document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=29577>Question 29577</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I can't figure this one out, can you help me.\r<BR>\n" );
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document.write( "Find the equation of the quadratic function with a vertex at V(1,3) passing thru the point P(2,1).Write answer in the form y=ax^2 + bx + c.\r<BR>\n" );
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document.write( "Thanx for your help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #16333 by </B><A HREF=/tutors/aboutme.mpl?userid=longjonsilver><B>longjonsilver(2297)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=longjonsilver><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=longjonsilver><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=16333\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are a couple of ways to do this. I shall show you the more visual method, so you can appreciate what quadratics are a bit more.\r<BR>\n" );
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document.write( "First, I shall show you the graph of the answer... <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-2%2C4%2C-4%2C4%2C-2x%5E2%2B4x%2B1%2C+1%29+ ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-2%2C4%2C-4%2C4%2C-2x%5E2%2B4x%2B1%2C+1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "How do i know the graph is n-shaped rather than u-shaped? Well, plot the 2 points you know. If the vertex is at (1,3), there is no way that a u-shaped curve can pass through point (2,1) as well.\r<BR>\n" );
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document.write( "I have drawn a line at y=1, since you are given the point (2,1) is on the curve. Now, a quadratic is ALWAYS symmetric about the vertex. The vertex is at x=1: a given in the question. We are also given the point on the curve at x=2, so we instantly know that the point one less than the vertex (x=0) also has the y-value of 1... points (0,1) and (2,1) are symmetric.\r<BR>\n" );
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document.write( "So we have 3 points. We have an equation with 3 unknowns too... a, b and c. So we can now solve.\r<BR>\n" );
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document.write( "So, starting with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2+%2B+bx+%2B+c+ ALIGN=MIDDLE  ALT=\"y+=+ax%5E2+%2B+bx+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have:\r<BR>\n" );
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document.write( "(0,1): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1+=+a%280%29%5E2+%2B+b%280%29+%2B+c+ ALIGN=MIDDLE  ALT=\"+1+=+a%280%29%5E2+%2B+b%280%29+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " --> 1 = c\r<BR>\n" );
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document.write( "(1,3): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3+=+a%281%29%5E2+%2B+b%281%29+%2B+c+ ALIGN=MIDDLE  ALT=\"+3+=+a%281%29%5E2+%2B+b%281%29+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " --> 3 = a + b + c<BR>\n" );
document.write( " --> 3 = a + b + 1<BR>\n" );
document.write( " --> a+b = 2 -->eqn1\r<BR>\n" );
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document.write( "(2,1): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1+=+a%282%29%5E2+%2B+b%282%29+%2B+c+ ALIGN=MIDDLE  ALT=\"+1+=+a%282%29%5E2+%2B+b%282%29+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " --> 1 = 4a + 2b + c<BR>\n" );
document.write( " --> 1 = 4a + 2b + 1<BR>\n" );
document.write( " --> 4a+2b = 0<BR>\n" );
document.write( " --> 2a+b = 0 -->eqn2\r<BR>\n" );
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document.write( "Subtract eqn1 from eqn2 to give a = -2\r<BR>\n" );
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document.write( "Hence, using 2a+b=0 we get that 2(-2)+b = 0<BR>\n" );
document.write( "-4 + b = 0<BR>\n" );
document.write( "--> b = 4\r<BR>\n" );
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document.write( "So the formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=-2x%5E2+%2B+4x+%2B+1+ ALIGN=MIDDLE  ALT=\"+y=-2x%5E2+%2B+4x+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "jon.<BR>\n" );
document.write( " </SPAN>\n" );
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