document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=332261>Question 332261</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the vertex of a parabola? <BR>\n" );
document.write( "How do you determine the vertex of the parabola? <BR>\n" );
document.write( "Demonstrate with an example. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #238163 by </B><A HREF=/tutors/aboutme.mpl?userid=jrfrunner><B>jrfrunner(365)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jrfrunner><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=jrfrunner><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=238163\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the equation for a parabola:  y=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ax%5E2%2Bb%2Ax%2Bc ALIGN=MIDDLE  ALT=\"a%2Ax%5E2%2Bb%2Ax%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The vertex is at  x=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, y=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284ac-b%5E2%29%2F%284a%29 ALIGN=MIDDLE  ALT=\"%284ac-b%5E2%29%2F%284a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this can be found from the quadratic solutions are<BR>\n" );
document.write( "x=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b-Sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-b-Sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <BR>\n" );
document.write( "x=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%2BSqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-b%2BSqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The center is at value x=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and then substituting this x value in the parabola equation to find y<BR>\n" );
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document.write( "Try it: for parabola  Y=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%5E2%2B8%2Ax%2B11 ALIGN=MIDDLE  ALT=\"2%2Ax%5E2%2B8%2Ax%2B11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "x=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = -8/4=-2<BR>\n" );
document.write( "y=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284ac-b%5E2%29%2F%284a%29 ALIGN=MIDDLE  ALT=\"%284ac-b%5E2%29%2F%284a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = (4*2*11-8^2)/(4*2) = 24/8=3<BR>\n" );
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document.write( "if you complete the square for Y=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%5E2%2B8%2Ax%2B11 ALIGN=MIDDLE  ALT=\"2%2Ax%5E2%2B8%2Ax%2B11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28x%2B2%29%5E2+%2B3 ALIGN=MIDDLE  ALT=\"2%2A%28x%2B2%29%5E2+%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > for which it is easy to see that the vertex is (-2,3)<BR>\n" );
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