SOLUTION: Form quadratic functions given the following information: Chapter Reference b f(3+ sqrt2 )=f(3−sqrt2 )=0; f(1)=−8

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Question 1109709: Form quadratic functions given the following information:
Chapter Reference
b
f(3+ sqrt2 )=f(3−sqrt2 )=0; f(1)=−8
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The roots are 3+sqrt(2) and 3-sqrt(2); that means the axis of symmetry is x=3; that means the vertex has x coordinate 3. Then the vertex form of the equation is

y+=+a%28x-3%29%5E2+%2B+k

Then since y is 0 when x is 3+sqrt(2),
0+=+a%28sqrt%282%29%29%5E2%2Bk
0+=+2a%2Bk (1)

And since y is -8 when x is 1,
-8+=+a%28-2%29%5E2%2Bk
-8+=+4a%2Bk (2)

Then from equations (1) and (2)

-8+=+2a
a+=+-4

And then from either equation (1) or (2), k = 8.

So the equation is

y+=+-4%28x-3%29%5E2%2B8
y+=+-4%28x%5E2-6x%2B9%29%2B8
y+=+-4x%5E2%2B24x-28

Solving this equation using the quadratic formula verifies that the roots are 3 +/- sqrt(2); evaluating the expression for x=1 verifies that the y value is -8.