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Exploring Exponential Growth & Decay: Real-World Applications and Comparisons

Dive into the world of exponential functions and their practical implications in finance, investing, and depreciation. Understand the impact of growth and decay rates on investments and asset values. Learn how to analyze and compare different scenarios to make informed decisions.

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Exploring Exponential Growth & Decay: Real-World Applications and Comparisons

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  1. Exponential Growth & Decay Applications that Apply to Me!

  2. Exponential Function • What do we know about exponents? • What do we know about functions?

  3. Exponential Functions • Always involves the equation: bx • Example: • 23 = 2 · 2 · 2 = 8

  4. Group investigation:Y = 2x • Create an x,y table. • Use x values of -1, 0, 1, 2, 3, • Graph the table • What do you observe.

  5. The Table: Results

  6. The Graph of y = 2x

  7. Observations • What did you notice? • What is the pattern? • What would happen if x= -2 • What would happen if x = 5 • What real-life applications are there?

  8. Group: Money Doubling? • You have a $100.00 • Your money doubles each year. • How much do you have in 5 years? • Show work.

  9. Money Doubling Year 1: $100 · 2 = $200 Year 2: $200 · 2 = $400 Year 3: $400 · 2 = $800 Year 4: $800 · 2 = $1600 Year 5: $1600 · 2 = $3200

  10. Earning Interest on • You have $100.00. • Each year you earn 10% interest. • How much $ do you have in 5 years? • Show Work.

  11. Earning 10% results Year 1: $100 + 100·(.10) = $110 Year 2: $110 + 110·(.10) = $121 Year 3: $121 + 121·(.10) = $133.10 Year 4: $133.10 + 133.10·(.10) = $146.41 Year 5: $146.41 + 1461.41·(.10) = $161.05

  12. Growth Models: Investing The Equation is: A = P (1+ r)t P = Principal r = Annual Rate t = Number of years

  13. Using the Equation • $100.00 • 10% interest • 5 years • 100(1+ 100·(.10))5 = $161.05 • What could we figure out now?

  14. Comparing Investments • Choice 1 • $10,000 • 5.5% interest • 9 years • Choice 2 • $8,000 • 6.5% interest • 10 years

  15. Choice 1 $10,000, 5.5% interest for 9 years. Equation: $10,000 (1 + .055)9 Balance after 9 years: $16,190.94

  16. Choice 2 $8,000 in an account that pays 6.5% interest for 10 years. Equation: $8,000 (1 + .065)10 Balance after 10 years: $15,071.10

  17. Which Investment? • The first one yields more money. • Choice 1: $16,190.94 • Choice 2: $15,071.10

  18. Exponential Decay Instead of increasing, it is decreasing. Formula: y = a (1 – r)t a = initial amount r = percent decrease t = Number of years

  19. Real-life Examples • What is car depreciation? • Car Value = $20,000 • Depreciates 10% a year • Figure out the following values: • After 2 years • After 5 years • After 8 years • After 10 years

  20. Exponential Decay: Car Depreciation Assume the car was purchased for $20,000 Formula: y = a (1 – r)t a = initial amount r = percent decrease t = Number of years

  21. What Else? • What happens when the depreciation rate changes. • What happens to the values after 20 or 30 years out – does it make sense? • What are the pros and cons of buying new or used cars.

  22. Assignment • 2 Worksheets: • Exponential Growth: Investing Worksheet (available at ttp://www.uen.org/Lessonplan/preview.cgi?LPid=24626) • Exponential Decay: Car Depreciation

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