SOLUTION: Find a quadratic model for each set of values. (-1, 1), (1, 1), (3, 9)

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Question 556161:
Find a quadratic model for each set of values.

(-1, 1), (1, 1), (3, 9)
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Two points determine a linear function; three points determine a quadratic function.
A quadratic function can be written as
f%28x%29=ax%5E2%2Bbx%2Bc or y=ax%5E2%2Bbx%2Bc
In general, you can always substitute the coordinates of each of 3 points to get 3 equations.
For your problem,
(-1,1) ---> 1=a%28-1%29%5E2%2Bb%28-1%29%2Bc ---> 1=a-b%2Bc
(1,1) ---> 1=a%281%29%5E2%2Bb%281%29%2Bc ---> 1=a%2Bb%2Bc
(3,9) ---> 9=a%283%29%5E2%2Bb%283%29%2Bc ---> 9=9a%2B3b%2Bc
You got a system of equations.
From there, you solve the system for a, b, and c and those coefficients determine your quadratic function.
In your case, symmetrical points (-1, 1), and (1, 1) tell you that the axis of symmetry will be the y-axis (the line x=0), making b=0.
In your case f%28x%29=ax%5E2%2Bc or y=ax%5E2%2Bc
The simplest quadratic function (the mother of all quadratic functions) is
f%28x%29=x%5E2 or y=x%5E2
Without grabbing your pencil (or pen), you can see that it passes through all 3 of your points. There is only one quadratic function that passes through any set of 3 points, so
Your function is f%28x%29=x%5E2 or y=x%5E2.
(But you can solve the system of equations if it makes you, or your teacher happy. It's an easy one.)