document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1008642>Question 1008642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the quadratic function y=ax^2+bx+c, the maximum value is a^2+4 at x=1 and the graph passes through the point (3,1). Find the values for a, b and c.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #624375 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=624375\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Given the quadratic function y=ax^2+bx+c, the maximum value is a^2+4 at x=1 and the graph passes through the point (3,1).<BR>\n" );
document.write( " Find the values for a, b and c.<BR>\n" );
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document.write( "At x^2 + 4, when x=1, find y<BR>\n" );
document.write( "y = 2^2 + 4<BR>\n" );
document.write( " y = 5<BR>\n" );
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document.write( "We know the axis of symmetry (max value) x=1<BR>\n" );
document.write( "The given point to the right is 3,1<BR>\n" );
document.write( "Therefore the point to left is -1,1<BR>\n" );
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document.write( "Three equations we can use to find a, b, c<BR>\n" );
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document.write( "-1, 1: a - b + c = 1<BR>\n" );
document.write( " 1, 5: a + b + c = 5<BR>\n" );
document.write( " 3, 1: 9a + 3b + c = 1 <BR>\n" );
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document.write( "We can use the 1st two equations to find b <BR>\n" );
document.write( "a + b + c = 5<BR>\n" );
document.write( "a - b + c = 1<BR>\n" );
document.write( "-----------------Subtraction eliminates a and c<BR>\n" );
document.write( "2b = 4<BR>\n" );
document.write( "b = 2<BR>\n" );
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document.write( "using 2nd and 3rd equations, replace b with 2<BR>\n" );
document.write( "9a + 3(2) + c = 1<BR>\n" );
document.write( "9a + 6 + c = 1<BR>\n" );
document.write( "9a + c = -5 (subtracted 6 from both sides)<BR>\n" );
document.write( "and<BR>\n" );
document.write( "a + 2 + c = 5<BR>\n" );
document.write( "a + c = 3<BR>\n" );
document.write( "Use elimination with theses two equations to find a<BR>\n" );
document.write( "9a + c = -5<BR>\n" );
document.write( " a + c = 3<BR>\n" );
document.write( "--------------Subtraction eliminates c, find a<BR>\n" );
document.write( "8a = -8<BR>\n" );
document.write( "a = -1<BR>\n" );
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document.write( "Use the 2nd original equation, a + b + c = 5; to find c, replace a and b<BR>\n" );
document.write( "-1 + 2 + c = 5<BR>\n" );
document.write( "c = 5 - 1<BR>\n" );
document.write( "c = 4<BR>\n" );
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document.write( "Our equation: y = -x^2 + 2x + 4, (green)<BR>\n" );
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document.write( "Plotting the given equation and the above equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-4%2C+4%2C+-4%2C+8%2C+x%5E2%2B4%2C+-x%5E2%2B2x%2B4%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-4%2C+4%2C+-4%2C+8%2C+x%5E2%2B4%2C+-x%5E2%2B2x%2B4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can see the max occurs where the two curves intersect; x=1; y=5\r<BR>\n" );
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