1. Intro to Exponential Functions Lesson 3.1

2. Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions Change at a changing rate Change at a constant percent rate View differences using spreadsheetspreadsheet View differences using spreadsheetspreadsheet

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 next value 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 YearOption AOption B 1$30,000$40,000 2$31,800$41,200 3$33,708$42,400 4$35,730$43,600 5$37,874$44,800 6$40,147$46,000 7$42,556$47,200 8$45,109$48,400 9$47,815$49,600 10$50,684$50,800 11$53,725$52,000 12$56,949$53,200 13$60,366$54,400 14$63,988$55,600

5. Example Consider a savings account with compounded yearly income  You have $100 in the account  You receive 5% annual interest At end of year Amount of interest earned New balance in account 1100 * 0.05 = $5.00$105.00 2105 * 0.05 = $5.25$110.25 3110.25 * 0.05 = $5.51$115.76 4 5 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 formula new balance = old balance + 0.05 * old balance Another way of writing the formula new 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 At end of hourAmount remaining 1100 – 0.15 * 100 = 85 285 – 0.15 * 85 = 72.25 3 4 5 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 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 function for 0 < t < 25 Use the graph to determine 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 View results of B>1, B<1 with spreadsheet

15. Assignment Lesson 3.1A Page 112 Exercises 1 – 23 odd Lesson 3.1B Pg 113 Exercises 25 – 37 odd