document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=131500>Question 131500</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the vertex, the line of symmetry, the maximum or minimum value of the quadratic function, and graph the function.\r<BR>\n" );
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document.write( "f(x) = -x^2 + 4x + 6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #95982 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=95982\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The x-coordinate of the vertex of a parabola in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a ALIGN=MIDDLE  ALT=\"-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The y-coordinate is then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-b%2F2a%29 ALIGN=MIDDLE  ALT=\"f%28-b%2F2a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The line of symmetry passes through the vertex, so the equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The maximum or minimum is the value of the function at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a ALIGN=MIDDLE  ALT=\"-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Whether it is a maximum or minimum depends on whether the parabola opens up or down.  If it is concave up (makes a valley rather than a hill), the point is a minimum, otherwise it is a maximum.  You can tell which way the parabola opens by the sign on the lead coefficient.  if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C0 ALIGN=MIDDLE  ALT=\"a%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, it is concave down, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E0 ALIGN=MIDDLE  ALT=\"a%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, it is concave up, and, of course, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=0 ALIGN=MIDDLE  ALT=\"a=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > you don't have a parabola at all.\r<BR>\n" );
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document.write( "Let's look at your specific problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+-2x%5E2%2B4x%2B6+ ALIGN=MIDDLE  ALT=\"f%28x%29+=+-2x%5E2%2B4x%2B6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First thing to note is that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C0 ALIGN=MIDDLE  ALT=\"a%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so this is a concave down parabola and the vertex is a maximum.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a=-4%2F%282%28-2%29%29=1 ALIGN=MIDDLE  ALT=\"-b%2F2a=-4%2F%282%28-2%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the x-coordinate of the vertex is <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 the equation of the line of symmetry is <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;\" >.\r<BR>\n" );
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document.write( "The value of the function at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a ALIGN=MIDDLE  ALT=\"-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, denoted <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-b%2F2a%29 ALIGN=MIDDLE  ALT=\"f%28-b%2F2a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for your problem is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=-2%281%29%5E2%2B4%281%29%2B6=%28-2%29%2B4%2B6=8 ALIGN=MIDDLE  ALT=\"f%281%29=-2%281%29%5E2%2B4%281%29%2B6=%28-2%29%2B4%2B6=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the y-coordinate of the vertex and the maximum value of f is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   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=drawing%28600%2C600%2C-10%2C10%2C-10%2C10%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28600%2C600%2C-10%2C10%2C-10%2C10%2C-2x%5E2%2B4x%2B6%29%2C%0D%0Acircle%281%2C8%2C.1%29%2C%0D%0Alocate%281.2%2C8.5%2CV%281%2C8%29%29%2C%0D%0Agreen%28line%281%2C-10%2C1%2C10%29%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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