functions? E XPONENTIAL F UNCTIONS Always involves the equation: b x - PowerPoint PPT Presentation

D AY 107 – E XPONENTIAL G ROWTH AND D ECAY

E XPONENTIAL F UNCTION  What do we know about exponents?  What do we know about functions?

E XPONENTIAL F UNCTIONS  Always involves the equation: b x  Example:  2 3 = 2 · 2 · 2 = 8

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

T HE T ABLE : R ESULTS x f(x) = 2 x 2 -1 = ½ -1 2 0 = 1 0 2 1 = 2 1 2 2 = 4 2 2 3 = 8 3

 x 2 T HE G RAPH OF y

O BSERVATIONS  What did you notice?  What is the pattern?    What would happen if 2 x   What would happen if 5 x  What real-life applications are there?

G ROUP : M ONEY D OUBLING ?  You have a $100.00  Your money doubles each year.  How much do you have in 5 years?  Show work.

M ONEY D OUBLING 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

E ARNING I NTEREST ON  You have $100.00.  Each year you earn 10% interest.  How much $ do you have in 5 years?  Show Work.

E ARNING 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

G ROWTH M ODELS : I NVESTING The Equation is:  (  t 1 ) a p r P = Principal r = Annual Rate t = Number of years

U SING THE E QUATION  $100.00  10% interest  5 years  100(1+ (.10)) 5 = $161.05  What could we figure out now?

C OMPARING I NVESTMENTS  Choice 1  $10,000  5.5% interest  9 years  Choice 2  $8,000  6.5% interest  10 years

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

C HOICE 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

W HICH I NVESTMENT ? The first one yields more money. Choice 1: $16,190.94 Choice 2: $15,071.10

E XPONENTIAL D ECAY Instead of increasing, it is decreasing.  (  Formula: t 1 ) y a r a = initial amount r = percent decrease t = Number of years

R EAL - LIFE E XAMPLES  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

E XPONENTIAL D ECAY : C AR D EPRECIATION Assume the car was purchased for $20,000 Depreciation Value after 2 Value after 5 Value after 8 Value after 10 Rate years years years years $16,200 $11,809.80 $8609.34 $6973.57 10%  (  Formula: t 1 5 ) y a a = initial amount r = percent decrease t = Number of years

W HAT E LSE ?  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.

A SSIGNMENT  2 Worksheets:  Exponential Growth: Investing Worksheet  Exponential Decay: Car Depreciation