document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=77060>Question 77060</A>: <I><SPAN STYLE=\"word-break: break-all;\"> can you help me understand how to find the vertex, the line of symmetry, and range with this problem\r<BR>\n" );
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document.write( "for the graph of the function f(x)=-x^2+6x-8\r<BR>\n" );
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document.write( "a. find the vertex\r<BR>\n" );
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document.write( "b. the line of symmetry\r<BR>\n" );
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document.write( "c.  find the range </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #55263 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=55263\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)\r<BR>\n" );
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document.write( "First factor out the negative 1\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=completing_the_square.solver>Completing the Square to Get a Quadratic into Vertex Form</A></TD></TR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1+x%5E2-6+x%2B8 ALIGN=MIDDLE  ALT=\"y=1+x%5E2-6+x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=1+x%5E2-6+x ALIGN=MIDDLE  ALT=\"y-8=1+x%5E2-6+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=1%28x%5E2-6x%29 ALIGN=MIDDLE  ALT=\"y-8=1%28x%5E2-6x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the leading coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Take half of the x coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28-6%29=-3 ALIGN=MIDDLE  ALT=\"%281%2F2%29%28-6%29=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
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document.write( "  Now square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%29%5E2=%28-3%29%28-3%29=9 ALIGN=MIDDLE  ALT=\"%28-3%29%5E2=%28-3%29%28-3%29=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=1%28x%5E2-6x%2B9-9%29 ALIGN=MIDDLE  ALT=\"y-8=1%28x%5E2-6x%2B9-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not change the equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=1%28%28x-3%29%5E2-9%29 ALIGN=MIDDLE  ALT=\"y-8=1%28%28x-3%29%5E2-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6x%2B9 ALIGN=MIDDLE  ALT=\"x%5E2-6x%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=1%28x-3%29%5E2-1%289%29 ALIGN=MIDDLE  ALT=\"y-8=1%28x-3%29%5E2-1%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-8=1%28x-3%29%5E2-9 ALIGN=MIDDLE  ALT=\"y-8=1%28x-3%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%28x-3%29%5E2-9%2B8 ALIGN=MIDDLE  ALT=\"y=1%28x-3%29%5E2-9%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides to isolate y<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%28x-3%29%5E2-1 ALIGN=MIDDLE  ALT=\"y=1%28x-3%29%5E2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms<BR>\n" );
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document.write( "  Now the quadratic is 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 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=3 ALIGN=MIDDLE  ALT=\"h=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-1 ALIGN=MIDDLE  ALT=\"k=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember (h,k) is the vertex and \"a\" is the stretch/compression factor.<BR>\n" );
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document.write( "  Check:<BR>\n" );
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document.write( "  Notice if we graph the original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1x%5E2-6x%2B8 ALIGN=MIDDLE  ALT=\"y=1x%5E2-6x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1x%5E2-6x%2B8%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1x%5E2-6x%2B8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1x%5E2-6x%2B8 ALIGN=MIDDLE  ALT=\"y=1x%5E2-6x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
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document.write( "  Notice if we graph the final equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%28x-3%29%5E2-1 ALIGN=MIDDLE  ALT=\"y=1%28x-3%29%5E2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1%28x-3%29%5E2-1%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1%28x-3%29%5E2-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%28x-3%29%5E2-1 ALIGN=MIDDLE  ALT=\"y=1%28x-3%29%5E2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is also (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
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document.write( "  So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.<BR>\n" );
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document.write( "So if we multiply everything by -1 we get\r<BR>\n" );
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document.write( "So the vertex is (3,1)\r<BR>\n" );
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document.write( "b)<BR>\n" );
document.write( "The line of symmetry will go through the vertex so the equation is<BR>\n" );
document.write( "x=3\r<BR>\n" );
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document.write( "c)<BR>\n" );
document.write( "The range is any value y can be, so by our graph we can see the range is<BR>\n" );
document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large -\infty\">, 1]\r<BR>\n" );
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document.write( "Here's our graph\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-6%2C+5%2C+-10%2C+10%2C+-x%5E2%2B6x-8%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-6%2C+5%2C+-10%2C+10%2C+-x%5E2%2B6x-8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%28x-3%29%5E2%2B1 ALIGN=MIDDLE  ALT=\"y=-%28x-3%29%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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