SOLUTION: Please help me find the quadratic function given by {{{f(x)=a(x-h)^2+k}}} that models this data: Year - 1940, 1968, 1997; Wage ($) - 0.25, 1.60, 5.15. I would love to know the ste

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Question 462509: Please help me find the quadratic function given by that models this data: Year - 1940, 1968, 1997; Wage ($) - 0.25, 1.60, 5.15. I would love to know the steps to do this, and then I can find the answer. I just don't know where to start. Thank you.
Answer by jsmallt9(3759) (Show Source):
You can put this solution on YOUR website!
Essentially, your problem boils down to finding the correct "a", "h" and "k". To find three unknowns, the general rule is that you need three equations. We will get these three equations using f(x) three times, once for each ordered pair you've been given:
(1940, 0.25)
(1968, 1.60)
(1997, 5.15)

Inserting each of these ordered pairs into f(x) we get:

Simplifying each of these we get:

Continuing to simplify:

A system like this can be difficult to solve. But since each of these equations has both "ah^2" and "k" as terms we can simplify quickly. Subtracting the second equation fron the first and the third equation from the first we get:

Now we have a system of two equations with two unknowns. It should be relatively simple to solve this system for "a" and "h". (Suggestion: Multiply the first equation by -114 and the second equation by 156 and then add the equations.)

Then take these values for "a" and "h" and use them with one of the earlier equations (that has "k" in it) to find "k".

Then you just write f(x) with these three values