Question

Find a quadratic function given by C(x) = a(x - 1959)2 + k that models the data

Answer 1

(1959,300), (1982,2000), (1990,4000)

Answer 2

I assume your expression is:\[C(x)=a(x-1959)^2+k\]

Answer 3

substitute the first pair of values into this, i.e. you are given C(x)=300 when x=1959

Answer 4

that will allow you to calculate the value of k.
then substitute the next pair of values in - i.e. C(x)=2000 when x=1982 and that should allow you to calculate the value for 'a'.

Answer 5

so
c(300)=a(1959-1959)^2+k
=a(0)+k
k=300

Answer 6

you have the value for k correct, but should have written as follows:

C(1959) = 300 = a1959-1959)^2 + k

Answer 7

missed a bracket:

C(1959) = 300 = a(1959-1959)^2 + k

Answer 8

c(1982)=2000=a(1982-1959)^2+k
=a(23)^2+k

i am stuck here

Answer 9

you know the value for k, so put that in

Answer 10

oh, gotcha. So,
c(1982)=2000=a(1982-1959)^2+k =a(23)^2+300
2000=a(23)^2+300
=529a+300-2000
a=-1700/529
so a=3.21?

Answer 11

almost, your steps have some mistakes in them, it should be:

2000 = a(23)^2 + 300
2000 - 300 = 529a
1700 = 529a
a = 1700 / 529 = 3.21

Answer 12

so a will change for every c(x)?

Answer 13

no - if you use the values of 'a' and 'k' that you found an re-write the expresion as:\[C(x)=3.21(x-1959)^2+300\]then you should find that this gives you the correct value for C(x) for every x value given to you in the question.
in other words, the values you found for 'a' and 'k' have given you a quadratic equation that satisfies all the data points.