document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=677673>Question 677673</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the vertex of the parabola\r<BR>\n" );
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document.write( "F(x)=2x^2+8x+3\r<BR>\n" );
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document.write( "Can you please show the work so that I can get a full understanding.\r<BR>\n" );
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document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #421002 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=421002\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+F%28x%29=2x%5E2%2B8x%2B3+ ALIGN=MIDDLE  ALT=\"+F%28x%29=2x%5E2%2B8x%2B3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is a parabola with a minimum and not a maximum<BR>\n" );
document.write( "I can tell because the squared term is positive. If I changed it<BR>\n" );
document.write( "to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+F%28x%29=-2x%5E2%2B8x%2B3+ ALIGN=MIDDLE  ALT=\"+F%28x%29=-2x%5E2%2B8x%2B3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > then it would have a maximum.<BR>\n" );
document.write( "The vertex is the (x,y) point at the minimum ( or maximum )<BR>\n" );
document.write( "---------------<BR>\n" );
document.write( "The x-coordinate of the vertex occurs halfway <BR>\n" );
document.write( "between the roots,  which are<BR>\n" );
document.write( "the two solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2x%5E2%2B8x%2B3+=+0+ ALIGN=MIDDLE  ALT=\"+2x%5E2%2B8x%2B3+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   <BR>\n" );
document.write( "---------------<BR>\n" );
document.write( "To get a deeper understanding of this look at the <BR>\n" );
document.write( "quadratic formula which gives these two solutions<BR>\n" );
document.write( "---------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B1%5D+=+%28-b+%2B+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"+x%5B1%5D+=+%28-b+%2B+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B2%5D+=+%28-b+-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"+x%5B2%5D+=+%28-b+-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "Now I'll separate the terms in each solution:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B1%5D+=+-b%2F%282a%29+%2B+%28+sqrt%28+b%5E2+-+4%2Aa%2Ac%29+%29+%2F+%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"+x%5B1%5D+=+-b%2F%282a%29+%2B+%28+sqrt%28+b%5E2+-+4%2Aa%2Ac%29+%29+%2F+%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B2%5D+=+-b%2F%282a%29+-+%28+sqrt%28+b%5E2+-+4%2Aa%2Ac%29+%29+%2F+%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"+x%5B2%5D+=+-b%2F%282a%29+-+%28+sqrt%28+b%5E2+-+4%2Aa%2Ac%29+%29+%2F+%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "-----------------------------------------<BR>\n" );
document.write( "Now what you have is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B1%5D+ ALIGN=MIDDLE  ALT=\"+x%5B1%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > = ( x coordinate of the vertex ) + ( the positive correction )<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B2%5D+ ALIGN=MIDDLE  ALT=\"+x%5B2%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > = ( x coordinate of the vertex ) - ( the negative correction )<BR>\n" );
document.write( "-------------------<BR>\n" );
document.write( "So, you can see that <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;\" > is the x-coordinate of the vertex<BR>\n" );
document.write( "when the form of the equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+ax%5E2+%2B+b%2Ax+%2B+c+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+ax%5E2+%2B+b%2Ax+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-------------------<BR>\n" );
document.write( "I can apply this to your problem<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+F%28x%29=2x%5E2%2B8x%2B3+ ALIGN=MIDDLE  ALT=\"+F%28x%29=2x%5E2%2B8x%2B3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+2+ ALIGN=MIDDLE  ALT=\"+a+=+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+=+8+ ALIGN=MIDDLE  ALT=\"+b+=+8+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+=+3+ ALIGN=MIDDLE  ALT=\"+c+=+3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-b%2F%282a%29+=+-8%2F%282%2A2%29+ ALIGN=MIDDLE  ALT=\"+-b%2F%282a%29+=+-8%2F%282%2A2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >       <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-b%2F%282a%29+=+-2+ ALIGN=MIDDLE  ALT=\"+-b%2F%282a%29+=+-2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "Now you know the vertex is at ( -2, y ). To find y, plug <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+-2+ ALIGN=MIDDLE  ALT=\"+x+=+-2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "into the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=2%2A%28-2%29%5E2%2B8%2A%28-2%29%2B3+ ALIGN=MIDDLE  ALT=\"+y+=2%2A%28-2%29%5E2%2B8%2A%28-2%29%2B3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+2%2A4+-+16+%2B+3+ ALIGN=MIDDLE  ALT=\"+y+=+2%2A4+-+16+%2B+3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+8+-+16+%2B+3+ ALIGN=MIDDLE  ALT=\"+y+=+8+-+16+%2B+3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+-5+ ALIGN=MIDDLE  ALT=\"+y+=+-5+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "---------------<BR>\n" );
document.write( "The vertex is at ( -2, -5 )<BR>\n" );
document.write( "Here's a plot of the equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+400%2C+400%2C+-7%2C+7%2C+-7%2C+7%2C+2x%5E2+%2B+8x+%2B+3+%29+ ALIGN=MIDDLE  ALT=\"+graph%28+400%2C+400%2C+-7%2C+7%2C+-7%2C+7%2C+2x%5E2+%2B+8x+%2B+3+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The vertex is where is should be\r<BR>\n" );
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document.write( " </SPAN>\n" );
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