rccl
y= P(1+r/n)^(n t),
& y&=& balance
& P&=& principal
& r&=& annual interest
&&& rate (a decimal)
& t&=& time (in years)
&n&=&number of times
&&&interest is
&&&compounded
&&&per year
We will use this information to make two functions to represent the different accounts.
Writing a Function for Investment Account
We deposit $ 300 into an investment account that earns 12 %, or 0.12, interest compounded quarterly. Let's put these values into the formula.
y= 300(1+0.12/4)^(4t)
We can simplify it.
This function represents the balance of the investment account after t years.
Writing a Function for Savings Account
In the given graph, we see that the initial balance is $ 300. It seems that the y-coordinates of the points divide the interval from 300 to 400 into four equal intervals.
The y-coordinates of the points should be 325, 350, and 375. Then, the points are (1,325), (2,350), and (3,375). Now let's write the function partially. We assume that the interest rate is compounded yearly.
y= 300(1+r)^t
We will use the point (1,325) to find the growth factor.
Let's replace it in the function.
y=300(1+r)^t ⇒ y=300(1.08)^t
This function represents the savings account balance.
b We will draw the graphs of the functions y=300(1.03)^(4t) and y=300(1.08)^t on the same axes. Since the graph of y=300(1.08)^t is given, we will draw the other function. Let's make a table of values.
x
300(1.03)^(4t)
y=300(1.03)^(4t)
1
300(1.03)^(4* 1)
≈ 337.7
2
300(1.03)^(4* 2)
≈ 380.0
3
300(1.03)^(4* 3)
≈ 427.7
4
300(1.03)^(4* 4)
≈ 481.4
The points ( 1, 337.7), ( 2, 380.0), ( 3, 427.7), and ( 7, 481.4) are on the graph of the function y=300(1.03)^(4t).
Let's now plot and connect them with smooth curves.
The balance of the investment account increases faster.
