how to write an exponential function

Question

how do i write an exponential function that models this situation where ,y, is the amount of Grace's investment and ,x, is time in years.

 

 

"Grace deposits $1000 in a mutual fund earning  9.5% annual interest, compounded monthly."

 

 

Then how would I complete a chart rounding to the nearest cent?

 

                   years                balance

                    5                            ??

                   10                           ??

                   15                           ??

 

Thanks!

Answer 1

Invest P dollars at an interest rate of (i), compounded annually, we get the following after one year.

 

A₁ = P + P i = P (1 + i)

 

after the 2nd year

 

A₂ = A₁ (1 + i)

 

substituting our expression for A₁ gives us

 

A₂ = P (1 + i) (1 + i) = P (1 + i)²

 

Similarly, we can extend this idea to t years and we get

 

A = P (1 + i)^t

 

now we expand to account for interest being compounded more than once per year say, n times per year.  Hence the number of time the interest is compounded will be nt, and the interest rate for a period is i/n.

 

A(t) = P (1 + i/n)^nt

 

now for your example... P = 1,000, i=9.5%, n=12

 

A(t) = 1,000 (1 + 0.095/12)^12t

 

and now look at the values for t = 5, 10, 15 years

 

A(5) = 1,000 (1 + .095/12)¹²⁽⁵⁾=1,605.01

A(10) = 2,576.06

A(15) = 4,134.59