document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=684539>Question 684539</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the vertex form for:y=-1/3x^2+8/3x-25/3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #424136 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=424136\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the vertex form:\r<BR>\n" );
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document.write( "<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;\" >\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=%28-1%2F3%29+x%5E2%2B%288%2F3%29+x-25%2F3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29+x%5E2%2B%288%2F3%29+x-25%2F3\"   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%2B25%2F3=%28-1%2F3%29+x%5E2%2B%288%2F3%29+x ALIGN=MIDDLE  ALT=\"y%2B25%2F3=%28-1%2F3%29+x%5E2%2B%288%2F3%29+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%2F3 ALIGN=MIDDLE  ALT=\"25%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B25%2F3=%28-1%2F3%29%28x%5E2-8x%29 ALIGN=MIDDLE  ALT=\"y%2B25%2F3=%28-1%2F3%29%28x%5E2-8x%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=%28-1%2F3%29 ALIGN=MIDDLE  ALT=\"%28-1%2F3%29\"   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=-8 ALIGN=MIDDLE  ALT=\"-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28-8%29=-4 ALIGN=MIDDLE  ALT=\"%281%2F2%29%28-8%29=-4\"   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=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-4%29%5E2=%28-4%29%28-4%29=16 ALIGN=MIDDLE  ALT=\"%28-4%29%5E2=%28-4%29%28-4%29=16\"   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%2B25%2F3=%28-1%2F3%29%28x%5E2-8x%2B16-16%29 ALIGN=MIDDLE  ALT=\"y%2B25%2F3=%28-1%2F3%29%28x%5E2-8x%2B16-16%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=16 ALIGN=MIDDLE  ALT=\"16\"   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%2B25%2F3=%28-1%2F3%29%28%28x-4%29%5E2-16%29 ALIGN=MIDDLE  ALT=\"y%2B25%2F3=%28-1%2F3%29%28%28x-4%29%5E2-16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-4%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%2B25%2F3=%28-1%2F3%29%28x-4%29%5E2-%28-1%2F3%29%2816%29 ALIGN=MIDDLE  ALT=\"y%2B25%2F3=%28-1%2F3%29%28x-4%29%5E2-%28-1%2F3%29%2816%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%2B25%2F3=%28-1%2F3%29%28x-4%29%5E2%2B16%2F3 ALIGN=MIDDLE  ALT=\"y%2B25%2F3=%28-1%2F3%29%28x-4%29%5E2%2B16%2F3\"   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=%28-1%2F3%29%28x-4%29%5E2%2B16%2F3-25%2F3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29%28x-4%29%5E2%2B16%2F3-25%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B25%2F3 ALIGN=MIDDLE  ALT=\"%2B25%2F3\"   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=%28-1%2F3%29%28x-4%29%5E2-3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29%28x-4%29%5E2-3\"   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%2F3 ALIGN=MIDDLE  ALT=\"a=-1%2F3\"   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=-3 ALIGN=MIDDLE  ALT=\"k=-3\"   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=%28-1%2F3%29x%5E2%2B%288%2F3%29x-25%2F3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29x%5E2%2B%288%2F3%29x-25%2F3\"   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%2C%28-1%2F3%29x%5E2%2B%288%2F3%29x-25%2F3%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-1%2F3%29x%5E2%2B%288%2F3%29x-25%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F3%29x%5E2%2B%288%2F3%29x-25%2F3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29x%5E2%2B%288%2F3%29x-25%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how the vertex is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   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=%28-1%2F3%29%28x-4%29%5E2-3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29%28x-4%29%5E2-3\"   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%2C%28-1%2F3%29%28x-4%29%5E2-3%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-1%2F3%29%28x-4%29%5E2-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F3%29%28x-4%29%5E2-3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29%28x-4%29%5E2-3\"   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=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   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, (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >)=(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >) </SPAN>\n" );
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