document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1042694>Question 1042694</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the range of the function:<BR>\n" );
document.write( "y=3x^2-6x+5<BR>\n" );
document.write( "Help, are we supposed to tell the range just by looking at it? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #657725 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52788)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=657725\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Find the range of the function:<BR>\n" );
document.write( "y=3x^2-6x+5<BR>\n" );
document.write( "Help, are we supposed to tell the range just by looking at it?'<BR>\n" );
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document.write( "No, you are supposed to find the minimum of this quadratic function.\r<BR>\n" );
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document.write( "The procedure is as follows.\r<BR>\n" );
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document.write( "First find the value of \"x\" where the minimum occurs.\r<BR>\n" );
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document.write( "For the general quadratic function y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > the minimum is at x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a%29 ALIGN=MIDDLE  ALT=\"-b%2F2a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In your case it is x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F%282%2A3%29 ALIGN=MIDDLE  ALT=\"6%2F%282%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1.\r<BR>\n" );
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document.write( "Now, to find the value of the minimum for \"y\" (the value of the minimum of the function), <BR>\n" );
document.write( "substitute this x=1 into the quadratic function.\r<BR>\n" );
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document.write( "You will get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5Bmin%5D ALIGN=MIDDLE  ALT=\"y%5Bmin%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A1%5E2+-+6%2A1+%2B+5 ALIGN=MIDDLE  ALT=\"3%2A1%5E2+-+6%2A1+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 3 - 6 + 5 = 2.\r<BR>\n" );
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document.write( "Now compare it with the plot below.\r<BR>\n" );
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document.write( "  <TR>\r\n" );
document.write( "  <TD > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+330%2C+330%2C+-5.5%2C+5.5%2C+-2.5%2C+8.5%2C%0D%0A++++++++++3x%5E2-6x%2B5%0D%0A%29 ALIGN=MIDDLE  ALT=\"graph%28+330%2C+330%2C+-5.5%2C+5.5%2C+-2.5%2C+8.5%2C%0D%0A++++++++++3x%5E2-6x%2B5%0D%0A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Plot y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-6x%2B5 ALIGN=MIDDLE  ALT=\"3x%5E2-6x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "  </TD>\r\n" );
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document.write( "Now the answer to the question about the range is: [<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=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "It is a semi-infinite segment from 2 to infinity.  The value of 2 is included into the segment.\r<BR>\n" );
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