Intro to Exponential FunctionsContrastGeneral FormulaSlide 4Which Job?ExampleCompounded InterestSlide 8Exponential ModelingGrowth FactorAssignmentDecreasing ExponentialsSlide 13Solving Exponential Equations GraphicallyTypical Exponential GraphsSlide 16Intro to Exponential FunctionsLesson 4.1ContrastLinearFunctions• Change at a constant rate• Rate of change (slope) is a constantExponentialFunctions• Change at a changing rate• Change at a constant percent rateGeneral Formula•All exponential functions have the general format:•WhereA = initial valueB = growth factort = number of time periods( )tf t A B= �Contrast•Suppose you have a choice of two different jobs at graduationStart at $30,000 with a 10% per year increaseStart at $40,000 with $1000 per year raise•Which should you choose?One is linear growthOne is exponential growth1 30,000 40,000 2 33,000 41,000 3 36,300 42,000 4 39,930 43,000 5 43,923 44,000 6 48,315 45,000 7 53,147 46,000 8 58,462 47,000 9 64,308 48,000 10 70,738 49,000 11 77,812 50,000 12 85,594 51,000 13 94,153 52,000 14 103,568 53,000 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 $1000•Rate of increase changing Percent of increase is a constant Ratio of successive years is 1.10Example•Consider a savings account with compounded yearly incomeYou have $100 in the accountYou receive 5% annual interestAt end of yearAmount of interest earnedNew balance in account1 100 * 0.05 = $5.00 $105.00 2 105 * 0.05 = $5.25 $110.25 3 110.25 * 0.05 = $5.51 $115.76 4 5 View completed tableCompounded Interest•Completed table( )tf t A B= �Compounded Interest•Table of results from calculatorSet y= screen y1(x)=100*1.05^xChoose Table (Diamond Y/F5)•Graph of resultsExponential Modeling•Population growth often modeled by exponential function•Half life of radioactive materials modeled by exponential functionGrowth 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 fractionAssignment•Lesson 3.1A•Page 112•Exercises1 – 23 oddDecreasing 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 hourAt end of hour Amount remaining1 100 – 0.15 * 100 = 852 85 – 0.15 * 85 = 72.25345Fill in the rest of the tableFill in the rest of the tableWhat is the growth factor?What is the growth factor?Decreasing Exponentials•Completed chart•GraphGrowth Factor = 0.85Note: when growth factor < 1, exponential is a decreasing functionGrowth Factor = 0.85Note: when growth factor < 1, exponential is a decreasing functionSolving 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( ) 100 0.85 5.0tM t = � <Typical Exponential Graphs•When B > 1•When B < 1( )tf t A B= �View results of B>1, B<1 with ExcelView results of B>1, B<1 with ExcelAssignment•Lesson 4.1•Pg 136•Exercises 1 – 53