Question

You will use a graphing calculator to compare savings plans. For instance, suppose you are depositing $1000 in a savings account

and are given the following options:

- 6.2% annual interest rate, compounded annually

- 6.1% annual interest rate, compounded quarterly

- 6.0% annual interest rate, compounded continuously

1. For each option, write a function that gives the balance as a function of time t (in years). Graph all three functions in the same viewing window, and within parameters that are applicable for this money problem. Can you find a viewing window that distinguishes among the graphs of the three functions? If so, describe the viewing window by stating xmin, xmax, ymin, and ymax

Answer 1

Let F(t) = Future value at time t, A = Principal = 1000, r = interest rate, m = No. of times compounded annually, t = No. of years

then F(t) = A(1+(r/m))^mt

For option 1:

6.2% annual interest rate compounded annually,

F₁(t) = A(1+r)^t

F₁(t) = 1000*1.062^t

For option 2:

6.1% annual interest rate, compounded quarterly,

F₂(t) = A(1+r/m)^mt

F₂(t) = 1000(1+(0.061/4))^4t

F₂(t) = 1000(1.01525)^4t

For option 3:

6% annual interest rate, compounded countinously,

Here the formula changes a little bit,

F₃(t) = Ae^rt

F₃(t) = 1000e^0.06t

Now I have graphed them on an app called Desmos on my phone, use y instead of F(t) and x instead of t

Graph in red describes F₁(t), the one in green describes F₂(t) and the one in purple describes F₃(t)

All three converge to (0,1000) and diverge on both sides afterwards.

You can see that F₃ > F₁ > F₂ in the region t<0

You can see that F₂> F₁>F₃ in the region t>0

You can use the third picture to find

X_min = 20, X_max = 24

Y_min = 3624, Y_max = 3632