SOLUTION: What does it mean when it says "find the quadratic equation where f(1)=2 and the maximum value of f is f(3)=-5"?

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Question 147835: What does it mean when it says "find the quadratic equation where f(1)=2 and the maximum value of f is f(3)=-5"?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
A quadratic function has a general form of
f%28x%29=ax%5E2%2Bbx%2Bc
A quadratic function has a maximum value where its derivative equals 0.
The derivative of f(x) is,
df%2Fdx=2ax%2Bb
So now add your values that you know,
f%281%29=2
a%281%29%5E2%2Bb%281%29%2Bc=2
1.a%2Bb%2Bc=2
and for the maximum value,
2ax%2Bb=0
2a%283%29%2Bb=0
2.6a%2Bb=0
Generally, you need three points to solve for a quadratic equation.
We can solve for the other two variables in terms of one variable.
From eq. 2,
2.6a%2Bb=0
b=-6a
Then from eq.1,
1.a%2Bb%2Bc=2
a-6a%2Bc=2
c=2%2B5a
So the general form of the solution is,
f%28x%29=ax%5E2-6ax%2B%282%2B5a%29
There are some limits on a also.
Since the value at x=3 is a maximum, then a<0.

Here are three graphs for a=-.5,-1,-2 just to give a sense of the family of solutions.