• Facilitator

● Go to your calendar paper.

● Select a skill to work on.

● Work on Unit 1 Deltamath.

When waiting for other tables to be ready...

Part 1: Exploring an
Exponential Situation

Key Features of Exponential Functions

How are horizontal asymptotes related to
exponential functions? What else do you know
about exponential functions? Record this in your
journal.

How are these
graphs similar
and different to
linear graphs?

A(t) = P(1 + r)^t

Car Value Depreciation

Most cars decrease in value over time. Kelly Blue Book is a
resource that consumers can use the determine the value of both new and used cars. But how do the editors of Kelley Blue Book calculate a car’s value? Suppose that an $18,000 new car decreases in value by 20% every year. Make a table of values that you can use to estimate the car’s value 10 years after purchase. Then use the table of values to write a function to model the value of the car t years after purchase.

Multiple Choice

What is the initial value of the car?

• $21,600

• $18,000

• $12,500

• cannot be determined

• $14,400

Multiple Choice

Can we model this scenario with a function? If yes, what would be the best model?

• Yes, exponential

• Yes, quadratic

• Yes, linear

• No

Multiple Choice

By how much does the value decrease every year? Is that amount constant or does it change?

• 80% decrease every year or 20% of the current value; it is constant

• 20% decrease every year or 80% of the current value; it changes

• 20% decrease every year or 80% of the current value; it is constant

• 80% decrease every year or 20% of the current value; it changes

Multiple Choice

After one year, what is the value of the car?

• $21,600

• $18,000

• $12,500

• $3,600

• $14,400

Table for first 3 years… Fill in rest

Fill in rows for the 4th through 10th year.

4​	​	​
5​	​	​
6​	​	​
7​	​	​
8​	​	​
9​	​	​
10​	​	​

​Not linear since 1st differences are not same.

​​Not quadratic since 2nd differences are not same.

Categorize

Options (2)

time in years

value of car in dollars

What are the independent and dependent variables for this scenario?

Independent

Dependent

Multiple Choice

What is the algebraic representation of this scenario?

• V(t) = 18000(0.2)^t

• V(t) = 18000(0.8)^t

• V(t) = 18000(1.8)^t

• V(t) = 18000(1.2)^t

If the value of the car decreases by

20%, why is the decay factor 0.80?

The decay factor is determined by the % of the value of the car that remains.

100% - 20% = 80% remaining.

Fill in the Blank

What is the predicted value of of the car after 10 years?

$

,

.

Part 2: Writing a Function Model with Two Data Pairs

Modeling the Value of a Classic Car

The Ford car company manufactured the “first generation” of Bronco SUVs from 1966
to 1977. Many automobile enthusiasts collect these classic cars as they often increase
in value over time. The average value of a first-generation Ford Bronco in perfect
condition, like the one shown here, was $47,025 in 2017 versus $23,400 five years
before. Assuming the average value continues to increase at the same percent growth
per year, estimate the average value of a first-generation Bronco in perfect condition in
2025 by answering each of the following questions.

Year

Years
Since
2012

Value

Linear Model
Prediction $

Exponential Model
Prediction $

2012

0

$23,400

2017

5

$47,025

2025

13

In your journal…

(a)

Construct a linear function model of the value of a first-generation Ford
Bronco. Explain the meaning of the slope of the line in context. Then use your linear function model to estimate the value of a first-generation Bronco in 2025.

(b)

Construct an exponential function model of the value of a first-generation Ford Bronco. Explain the meaning of the growth factor of the function in context. Then use your exponential model to estimate the value of a first-generation Bronco in 2025.

Categorize

Options (2)

Time in years since 2012

Value of the
classic Bronco in dollars

What could be identified as the independent and dependent variables?

Independent Variable

Dependent Variable

Drag and Drop

What does the slope of the line mean in the context of the problem? Change in
​

Placeholder

of car in ​

Placeholder

per ​

Placeholder

Drag these tiles and drop them in the correct blank above

dollars

year.

value

price

Multiple Choice

Is the percent of change an increase or a decrease?

• Increase

• Decrease

Multiple Choice

If the growth factor over one year is about 1.15, what is the percent increase in the
value of the car over one year?

• 15% decrease

• 15% increase

• 115% increase

• 115% decrease

Multiple Choice

Let’s use the information we found to write an exponential function model to estimate the average value of a first-generation Bronco in 2025.

• V(t) =
  23400(1.15)^t

• V(t) =
  23400(2.15)^t

• V(t) =
  23400(0.85)^t

Multiple Select

Using your exponential model, what would be the estimated value?

• $143,651.29

• $143,975.23

• $157,975.31

• $157,651.37

Linear Functions:

Slope

d = common difference (m)

Points: (x₁, y₁) and (x₂, y₂)

Exponential Functions:

Growth Factor

r = common ratio (B)

Points: (x₁, y₁) and (x₂, y₂)

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.

Part 3: Understanding
Compounding Interest

Computer SuperStore

You want to buy a new computer for $1,500 from Computer SuperStore. Since you
don’t actually have $1,500 for the computer, you decide to apply for a store credit
card. When a person makes a purchase with a credit card, they are often charged
interest (for the companies to make profits and to increase inflation). Interest is a
fee charged in exchange for allowing you to borrow money and is usually
calculated as a percent of the balance you owe. There are 3 different credit cards
available at Computer SuperStore:

Option 1: Interest charged 24% annually (yearly)

Option 2: Interest charged 6% quarterly (seasonally)

Option 3: Interest charged 2% monthly

Complete a table of values for each option. Then justify which option you would
choose using a mathematical argument involving exponential functions.

Compound Interest Formulas

A = New Amount

P = Principle Amount

r = rate

n = # times compounded

y = year

e ~ 2.718…

Ln = natural log operator

Continuous
Compounding

Categorize

Options (5)

f(t) = 1,500(1 + 0.24)^t

g(t) = 1,500(1 + 0.06/4)^4t

h(t) = 1,500(1 + 0.02/12)^12t

h(t) = 1,500(1 + 0.2/12)^12t

h(t) = 1,500(1 + 0.6/25)^25t

Match each formula with the option

Option 1: Interest charged 24% annually

Option 2: Interest charged 6% quarterly

Option 3: Interest charged 2% monthly

Option 1

Option 2

Option 3

None of These

Open Ended

Summarize the topic and answer the essential question: How do I formulate exponential equations and use them to solve given a verbal description?