Intro to Exponential Functions

Contrast Linear Functions Change at a constant rate View differences using spreadsheet 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

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

Which Job? How do we get each next value for Option A? Year Option A Option 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 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

Amount of interest earned 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 1 100 * 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 table

Compounded Interest Completed table

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

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

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

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 hour Amount remaining 1 100 – 0.15 * 100 = 85 2 85 – 0.15 * 85 = 72.25 3 4 5 Fill in the rest of the table What is the growth factor?

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

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

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

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