document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=118377>Question 118377</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sketch the graph, using the function and three points:  two roots and the turning point. <BR>\n" );
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document.write( "                                a.  f(x) = x^2 + 6x - 16 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #86497 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=86497\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2%2B6x-16 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2B6x-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >, set the entire function equal to zero\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x-16=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x-16=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=%28x%2B8%29%28x-2%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B8%29%28x-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side (note: if you need help with factoring, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)\r<BR>\n" );
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document.write( "Now set each factor equal to zero:\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-2=0 ALIGN=MIDDLE  ALT=\"x-2=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=-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=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >    Now solve for x in each case\r<BR>\n" );
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document.write( "So the roots are\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=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now let's find the vertex (ie the turning point)\r<BR>\n" );
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document.write( "To find the x coordinate of the vertex, 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( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2%2B6x-16 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2B6x-16\"   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=6 ALIGN=MIDDLE  ALT=\"b=6\"   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=-6%2F2%281%29 ALIGN=MIDDLE  ALT=\"x=-6%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=6 ALIGN=MIDDLE  ALT=\"b=6\"   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=-6%2F2 ALIGN=MIDDLE  ALT=\"x=-6%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=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\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 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "To find the y coordinate, simply plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > into  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2%2B6x-16 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2B6x-16\"   DISABLED_style= \"max-width:90%; height:auto;\" > to evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29 ALIGN=MIDDLE  ALT=\"f%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29 ALIGN=MIDDLE  ALT=\"f%28-3%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%2B6x-16 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2%2B6x-16\"   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%28-3%29=%28-3%29%5E2%2B6%28-3%29-16 ALIGN=MIDDLE  ALT=\"f%28-3%29=%28-3%29%5E2%2B6%28-3%29-16\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, replace each x with -3.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29=%289%29%2B6%28-3%29-16 ALIGN=MIDDLE  ALT=\"f%28-3%29=%289%29%2B6%28-3%29-16\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%29%5E2 ALIGN=MIDDLE  ALT=\"%28-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 9.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29=%289%29%2B-18-16 ALIGN=MIDDLE  ALT=\"f%28-3%29=%289%29%2B-18-16\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 6 and -3 to get  -18\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29=-25 ALIGN=MIDDLE  ALT=\"f%28-3%29=-25\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now combine like terms\r<BR>\n" );
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document.write( "So the vertex is: (-3,-25)\r<BR>\n" );
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document.write( "-----------------------------\r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "So our roots are \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=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and our vertex is at the point \r<BR>\n" );
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document.write( "(-3,-25)\r<BR>\n" );
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document.write( "Now let's graph these three points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-15%2C15%2C-30%2C5%2C+%0D%0Agraph%28500%2C500%2C-15%2C15%2C-30%2C5%2C0%29%2C%0D%0Agrid%281%29%2C%0D%0Acircle%28-8%2C0%2C0.12%29%2C%0D%0Acircle%28-8%2C0%2C0.15%29%2C%0D%0Acircle%28-8%2C0%2C0.18%29%2C%0D%0A%0D%0Acircle%282%2C0%2C0.12%29%2C%0D%0Acircle%282%2C0%2C0.15%29%2C%0D%0Acircle%282%2C0%2C0.18%29%2C%0D%0A%0D%0A%0D%0Acircle%28-3%2C-25%2C0.12%29%2C%0D%0Acircle%28-3%2C-25%2C0.15%29%2C%0D%0Acircle%28-3%2C-25%2C0.18%29%0D%0A%0D%0A%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now connect the points with a parabola (note: the more points you plot the more accurate the graph is)\r<BR>\n" );
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