Your grandfather purchased a house for 50000 in 1952 and it

Your grandfather purchased a house for $50,000 in 1952 and it has increased in value according to a function

y = v(x),

where x is the number of years owned. These questions probe the future value of the house under various mathematical models. (Let x = 0 represent the year 1952.)

Suppose the value of the house is $75,000 in 1962 and $120,000 in 1967. Assume v(x) is a quadratic function. Find a formula for v(x). (Round your values to two decimal places.)    

Solution

genreal form of quadratic function is y= ax²+bx +c

let v(x)= ax²+bx +c

x=0 in year 1952 ,x=1962-1952=10 in year 1962 ,x=1967-1952=15 in year 1967 ,

in 1952

50000=a*0²+b*0+c

c=50000

so y=ax²+bx +50000

in 1962

a*10²+b*10 +50000=75000

100a+10b=25000

10a+b=2500

=>b=2500-10a-------->(1)

in 1967

a*15²+b*15 +50000=12000

225a+15b=70000------>(2)

put (1) in(2)

225a+15(2500-10a)=70000

225a +37500 -150a =70000

75a=32500

a=32500/75

a=1300/3

b=2500- 10a

b=2500 -10(1300)/3

b=(7500-13000)/3

b=-5500/3

v(x)= (1300/3)x²-(5500/3)x +50000 is the quartatic equation

v(x)=433.33x²-1833.33x +50000 is the quartatic equation