A $$125,000$ mortgage is financed through a credit union for $20$ years at an interest rate of $6\mathrm{.}25\%,$ with interest compounded monthly.

a$)$ Find the monthly payment.

b$)$ As seen in our textbook the formula $P(1+\frac{r}{12}{)}^{12(20-t)}=\frac{m((1+\frac{r}{12}{)}^{12(20-t)}-1)}{\frac{r}{12}}$ is used to find the unpaid loan balance $P$ of the $20$ year loan after $t$ years. Find the unpaid loan balance after $5,10$ and $15$ years.

c$)$ If the unpaid loan balance $P$ is a function of time $t$ graph the function from b$)$ using technology and determine from the graph how long it will take for your balance from your original mortgage to decrease by one half of the original amount.

d$)$ If $$500$ extra is paid per month towards the mortgage, how long will it take to pay it in full? What is your savings in interest?

e$)$ Use the present value for an annuity formula with $m=($initial monthly payment$)+500$ to find out how long it will take to pay off the mortgage by paying an extra $$500$ per month.

f$)$ The same formula from b$)$ will give you the unpaid loan balance after $t$ years for the extra payment option, paying the extra $$500$ per month. Using the time found in e$)$ in place of the $20$ years on the original loan and your monthly payment $m$ with the extra $$500$ added in$,$ graph the new function on the same axes as the one found in b$)$ and determine how many years it will take for this balance to decrease by one half of the original amount.