document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=830938>Question 830938</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the rule of a quadratic function whose graph has the given vertex and passes through the given point.\r<BR>\n" );
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document.write( "Vertex (-1,-6) point (5,18)\r<BR>\n" );
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document.write( "Please help very confused. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #501019 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=501019\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The function is a parabola. If you plot <BR>\n" );
document.write( "the given points, the vertex (-1,-6) is<BR>\n" );
document.write( "in the 3rd quadrant, and (5,18) is in the<BR>\n" );
document.write( "1st quadrant.<BR>\n" );
document.write( "----------------------<BR>\n" );
document.write( "It is important to see that the parabola<BR>\n" );
document.write( "looks like a cup which is holding water<BR>\n" );
document.write( "because the vertex is below the other<BR>\n" );
document.write( "point<BR>\n" );
document.write( "-----------------------<BR>\n" );
document.write( "That means that the general form of the<BR>\n" );
document.write( "equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "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;\" > has to be positive for a parabola<BR>\n" );
document.write( "that holds water.<BR>\n" );
document.write( "------------------------<BR>\n" );
document.write( "The formula for the vertex point is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5Bv%5D+=+-b%2F%282a%29+ ALIGN=MIDDLE  ALT=\"+x%5Bv%5D+=+-b%2F%282a%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You are given <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5Bv%5D+=+-1+ ALIGN=MIDDLE  ALT=\"+x%5Bv%5D+=+-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-1+=+-b%2F2a+ ALIGN=MIDDLE  ALT=\"+-1+=+-b%2F2a+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+=+2a+ ALIGN=MIDDLE  ALT=\"+b+=+2a+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------------------------<BR>\n" );
document.write( "Also, I can plug the vertex point ( -1,-6)into the equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-6+=+a%2A%28-1%29%5E2+%2B+b%2A%28-1%29+%2B+c+ ALIGN=MIDDLE  ALT=\"+-6+=+a%2A%28-1%29%5E2+%2B+b%2A%28-1%29+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-6+=+a+-+b+%2B+c+ ALIGN=MIDDLE  ALT=\"+-6+=+a+-+b+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and, substituting for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-6+=+a+-+2a+%2B+c+ ALIGN=MIDDLE  ALT=\"+-6+=+a+-+2a+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+=+a+-+6+ ALIGN=MIDDLE  ALT=\"+c+=+a+-+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-------------------<BR>\n" );
document.write( "So, now I have:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+a%2Ax%5E2+%2B+2a%2Ax+%2B+a+-+6+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+a%2Ax%5E2+%2B+2a%2Ax+%2B+a+-+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "Now you can use the other given point, (5,18)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+18+=+a%2A5%5E2+%2B+2a%2A5+%2B+a+-+6+ ALIGN=MIDDLE  ALT=\"+18+=+a%2A5%5E2+%2B+2a%2A5+%2B+a+-+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+18+=+25a+%2B+10a+%2B+a+-+6+ ALIGN=MIDDLE  ALT=\"+18+=+25a+%2B+10a+%2B+a+-+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+36a+=+24+ ALIGN=MIDDLE  ALT=\"+36a+=+24+\"   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%2F3+ ALIGN=MIDDLE  ALT=\"+a+=+2%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------<BR>\n" );
document.write( "Then it follows that <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+=+a+-+6+ ALIGN=MIDDLE  ALT=\"+c+=+a+-+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+=+2%2F3+-+18%2F3+ ALIGN=MIDDLE  ALT=\"+c+=+2%2F3+-+18%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+=+-16%2F3+ ALIGN=MIDDLE  ALT=\"+c+=+-16%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+=+2a+ ALIGN=MIDDLE  ALT=\"+b+=+2a+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+=+2%2A%282%2F3%29+ ALIGN=MIDDLE  ALT=\"+b+=+2%2A%282%2F3%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+=+4%2F3+ ALIGN=MIDDLE  ALT=\"+b+=+4%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "So, the equation is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+%282%2F3%29%2Ax%5E2+%2B+%284%2F3%29%2Ax+-+16%2F3+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+%282%2F3%29%2Ax%5E2+%2B+%284%2F3%29%2Ax+-+16%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "check:<BR>\n" );
document.write( "does it go through (-1,-6) ?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-6+=+%282%2F3%29%2A%28-1%29%5E2+%2B+%284%2F3%29%2A%28-1%29+-+16%2F3+ ALIGN=MIDDLE  ALT=\"+-6+=+%282%2F3%29%2A%28-1%29%5E2+%2B+%284%2F3%29%2A%28-1%29+-+16%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-6+=+2%2F3+-+4%2F3+-16%2F3+ ALIGN=MIDDLE  ALT=\"+-6+=+2%2F3+-+4%2F3+-16%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-18+=+2+-+4+-16+ ALIGN=MIDDLE  ALT=\"+-18+=+2+-+4+-16+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-18+=+-18+ ALIGN=MIDDLE  ALT=\"+-18+=+-18+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "OK<BR>\n" );
document.write( "-----------------------<BR>\n" );
document.write( "Does it go through (5,18) ?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+18+=+%282%2F3%29%2A5%5E2+%2B+%284%2F3%29%2A5+-+16%2F3+ ALIGN=MIDDLE  ALT=\"+18+=+%282%2F3%29%2A5%5E2+%2B+%284%2F3%29%2A5+-+16%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+18+=+50%2F3+%2B+20%2F3+-+16%2F3+ ALIGN=MIDDLE  ALT=\"+18+=+50%2F3+%2B+20%2F3+-+16%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+54+=+50+%2B+20+-+16+ ALIGN=MIDDLE  ALT=\"+54+=+50+%2B+20+-+16+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+54+=+54+ ALIGN=MIDDLE  ALT=\"+54+=+54+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "OK<BR>\n" );
document.write( "-------<BR>\n" );
document.write( "Hope this helps. Here's the plot:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C+%282%2F3%29%2Ax%5E2+%2B+%284%2F3%29%2Ax+-+16%2F3+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "   <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );