document.write( "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\"? \n" ); document.write( "
Algebra.Com's Answer #108212 by Fombitz(32388)\"\" \"About 
You can put this solution on YOUR website!
A quadratic function has a general form of
\n" ); document.write( "\"f%28x%29=ax%5E2%2Bbx%2Bc\"
\n" ); document.write( "A quadratic function has a maximum value where its derivative equals 0.
\n" ); document.write( "The derivative of f(x) is,
\n" ); document.write( "\"df%2Fdx=2ax%2Bb\"
\n" ); document.write( "So now add your values that you know,
\n" ); document.write( "\"f%281%29=2\"
\n" ); document.write( "\"a%281%29%5E2%2Bb%281%29%2Bc=2\"
\n" ); document.write( "1.\"a%2Bb%2Bc=2\"
\n" ); document.write( "and for the maximum value,
\n" ); document.write( "\"2ax%2Bb=0\"
\n" ); document.write( "\"2a%283%29%2Bb=0\"
\n" ); document.write( "2.\"6a%2Bb=0\"
\n" ); document.write( "Generally, you need three points to solve for a quadratic equation.
\n" ); document.write( "We can solve for the other two variables in terms of one variable.
\n" ); document.write( "From eq. 2,
\n" ); document.write( "2.\"6a%2Bb=0\"
\n" ); document.write( "\"b=-6a\"
\n" ); document.write( "Then from eq.1,
\n" ); document.write( "1.\"a%2Bb%2Bc=2\"
\n" ); document.write( "\"a-6a%2Bc=2\"
\n" ); document.write( "\"c=2%2B5a\"
\n" ); document.write( "So the general form of the solution is,
\n" ); document.write( "\"f%28x%29=ax%5E2-6ax%2B%282%2B5a%29\"
\n" ); document.write( "There are some limits on a also.
\n" ); document.write( "Since the value at x=3 is a maximum, then a<0.
\n" ); document.write( "
\n" ); document.write( "Here are three graphs for a=-.5,-1,-2 just to give a sense of the family of solutions. \n" ); document.write( "
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