document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186906>Question 1186906</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A parabola has a y-intercept of -4 and passes through the points (-2, 8) and (1, -1). Determine the vertex of the<BR>\n" );
document.write( "parabola. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #817871 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=817871\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "A parabola has a y-intercept of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4+ ALIGN=MIDDLE  ALT=\"-4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=>(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4+ ALIGN=MIDDLE  ALT=\"-4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >) =><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-4 ALIGN=MIDDLE  ALT=\"k=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and passes through the points (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >). \r<BR>\n" );
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document.write( "Determine the vertex of the parabola.\r<BR>\n" );
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document.write( "use  the vertex form of a quadratic equation: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a+%28+x-+h+%29%5E2+%2B+k ALIGN=MIDDLE  ALT=\"y+=+a+%28+x-+h+%29%5E2+%2B+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >........use given point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >)  and intercept <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-4 ALIGN=MIDDLE  ALT=\"k=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8=+a+%28+-2-+h+%29%5E2+-4 ALIGN=MIDDLE  ALT=\"8=+a+%28+-2-+h+%29%5E2+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >......solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288%2B4%29%2F%28+-2-+h+%29%5E2+=+a+ ALIGN=MIDDLE  ALT=\"%288%2B4%29%2F%28+-2-+h+%29%5E2+=+a+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F%28+-2-+h+%29%5E2+=+a+ ALIGN=MIDDLE  ALT=\"12%2F%28+-2-+h+%29%5E2+=+a+\"   DISABLED_style= \"max-width:90%; height:auto;\" >........eq.1\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a+%28+x-+h+%29%5E2+%2B+k ALIGN=MIDDLE  ALT=\"y+=+a+%28+x-+h+%29%5E2+%2B+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >........use given point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)  and intercept <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-4 ALIGN=MIDDLE  ALT=\"k=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+a+%281-+h+%29%5E2+-4 ALIGN=MIDDLE  ALT=\"-1+=+a+%281-+h+%29%5E2+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2B4%29+%2F%281-+h+%29%5E2=+a++ ALIGN=MIDDLE  ALT=\"%28-1%2B4%29+%2F%281-+h+%29%5E2=+a++\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%2F%281-+h+%29%5E2=+a++ ALIGN=MIDDLE  ALT=\"3+%2F%281-+h+%29%5E2=+a++\"   DISABLED_style= \"max-width:90%; height:auto;\" >...........eq.2\r<BR>\n" );
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document.write( "from eq.1 and eq.2 we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F%28+-2-+h+%29%5E2+=+3+%2F%281-+h+%29%5E2 ALIGN=MIDDLE  ALT=\"12%2F%28+-2-+h+%29%5E2+=+3+%2F%281-+h+%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%281-+h+%29%5E2+=+3%28+-2-+h+%29%5E2 ALIGN=MIDDLE  ALT=\"12%281-+h+%29%5E2+=+3%28+-2-+h+%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.....simplify\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%281-+h+%29%5E2+=+%28+-2-+h+%29%5E2 ALIGN=MIDDLE  ALT=\"4%281-+h+%29%5E2+=+%28+-2-+h+%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28h%5E2+-+2+h+%2B+1+%29+=+h%5E2+%2B+4+h+%2B+4 ALIGN=MIDDLE  ALT=\"4%28h%5E2+-+2+h+%2B+1+%29+=+h%5E2+%2B+4+h+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4h%5E2+-+8+h+%2B+4+=+h%5E2+%2B+4+h+%2B+4 ALIGN=MIDDLE  ALT=\"4h%5E2+-+8+h+%2B+4+=+h%5E2+%2B+4+h+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4h%5E2+-+8+h+%2B+4+-+h%5E2+-+4+h+-4=0 ALIGN=MIDDLE  ALT=\"4h%5E2+-+8+h+%2B+4+-+h%5E2+-+4+h+-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3h%5E2+-+12h+++=0 ALIGN=MIDDLE  ALT=\"3h%5E2+-+12h+++=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283h+-+12%29h+++=0 ALIGN=MIDDLE  ALT=\"%283h+-+12%29h+++=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "solutions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=0 ALIGN=MIDDLE  ALT=\"h=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3h+-+12+++=0 ALIGN=MIDDLE  ALT=\"3h+-+12+++=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > =><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3h=12 ALIGN=MIDDLE  ALT=\"3h=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=4 ALIGN=MIDDLE  ALT=\"h=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "the vertex of the parabola is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >)  or  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "go to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F%28+-2-+h+%29%5E2+=+a+ ALIGN=MIDDLE  ALT=\"12%2F%28+-2-+h+%29%5E2+=+a+\"   DISABLED_style= \"max-width:90%; height:auto;\" >........eq.1, substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=0 ALIGN=MIDDLE  ALT=\"h=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F%28+-2-+0+%29%5E2+=+a+ ALIGN=MIDDLE  ALT=\"12%2F%28+-2-+0+%29%5E2+=+a+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F4+=+a+ ALIGN=MIDDLE  ALT=\"12%2F4+=+a+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "or\r<BR>\n" );
document.write( "\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=4 ALIGN=MIDDLE  ALT=\"h=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F%28+-2-+4+%29%5E2+=+a+ ALIGN=MIDDLE  ALT=\"12%2F%28+-2-+4+%29%5E2+=+a+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F36+=+a+ ALIGN=MIDDLE  ALT=\"12%2F36+=+a+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1%2F3 ALIGN=MIDDLE  ALT=\"a=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<BR>\n" );
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document.write( "so, your parabola is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3+%28+x-+0+%29%5E2+-4 ALIGN=MIDDLE  ALT=\"y+=+3+%28+x-+0+%29%5E2+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3+x%5E2+-4 ALIGN=MIDDLE  ALT=\"y+=+3+x%5E2+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%281%2F3%29+%28+x-+4+%29%5E2+-4 ALIGN=MIDDLE  ALT=\"y+=+%281%2F3%29+%28+x-+4+%29%5E2+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "let's see both solutions on the graph:\r<BR>\n" );
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document.write( "1. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3+x%5E2+-4 ALIGN=MIDDLE  ALT=\"y+=+3+x%5E2+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Acircle%280%2C-4%2C.12%29%2C+locate%280%2C-4%2CV%280%2C-4%29%29%2C%0D%0Acircle%281%2C-1%2C.12%29%2C+locate%281%2C-1%2Cp%281%2C-1%29%29%2C%0D%0Acircle%28-2%2C8%2C.12%29%2C+locate%28-2%2C8%2Cp%28-2%2C8%29%29%2C%0D%0Agraph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+3x%5E2+-4%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "2.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Acircle%284%2C-4%2C.12%29%2C+locate%284%2C-4%2CV%284%2C-4%29%29%2C%0D%0Acircle%281%2C-1%2C.12%29%2C+locate%281%2C-1%2Cp%281%2C-1%29%29%2C%0D%0Acircle%28-2%2C8%2C.12%29%2C+locate%28-2%2C8%2Cp%28-2%2C8%29%29%2C%0D%0Agraph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%281%2F3%29%28+x-+4+%29%5E2+-4%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );