document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=98523>Question 98523</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value.\r<BR>\n" );
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document.write( "f(x) = -x2 - 3x - 9<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #71681 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=71681\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The function has a maximum since the first term <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;\" > has a negative coefficient. \r<BR>\n" );
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document.write( "To find the max, you need to find the axis of symmetry first\r<BR>\n" );
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document.write( "To find the axis of symmetry, use this formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"x=-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%5E2-3x-9 ALIGN=MIDDLE  ALT=\"y=-x%5E2-3x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that a=-1 and b=-3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28--3%29%2F%282%2A-1%29 ALIGN=MIDDLE  ALT=\"x=%28--3%29%2F%282%2A-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in b=-3 and a=-1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2F%282%2A-1%29 ALIGN=MIDDLE  ALT=\"x=3%2F%282%2A-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -3 to get 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%283%29%2F-2 ALIGN=MIDDLE  ALT=\"x=%283%29%2F-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and -1 to get -2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3%2F2 ALIGN=MIDDLE  ALT=\"x=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "So the axis of symmetry is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3%2F2 ALIGN=MIDDLE  ALT=\"x=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the x-coordinate of the vertex is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3%2F2 ALIGN=MIDDLE  ALT=\"x=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.5 ALIGN=MIDDLE  ALT=\"x=-1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > in decimal form). Lets plug this into the equation to find the y-coordinate of the vertex.\r<BR>\n" );
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document.write( "Lets evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1.5%29 ALIGN=MIDDLE  ALT=\"f%28-1.5%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=f%28x%29=-x%5E2-3x-9 ALIGN=MIDDLE  ALT=\"f%28x%29=-x%5E2-3x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given polynomial\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1.5%29=-%28-1.5%29%5E2-3%28-1.5%29-9 ALIGN=MIDDLE  ALT=\"f%28-1.5%29=-%28-1.5%29%5E2-3%28-1.5%29-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.5 ALIGN=MIDDLE  ALT=\"x=-1.5\"   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=f%28-1.5%29=-%282.25%29-3%28-1.5%29-9 ALIGN=MIDDLE  ALT=\"f%28-1.5%29=-%282.25%29-3%28-1.5%29-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Raise -1.5 to the second power to get 2.25\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1.5%29=-6.75 ALIGN=MIDDLE  ALT=\"f%28-1.5%29=-6.75\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now combine like terms\r<BR>\n" );
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document.write( "So the vertex is (-1.5,-6.75)\r<BR>\n" );
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document.write( "So that means the functions highest value is -6.75 which means the functions maximum is -6.75\r<BR>\n" );
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document.write( "Notice if you graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%5E2-3x-9 ALIGN=MIDDLE  ALT=\"y=-x%5E2-3x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > you get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28900%2C900%2C-11.5%2C8.5%2C-16.75%2C3.25%2C%0D%0Agraph%28900%2C900%2C-11.5%2C8.5%2C-16.75%2C3.25%2C+-x%5E2-3x-9%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice the highest y value is -6.75. So our answer is graphically verified. </SPAN>\n" );
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