DAY 107 EXPONENTIAL GROWTH AND DECAY EXPONENTIAL FUNCTION

DAY 107 – EXPONENTIAL GROWTH AND DECAY

EXPONENTIAL FUNCTION v. What do we know about exponents? v. What do we know about functions?

EXPONENTIAL FUNCTIONS �Always involves the equation: bx �Example: 2 =2· 2· 2=8 3

GROUP INVESTIGATION: Create an x, y table. Use x values of -1, 0, 1, 2, 3, Graph the table What do you observe.

THE TABLE: RESULTS x -1 0 1 2 3 f(x) = 2 x 2 -1 = ½ 20 = 1 21 = 2 22 = 4 23 = 8

THE GRAPH OF

OBSERVATIONS What did you notice? What is the pattern? What would happen if What real-life applications are there?

GROUP: MONEY DOUBLING? You have a $100. 00 Your money doubles each year. How much do you have in 5 years? Show work.

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

EARNING INTEREST ON You have $100. Each year you earn 10% interest. How much $ do you have in 5 years? Show Work.

EARNING 10% RESULTS Year 1: $100 + 100*(0. 10) = $110 Year 2: $110 + 110*(0. 10) = $121 Year 3: $121 + 121*(0. 10) = $133. 10 Year 4: $133. 10 + 133. 10*(0. 10) = $146. 41 Year 5: $146. 41 + 1461. 41*(0. 10) = $161. 05

GROWTH MODELS: INVESTING The Equation is: P = Principal r = Annual Rate t = Number of years

USING THE EQUATION $100. 00 10% interest 5 years 100(1+ (. 10))5 = $161. 05 What could we figure out now?

COMPARING INVESTMENTS Choice 1 �$10, 000 � 5. 5% interest � 9 years Choice 2 �$8, 000 � 6. 5% interest � 10 years

CHOICE 1 $10, 000, 5. 5% interest for 9 years. Equation: $10, 000 (1 +. 055)9 Balance after 9 years: $16, 190. 94

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

WHICH INVESTMENT? The first one yields more money. Choice 1: $16, 190. 94 Choice 2: $15, 071. 10

EXPONENTIAL DECAY Instead of increasing, it is decreasing. Formula: a = initial amount r = percent decrease t = Number of years

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

EXPONENTIAL DECAY: CAR DEPRECIATION Assume the car was purchased for $20, 000 Depreciation Value after 2 Rate years 10% $16, 200 Value after 5 years Value after 8 years Value after 10 years $11, 809. 80 $8609. 34 $6973. 57 Formula: a = initial amount r = percent decrease t = Number of years

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.

ASSIGNMENT 2 Worksheets: �Exponential Growth: Investing Worksheet �Exponential Decay: Car Depreciation