document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=143034>Question 143034</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3.	Given:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2%2B4x%2B5 ALIGN=MIDDLE  ALT=\"y=x%5E2%2B4x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "a.	Find the vertex.<BR>\n" );
document.write( "b.	Determine whether there is a minimum or maximum value and find that value.<BR>\n" );
document.write( "c.	Find the range.<BR>\n" );
document.write( "d.	Find the intervals on which the function is decreasing and the intervals on which the function is increasing.\r<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 #104075 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=104075\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)\r<BR>\n" );
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document.write( "To find the vertex, we first need to find the axis of symmetry (ie the x-coordinate of the vertex)\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%2B4x%2B5 ALIGN=MIDDLE  ALT=\"y=x%5E2%2B4x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that a=1 and b=4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-4%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x=%28-4%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in b=4 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=%28-4%29%2F2 ALIGN=MIDDLE  ALT=\"x=%28-4%29%2F2\"   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=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   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=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   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=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. 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%28-2%29 ALIGN=MIDDLE  ALT=\"f%28-2%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%2B4x%2B5 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2B4x%2B5\"   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%28-2%29=%28-2%29%5E2%2B4%28-2%29%2B5 ALIGN=MIDDLE  ALT=\"f%28-2%29=%28-2%29%5E2%2B4%28-2%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-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=f%28-2%29=%284%29%2B4%28-2%29%2B5 ALIGN=MIDDLE  ALT=\"f%28-2%29=%284%29%2B4%28-2%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Raise -2 to the second power to get 4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-2%29=%284%29%2B-8%2B5 ALIGN=MIDDLE  ALT=\"f%28-2%29=%284%29%2B-8%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 4 by -2 to get -8\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-2%29=1 ALIGN=MIDDLE  ALT=\"f%28-2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now combine like terms\r<BR>\n" );
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document.write( "So the vertex is (-2,1)\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Since <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;\" >, this tells us that the parabola is opening upward and that there is a minimum. At the minimum, there is the vertex. So this means that the minimum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which occurs at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "c)\r<BR>\n" );
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document.write( "Because the min is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the range is from  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to infinity. So the range is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=1 ALIGN=MIDDLE  ALT=\"y%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which in interval notation is: <font size=\"8\">[</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1,\infty\"><font size=\"8\">)</font>\r<BR>\n" );
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document.write( "d)\r<BR>\n" );
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document.write( "At the axis of symmetry, the graph changes from increasing to decreasing (or vice versa). Also, since the graph opens upward, this means that the interval that the function decreases is: <font size=\"8\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \bf{-\infty,-2}\"><font size=\"8\">)</font>\r<BR>\n" );
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document.write( "Also, the  interval that the function increases is: <font size=\"8\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -2,\infty\"><font size=\"8\">)</font>\r<BR>\n" );
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document.write( "Here is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2%2B4x%2B5 ALIGN=MIDDLE  ALT=\"y=x%5E2%2B4x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to visually verify our answer\r<BR>\n" );
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