document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=116123>Question 116123</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0131492624/algebrahomeworkh>Introductry Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> I have tried to do this equation but I'm having trouble with it. Can you please help me? Solve and graph the equation.\r<BR>\n" );
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document.write( "y= -3x^2 + 6x - 2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84424 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=84424\">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=y=+-3x%5E2+%2B+6x+-+2 ALIGN=MIDDLE  ALT=\"y=+-3x%5E2+%2B+6x+-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  By 'solve' I presume you mean to find the roots of the equation where y = 0.\r<BR>\n" );
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document.write( "Since this cannot be factored, you must either complete the square or use the quadratic formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >(which is really the same thing).\r<BR>\n" );
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document.write( "In this equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2+%2B+6x+-+2=0 ALIGN=MIDDLE  ALT=\"-3x%5E2+%2B+6x+-+2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=6 ALIGN=MIDDLE  ALT=\"b=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-2 ALIGN=MIDDLE  ALT=\"c=-2\"   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=x+=+%28-6+%2B-+sqrt%28+6%5E2-4%2A%28-3%29%2A%28-2%29+%29%29%2F%282%2A%28-3%29%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+6%5E2-4%2A%28-3%29%2A%28-2%29+%29%29%2F%282%2A%28-3%29%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=x=+%28-6+%2B-+sqrt%2812%29%29%2F%28-6%29 ALIGN=MIDDLE  ALT=\"x=+%28-6+%2B-+sqrt%2812%29%29%2F%28-6%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=x=%28-6%2B2sqrt%283%29%29%2F-6 ALIGN=MIDDLE  ALT=\"x=%28-6%2B2sqrt%283%29%29%2F-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-6-2sqrt%283%29%29%2F-6 ALIGN=MIDDLE  ALT=\"x=%28-6-2sqrt%283%29%29%2F-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%283-sqrt%283%29%29%2F3 ALIGN=MIDDLE  ALT=\"x=%283-sqrt%283%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%283%2Bsqrt%283%29%29%2F3 ALIGN=MIDDLE  ALT=\"x=%283%2Bsqrt%283%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To graph the equation, first find the x-coordinate of the vertex given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a=-6%2F-6=1 ALIGN=MIDDLE  ALT=\"-b%2F2a=-6%2F-6=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then evaluate the function f(x)=y at this x value:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=-3%2B6-2=1 ALIGN=MIDDLE  ALT=\"f%281%29=-3%2B6-2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Therefore the vertex is at (1, 1).  Since the coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term is negative, you know the graph is convex down.  With the vertex at (1,1), we have an axis of symmetry at the vertical line <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;\" >.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=0%2B0-2=-2 ALIGN=MIDDLE  ALT=\"f%280%29=0%2B0-2=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so we have the y-axis intercept at (0,-2).  The roots of the equation that we calculated above come out to a little less than .5 for one and a little more than 1.5 for the other.  That's close enough for graphing purposes.  If you want to pick a few more values for x to give you some additional points to smooth the curve, pick them in the neighborhood of .5 and 1.5.  Plot all of these points and then draw a smooth curve.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C-3x%5E2%2B6x-2%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C-3x%5E2%2B6x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Super-Double-Plus Extra Credit:  Why did I say that using the quadratic formula and completing the square are really the same thing?\r<BR>\n" );
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document.write( "Hope that helps,<BR>\n" );
document.write( "John </SPAN>\n" );
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