document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=929710>Question 929710</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the quadratic function y = ax ^ 2 + bx + c whose graph passes through the given points.\r<BR>\n" );
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document.write( "(1,-1)(-1,-9)(-2,-10)\r<BR>\n" );
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document.write( "thanks in advance for the help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #564440 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=564440\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> to find the quadratic function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax+%5E+2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"y+=+ax+%5E+2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > whose graph passes through the given points (<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;\" >)(<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=-9 ALIGN=MIDDLE  ALT=\"-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >)(<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=-10 ALIGN=MIDDLE  ALT=\"-10\"   DISABLED_style= \"max-width:90%; height:auto;\" >), we need to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >,and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we can do it using given points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax+%5E+2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"y+=+ax+%5E+2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1 ALIGN=MIDDLE  ALT=\"y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > from  (<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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+a%2A1+%5E+2+%2B+b%2A1+%2B+c ALIGN=MIDDLE  ALT=\"-1+=+a%2A1+%5E+2+%2B+b%2A1+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+a+%2B+b+%2B+c ALIGN=MIDDLE  ALT=\"-1+=+a+%2B+b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........eq.1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax+%5E+2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"y+=+ax+%5E+2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-9 ALIGN=MIDDLE  ALT=\"y=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > from (<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=-9 ALIGN=MIDDLE  ALT=\"-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9=+a%28-1%29%5E2+%2Bb%28-1%29+%2B+c ALIGN=MIDDLE  ALT=\"-9=+a%28-1%29%5E2+%2Bb%28-1%29+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9=+a+-b+%2B+c ALIGN=MIDDLE  ALT=\"-9=+a+-b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........eq.2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax+%5E+2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"y+=+ax+%5E+2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...plug in <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-10 ALIGN=MIDDLE  ALT=\"y=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > from (<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=-10 ALIGN=MIDDLE  ALT=\"-10\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10=+a%28-2%29+%5E+2+%2B+b%28-2%29+%2B+c ALIGN=MIDDLE  ALT=\"-10=+a%28-2%29+%5E+2+%2B+b%28-2%29+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10=+4a+-2b+%2B+c ALIGN=MIDDLE  ALT=\"-10=+4a+-2b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........eq.3\r<BR>\n" );
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document.write( "you have a system:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+a+%2B+b+%2B+c ALIGN=MIDDLE  ALT=\"-1+=+a+%2B+b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........eq.1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9=+a+-b+%2B+c ALIGN=MIDDLE  ALT=\"-9=+a+-b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........eq.2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10=+4a+-2b+%2B+c ALIGN=MIDDLE  ALT=\"-10=+4a+-2b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........eq.3<BR>\n" );
document.write( "_______________________________\r<BR>\n" );
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document.write( "start with\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+a+%2B+b+%2B+c ALIGN=MIDDLE  ALT=\"-1+=+a+%2B+b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........eq.1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9=+a+-b+%2B+c ALIGN=MIDDLE  ALT=\"-9=+a+-b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........eq.2<BR>\n" );
document.write( "___________________________subtract 2 from 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1-%28-9%29+=+a-a+%2B+b-%28-b%29+%2B+c-c ALIGN=MIDDLE  ALT=\"-1-%28-9%29+=+a-a+%2B+b-%28-b%29+%2B+c-c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2B9+=+cross%28a%29-cross%28a%29+%2B+b%2Bb+%2B+cross%28c%29-cross%28c%29 ALIGN=MIDDLE  ALT=\"-1%2B9+=+cross%28a%29-cross%28a%29+%2B+b%2Bb+%2B+cross%28c%29-cross%28c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8+=+2b ALIGN=MIDDLE  ALT=\"8+=+2b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2F2+=+b ALIGN=MIDDLE  ALT=\"8%2F2+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=4%29 ALIGN=MIDDLE  ALT=\"highlight%28b=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "go to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10=+4a+-2b+%2B+c ALIGN=MIDDLE  ALT=\"-10=+4a+-2b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........eq.3, plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10=+4a+-2%2A4+%2B+c ALIGN=MIDDLE  ALT=\"-10=+4a+-2%2A4+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10=+4a+-8+%2B+c ALIGN=MIDDLE  ALT=\"-10=+4a+-8+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10%2B8=+4a%2B+c ALIGN=MIDDLE  ALT=\"-10%2B8=+4a%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2=+4a%2B+c ALIGN=MIDDLE  ALT=\"-2=+4a%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2-4a=c ALIGN=MIDDLE  ALT=\"-2-4a=c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-2-4a ALIGN=MIDDLE  ALT=\"c=-2-4a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "go to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+a+%2B+b+%2B+c ALIGN=MIDDLE  ALT=\"-1+=+a+%2B+b+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........eq.1 substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+a+%2B+4+-2-4a ALIGN=MIDDLE  ALT=\"-1+=+a+%2B+4+-2-4a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+2+-3a ALIGN=MIDDLE  ALT=\"-1+=+2+-3a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a+=2%2B1 ALIGN=MIDDLE  ALT=\"3a+=2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a+=+3 ALIGN=MIDDLE  ALT=\"3a+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a+=1%29 ALIGN=MIDDLE  ALT=\"highlight%28a+=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "go to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-2-4a ALIGN=MIDDLE  ALT=\"c=-2-4a\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-2-4%281%29 ALIGN=MIDDLE  ALT=\"c=-2-4%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-2-4 ALIGN=MIDDLE  ALT=\"c=-2-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=-6%29 ALIGN=MIDDLE  ALT=\"highlight%28c=-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, your equation is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x+%5E+2+%2B+4x+-6 ALIGN=MIDDLE  ALT=\"y+=+x+%5E+2+%2B+4x+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(1,-1)(-1,-9)(-2,-10)<BR>\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+-15%2C+10%2Clocate%281%2C-1%2Cp%281%2C-1%29%29%2Clocate%28-1%2C-9%2Cp%28-1%2C-9%29%29%2Clocate%28-2%2C-10%2Cp%28-2%2C-10%29%29%2Ccircle%28-2%2C-10%2C.12%29%2Ccircle%28-1%2C-9%2C.12%29%2Ccircle%281%2C-1%2C.12%29%2C+graph%28+600%2C+600%2C+-10%2C+10%2C+-15%2C+10%2C+x+%5E+2+%2B+4x+-6%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );