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5.5 Exponential Change

Last updated 10 months ago
26 questions
Note from the author:
OBJECTIVES & STANDARDS
Math Objectives
  • Write percent change exponential functions given the necessary values
  • Model a real world situation using exponential functions
  • Find the rate of growth and decay given the equation
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.PRACTICE.MP2
  • CCSS.PRACTICE.MP4
  • CCSS.HSF.LE.A.1.C
  • CCSS.HSM
Personal Finance Objectives
  • Analyze return on an investment given starting investment, rate of return and time periods
  • Explain what inflation is and how it affects purchasing power and investments
  • Explain how depreciation affects the value of a car
National Standards for Personal Financial Education
Investing
  • 2b: Compare nominal annual rates of return over time on different types of investments, including cash flow and price changes
  • 4a: Describe the impact of inflation on prices over time
DISTRIBUTION & PLANNING
Distribute to students
  • Student Activity Packet
  • Application Problems
OBJECTIVES & STANDARDS
Math Objectives
  • Write percent change exponential functions given the necessary values
  • Model a real world situation using exponential functions
  • Find the rate of growth and decay given the equation
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.PRACTICE.MP2
  • CCSS.PRACTICE.MP4
  • CCSS.HSF.LE.A.1.C
  • CCSS.HSM
Personal Finance Objectives
  • Analyze return on an investment given starting investment, rate of return and time periods
  • Explain what inflation is and how it affects purchasing power and investments
  • Explain how depreciation affects the value of a car
National Standards for Personal Financial Education
Investing
  • 2b: Compare nominal annual rates of return over time on different types of investments, including cash flow and price changes
  • 4a: Describe the impact of inflation on prices over time
DISTRIBUTION & PLANNING
Distribute to students
  • Student Activity Packet
  • Application Problems
Intro/Warm-Up:
REVIEW: Calculating Percent Change
Let’s review percent increase and decrease. Try to perform each of the following calculations in one step.
1
Sierra walks into a store and spends $12. If there is a 7% sales tax, how can Sierra calculate her total bill in one step?_______
1
Ang finds a great sale on a new pair of headphones.  The original cost was $89 but the item is 25% off.  How can Ang calculate his total bill in one step?_______
Learn It
Exponential Growth and Decay using Percents
Remember that the b value in the general form
 represents the growth or decay of the initial value.  If the b value is 1, then no growth or decay happens.
If you are told that the initial value changes by a percent, then we can represent the b value as growth or decay where:
  • Growth is 1 + r
  • Decay is 1 - r
  • r is the decimal form of the percent
1
Write an equation that represents the following situation
  • Starting value = 125
  • Rate = 6% growth per year
  • Years = 10 years
_______
1
Write an equation that represents the following situation
  • Starting value = 200
  • Rate = 2.5% decay per year
  • Years = 5 years
_______
Practice It
Use what you know about writing exponential functions with percent growth or decay to complete the following problems.
1
Farid makes a $12,000 investment that yields an average yearly return of 4%. If he makes no additional investments, how much will his investment be worth in 15 years?_______ Screen shot desmos
1
You purchase a car for $36,000. You did some research and found that this particular model of car depreciates in value by around 15% per year. What is the value of your car after 6 years?_______
ACTIVITY: MATH: Exponential Decay in Purchasing Power
2
When Sarah turned 18, she took $500 from her savings and opened a ROTH IRA.  Sarah chose to invest her money in a low fee index fund.  Based on historical data, Sarah predicts she can expect an average 7% return on her money each year.  If Sarah’s prediction is correct, how much will her $500 investment be worth when she is 68 years old?  (Assume she makes no other contributions to the account after the first time)
Equation:_______
Answer:_______
Screen shot Desmos for full credit
2
Before Sarah does open her account, she sees her grandmother for the holidays. Grandma wants to contribute to Sarah’s ROTH and even matches her contribution, giving her an additional $500 to invest. Now that Sarah has an initial investment of $1000, how much would Sarah have at age 68 based on her predicted rate of return from historical data?
Equation: _______
Answer: _______
Screen shot Desmos for full credit
2
Over the last 10 years, a very popular tech company has averaged a 43.5% rate of return on their stock prices.  If someone had invested $200 with this company ten years ago, how much would their investment be worth today?
Equation:_______
Answer: _______
Screen shot Desmos for full credit
5
In 1999 your Aunt invested $5,000 in a Dow Jones index fund and $5,000 in a S&P 500 index fund.  Twenty years later, she is looking to see how much these investments are worth.  From 1999 to 2019 the S&P 500 averaged an increase of 5.90% and the Dow Jones averaged 7.03%.  What was the total value of these investments in 2019?
Equation: S&P500_______
S&P 500 Total: _______
Equation: Dow Jones_______
Dow Jones Total: _______
S&P 500 & Dow Jones Totals: _______
1

Refer back to questions 7 and 8. Plot the value of Sara’s ROTH at age 68 when she invested $500 and $1,000 to start.  Pick one additional initial investment amount and add that to your graph then draw a curve to represent the exponential growth.

1

Your friend is considering how much he should invest in his new IRA.  He is debating if he should invest the full $800 he was planning to or just $400 and use the extra money to buy a couple of pairs of sneakers he had been eyeing.  What advice would you give?  Use math to justify your answer.

Now that we know how to use 1 + r and 1 - r to write percent growth and decay equations, we can use this to determine the rate of change of a given equation.
1
You make an investment where the balance over time can be modeled by the equation
, where x represents the number of years since the investment started and y represents your total balance after x years.
a. What is the starting balance of your investment?%_______
b. What is the rate of growth of your investment?_______ %
1

is an equation that can be used to represent the purchasing power of $100 after x years of inflation. What is the rate of inflation used to make this calculation?_______ %
APPLICATION: Analyze Investments for Exponential Change
You make an investment of $250 and know that it has an average annual growth rate of 7%. Let’s do an analysis to see how much money you’ll have over a period of time and how much that return has grown from your initial investment, assuming your growth rate remains unchanged.
1
Write an equation that represents your investment where x represents the number of years that has passed since you started and y represents your account balance after x years._______
Review
To calculate return on investment, use the formula:

8
Complete the table below by using your equation from question #15 to calculate the Money column and the percent change formula for the % Growth column. The original value for every row will be $250 when calculating percent growth.
Years Money % growth
0 250 0
10 _______ _______
20 _______ _______
30 _______ _______
40 _______ _______
4
How much did your % Growth from Initial Investment change for each 10 year period?
Time Interval % Growth
0 to 10 _______
10 to 20 _______
20 to 30 _______
30 to 40 _______
1

After seeing the growth of this investment over 4 ten year periods, what conclusion can you draw about how time affects an investment with an exponential rate of growth?

Level 2 *DESMOS**
There are several factors to consider when deciding what investment is best for you. Let’s analyze three investments with different starting values, growth rates, and investment periods.
You can make one of the following investments:
  1. Option 1: $50 initial investment with an average annual growth rate of 9% starting at age 30
  2. Option 2: $50 initial investment with an average annual growth rate of 8% starting at age 20
  3. Option 3: $100 initial investment with an average annual growth rate of 7% starting at age 25
0

Before we do an analysis, which option do you think will have the greatest return after
a. 10 years?
b. 20 years?
c. 30 years?

0
Complete the following table to show the growth of each investment over time.  Start by writing the general equation in the top row then use it to complete the rest of the rows.  Be sure to start on the appropriate age row for each investment option.
Equation for Option 1:_______
Equation for Option 2:_______
Equation for Option 3:_______

Age Opt 1 Opt 2 Opt 3

20 _______
25 _______ _______
30 _______ _______ _______
35 _______ _______ _______
40 _______ _______ _______
45 _______ _______ _______
50 _______ _______ _______
55 _______ _______ _______
60 _______ _______ _______
0

Which investment option made the most money after 10 years?
A. Option 1
B. Option 2
C. Option 3

0

Which option yielded the most total money by the time you were 60 years old?

0

After seeing the growth of these investments over time, what is one piece of advice you would give to a friend who is making a long-term investment like this?

Level 3 -BONUS
Every investment that we’ve seen so far involved making a one time deposit and then letting your investment grow.  What if we wanted to keep contributing to the investment account every month like many people do for retirement? In this scenario, the initial value will be changing every year so it’s easier to use a spreadsheet.
You are going to invest $1200 at the start of every year into an account that has an annual rate of return of 3%.
0

  1. Make a copy of this spreadsheet
  2. Complete column C for each year.  The initial value will be the total amount in the account from the previous year plus any additional deposits that you’ve made.  Think carefully about what the exponent should be for this calculation.
  3. Complete column D by subtracting the total account investment (the money that you deposited) from the total in the account to find how much interest you’ve earned on your investment.
  4. Share and Paste a copy of your spread sheet (and screen shot)

0

Why are you not able to use an exponent of 10 in this situation?

0

How would you change this situation if you wanted to use one exponential equation with an exponent of 10?