Exponential Equations

Developing a Polynomial Equation

• 
• You won the lottery! $100 - woot-woot!
• You invest the money in a fund earning 20% interest per year.
• How much money do you have at the end of:
• year 1?
• year 2?
• year 5?
• year 35?

YEAR ONE

• 
• 
• Legenda: iNITIAL INVESTMENT
• Legenda: INITIAL INVESTMENT MULTIPLIED BY 20%

YEAR ONE IN NUMBERS

• How did we calculate the balance after one year?
• We took our $100 and multiplied it by the interest rate of 20%
• We then added our initial investment to the interest amount
• ==> $100 x 20% = $20
• ==> $100 + $20 = $120
• ==> OR $100 X 1.2 = $120

YEAR TWO

• 
• Legenda: NOW OUR INVESTMENT STARTS AT $120
• 
• Legenda: SO WE EARN $4 MORE THAN $20

YEAR TWO IN NUMBERS & EQUATIONS

• Because our investment value increased after year one, our interest in year two grows too.
• We call this EXPONENTIAL growth
• Let's apply exponents to the calculation:
• We started with $100 and at the end of year one had $120
• Now we have $120 which turns into $144 which is just the 1.2 to the exponent 2 (the number of years) multiplied by the initial investment
• We can now make an equation to use to calculate the value after 15 years
• ==> $100 X (120%) = $100 X (1.2) = $120
• ==> ($100 X (1.2 x 1.2) = $100 x (1.2 ^ 2) = $144
• the symbol ^ means to the exponent
• ==> $100 X (1.2 E 15) = $1,540.70

A POLYNOMIAL EQUATION

• We can also take this formula further
• Let's say we don't know how much money we have to invest; we'll call this amount i (for investment)
• And we don't know how many years we're going to invest for; we'll call this n (for number of years)
• And we don't know how much we'll earn every year;
• we'll call this r (for rate of interest)
• Now we can create an equation of variables that will represent the amount of our investment after n years;
• we'll call this T (for Total $)
• In our example i = $100
• In our example n = 2 and then we changed it to n = 15
• In our example r = 20%
• T = i X (1 + r) ^ n

LET'S TRY IT!

• Say we really hit the jackpot at the slot machine and win
• $54, 796
• And say we find a sucker that will pay us 15% interest if we lend him that exact amount
• And say that he needs the loan for 5 years
• ==> i = 54,796
• ==> r = 15%
• ==> n = 5
• ==> What is T?