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Analyzing Job Choices: Linear vs. Exponential Growth in Career Salaries

Contrast the impact of linear and exponential growth on career salaries by comparing two job offers with different growth structures. Learn how to calculate future values for each option, and determine when one offer becomes more beneficial than the other.

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Analyzing Job Choices: Linear vs. Exponential Growth in Career Salaries

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  1. Intro to Exponential Functions Lesson 3.1

  2. Contrast View differences using spreadsheet

  3. Contrast • Suppose you have a choice of two different jobs at graduation • Start at $30,000 with a 6% per year increase • Start at $40,000 with $1200 per year raise • Which should you choose? • One is linear growth • One is exponential growth

  4. Which Job? • How do we get each nextvalue for Option A? • When is Option A better? • When is Option B better? • Rate of increase a constant $1200 • Rate of increase changing • Percent of increase is a constant • Ratio of successive years is 1.06

  5. Example • Consider a savings account with compounded yearly income • You have $100 in the account • You receive 5% annual interest View completed table

  6. Compounded Interest • Completed table

  7. Compounded Interest • Table of results from calculator • Set y= screen y1(x)=100*1.05^x • Choose Table (Diamond Y) • Graph of results

  8. Exponential Modeling • Population growth often modeled by exponential function • Half life of radioactive materials modeled by exponential function

  9. Growth Factor • Recall formulanew balance = old balance + 0.05 * old balance • Another way of writing the formulanew balance = 1.05 * old balance • Why equivalent? • Growth factor: 1 + interest rate as a fraction

  10. Decreasing Exponentials • Consider a medication • Patient takes 100 mg • Once it is taken, body filters medication out over period of time • Suppose it removes 15% of what is present in the blood stream every hour Fill in the rest of the table What is the growth factor?

  11. Decreasing Exponentials • Completed chart • Graph Growth Factor = 0.85 Note: when growth factor < 1, exponential is a decreasing function

  12. Solving Exponential Equations Graphically • For our medication example when does the amount of medication amount to less than 5 mg • Graph the functionfor 0 < t < 25 • Use the graph todetermine when

  13. General Formula • All exponential functions have the general format: • Where • A = initial value • B = growth factor • t = number of time periods

  14. Typical Exponential Graphs • When B > 1 • When B < 1 View results of B>1, B<1 with spreadsheet

  15. Assignment • Lesson 3.1A • Page 112 • Exercises1 – 23 odd • Lesson 3.1B • Pg 113 • Exercises25 – 37 odd

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