document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=133003>Question 133003</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the vertex, line of symmetry, maximum/minimum value and then graph the function.\r<BR>\n" );
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document.write( "f(x) = 1/2 (x+4)^2 + 6\r<BR>\n" );
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document.write( "I'm thrown off by the fractions, and the whole equation.  Can someone help me figure out how to solve this type of problem. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #97227 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=97227\">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=f%28x%29+=+%281%2F2%29+%28x%2B4%29%5E2+%2B+6 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%281%2F2%29+%28x%2B4%29%5E2+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%281%2F2%29+%28x-%28-4%29%29%5E2+%2B+6 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%281%2F2%29+%28x-%28-4%29%29%5E2+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B4 ALIGN=MIDDLE  ALT=\"x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%28-4%29 ALIGN=MIDDLE  ALT=\"x-%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice how the equation is now in vertex form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1%2F2 ALIGN=MIDDLE  ALT=\"a=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-4 ALIGN=MIDDLE  ALT=\"h=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=6 ALIGN=MIDDLE  ALT=\"k=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Remember, the vertex is (h,k). So in this case the vertex is (-4,6). \r<BR>\n" );
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document.write( "The line of symmetry is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=h ALIGN=MIDDLE  ALT=\"x=h\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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;\" > (ie it's positive), the parabola opens up. So this means that a min will occur at the vertex. So in this case the minimum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=6 ALIGN=MIDDLE  ALT=\"y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we graph the function, we can visually verify our answer.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+%281%2F2%29+%28x%2B4%29%5E2+%2B+6%29%2C%0D%0Ablue%28line%28-4%2C-20%2C-4%2C20%29%29%2C%0D%0Acircle%28-4%2C6%2C0.12%29%2C%0D%0Acircle%28-4%2C6%2C0.14%29%2C%0D%0Acircle%28-4%2C6%2C0.16%29%0D%0A%29%0D%0A+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%281%2F2%29+%28x%2B4%29%5E2+%2B+6 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%281%2F2%29+%28x%2B4%29%5E2+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the axis of symmetry <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (blue line) and the vertex (-4,6) </SPAN>\n" );
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