document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=118382>Question 118382</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sketch the graphs, using the given roots to determine the function and the turning point.\r<BR>\n" );
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document.write( "                    b.  x = -8,  x = -6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #86501 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=86501\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-8 ALIGN=MIDDLE  ALT=\"x=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6 ALIGN=MIDDLE  ALT=\"x=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Start with the given roots\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B8=0 ALIGN=MIDDLE  ALT=\"x%2B8=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B6=0 ALIGN=MIDDLE  ALT=\"x%2B6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Add 8 to both sides for the first equation. Add 6 to both sides for the second equation. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B8%29%28x%2B6%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B8%29%28x%2B6%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now use the zero product property in reverse. Remember the zero product property says that if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2AB=0 ALIGN=MIDDLE  ALT=\"A%2AB=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > then <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;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=0 ALIGN=MIDDLE  ALT=\"B=0\"   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%5E2%2B14x%2B48=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B14x%2B48=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now foil. note: Let me know if you need help with foiling.\r<BR>\n" );
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document.write( "-----------------------\r<BR>\n" );
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document.write( "Now to find the turning point, we need to find the x-coordinate of the turning point. To do that, simply use this formula: <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( "So from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2%2B14x%2B48 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2B14x%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=14 ALIGN=MIDDLE  ALT=\"b=14\"   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=-14%2F2%281%29 ALIGN=MIDDLE  ALT=\"x=-14%2F2%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Plug <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the formula. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-14%2F2 ALIGN=MIDDLE  ALT=\"x=-14%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-7 ALIGN=MIDDLE  ALT=\"x=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "So the x-coordinate of the turning point is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-7 ALIGN=MIDDLE  ALT=\"x=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now we have to evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-7%29 ALIGN=MIDDLE  ALT=\"f%28-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to determine the y-coordinate of the turning point.\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%2B14x%2B48 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2B14x%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-7%29=%28-7%29%5E2%2B14%28-7%29%2B48 ALIGN=MIDDLE  ALT=\"f%28-7%29=%28-7%29%5E2%2B14%28-7%29%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-7 ALIGN=MIDDLE  ALT=\"x=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, replace each x with -7.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-7%29=%2849%29%2B14%28-7%29%2B48 ALIGN=MIDDLE  ALT=\"f%28-7%29=%2849%29%2B14%28-7%29%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-7%29%5E2 ALIGN=MIDDLE  ALT=\"%28-7%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 49.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-7%29=%2849%29%2B-98%2B48 ALIGN=MIDDLE  ALT=\"f%28-7%29=%2849%29%2B-98%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 14 and -7 to get  -98\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-7%29=-1 ALIGN=MIDDLE  ALT=\"f%28-7%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now combine like terms\r<BR>\n" );
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document.write( "So the turning point is at (-7,-1)\r<BR>\n" );
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document.write( "Notice if we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2%2B14x%2B48 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2B14x%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that the roots are<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-8 ALIGN=MIDDLE  ALT=\"x=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6 ALIGN=MIDDLE  ALT=\"x=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We can also see that the turning point is (-7,-1). So this visually verifies our answer.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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