SOLUTION: Write a quadratic model for (-1,1) (1,1) (3,9)

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Question 360690: Write a quadratic model for (-1,1) (1,1) (3,9)
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
I can call the points (x1,y1), (x2,y2), and (x3,y3)
y%5B1%5D+=+f%28x%5B1%5D%29
y%5B2%5D+=+f%28x%5B2%5D%29
y%5B3%5D+=+f%28x%5B3%5D%29
Now, plugging in the actual x's and y's
1+=+f%28-1%29
1+=+f%281%29
9+=+f%283%29
The general form for a quadratic is
y+=+ax%5E2+%2B+bx+%2B+c
Now I can write
1+=+a%2A%28-1%29%5E2+%2B+b%2A%28-1%29+%2B+c
(1) 1+=+a+-+b+%2B+c
and
1+=+a%2A1%5E2+%2B+b%2A1++%2B+c
(2) 1+=+a+%2B+b+%2B+c
and
9+=+a%2A3%5E2+%2B+b%2A3+%2B+c
(3) 9+=+9a+%2B+3b+%2B+c
This is 3 equations and
3 unknowns, so it's solvable
Add (1) and (2) to get
2+=+2a+%2B+2c
a+%2B+c+=+1
c+=+1+-+a
Substitute this in (3)
9+=+9a+%2B+3b+%2B+1+-+a
9+=+8a+%2B+3b+%2B+1
8a+%2B+3b+=+8
3b+=+-8a+%2B+8
b+=+%281%2F3%29%2A%28-8a+%2B+8%29
Now substitute c and b back into (2)
(2) 1+=+a+%2B+b+%2B+c
(2) 1+=+a+%2B++%281%2F3%29%2A%28-8a+%2B+8%29+%2B+1+-+a
Now solve for a, then do more substitutions to get b and c
3+=+3a+-+8a+%2B+8+%2B+3+-+3a
8a+=+8
a+=+1
c+=+1+-+a
c+=+0
b+=+%281%2F3%29%2A%28-8a+%2B+8%29
b+=+%281%2F3%29%2A%28-8+%2B+8%29
b+=+0
The solution is
y+=+1%2Ax%5E2+%2B+0%2Ax+%2B+0
y+=+x%5E2
This works if I plug the given points back in
The plot is
+graph%28+400%2C+400%2C+-5%2C+5%2C+-2%2C+10%2C+x%5E2%29+