document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=341749>Question 341749</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%282%2F3%29x%5E2-3x%2B6=y+ ALIGN=MIDDLE  ALT=\"+%282%2F3%29x%5E2-3x%2B6=y+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  I need to find the vertex and axis of symmetry to graph the quadratic form, I understand how to graph quadratic forms but the fraction in this specific problem has me all jumbled up. I tried solving it and got x= 9/4 for the axis of symmetry and (9/4,87/16)for the vertex but im almost 100% certain that is wrong. I would really appreciate some help (PLEASE HELP ME). Thanks in advance.<BR>\n" );
document.write( "                   -Kay </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #244668 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=244668\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the vertex and axis of symmetry, we need to complete the square for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%282%2F3%29x%5E2-3x%2B6 ALIGN=MIDDLE  ALT=\"+%282%2F3%29x%5E2-3x%2B6\"   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=%282%2F3%29x%5E2-3x%2B6 ALIGN=MIDDLE  ALT=\"%282%2F3%29x%5E2-3x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29%28x%5E2-%289%2F2%29x%2B9%29 ALIGN=MIDDLE  ALT=\"%282%2F3%29%28x%5E2-%289%2F2%29x%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This step is very important: the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient <font size=4><b>must</b></font> be equal to 1.\r<BR>\n" );
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document.write( "Take half of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9%2F2 ALIGN=MIDDLE  ALT=\"-9%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9%2F4 ALIGN=MIDDLE  ALT=\"-9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28-9%2F2%29=-9%2F4 ALIGN=MIDDLE  ALT=\"%281%2F2%29%28-9%2F2%29=-9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9%2F4 ALIGN=MIDDLE  ALT=\"-9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81%2F16 ALIGN=MIDDLE  ALT=\"81%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-9%2F4%29%5E2=%28-9%2F4%29%28-9%2F4%29=81%2F16 ALIGN=MIDDLE  ALT=\"%28-9%2F4%29%5E2=%28-9%2F4%29%28-9%2F4%29=81%2F16\"   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=%282%2F3%29%28x%5E2-%289%2F2%29x%2Bhighlight%2881%2F16-81%2F16%29%2B9%29 ALIGN=MIDDLE  ALT=\"%282%2F3%29%28x%5E2-%289%2F2%29x%2Bhighlight%2881%2F16-81%2F16%29%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <font size=4><b>and</b></font> subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81%2F16 ALIGN=MIDDLE  ALT=\"81%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" > inside the parenthesis. Make sure to place this after the \"x\" term. Notice how <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81%2F16-81%2F16=0 ALIGN=MIDDLE  ALT=\"81%2F16-81%2F16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the expression is not changed.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29%28%28x%5E2-%289%2F2%29x%2B81%2F16%29-81%2F16%2B9%29 ALIGN=MIDDLE  ALT=\"%282%2F3%29%28%28x%5E2-%289%2F2%29x%2B81%2F16%29-81%2F16%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group the first three terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29%28%28x-9%2F4%29%5E2-81%2F16%2B9%29 ALIGN=MIDDLE  ALT=\"%282%2F3%29%28%28x-9%2F4%29%5E2-81%2F16%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-%289%2F2%29x%2B81%2F16 ALIGN=MIDDLE  ALT=\"x%5E2-%289%2F2%29x%2B81%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-9%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-9%2F4%29%5E2\"   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=%282%2F3%29%28%28x-9%2F4%29%5E2%2B63%2F16%29 ALIGN=MIDDLE  ALT=\"%282%2F3%29%28%28x-9%2F4%29%5E2%2B63%2F16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29%28x-9%2F4%29%5E2%2B%282%2F3%29%2863%2F16%29 ALIGN=MIDDLE  ALT=\"%282%2F3%29%28x-9%2F4%29%5E2%2B%282%2F3%29%2863%2F16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8 ALIGN=MIDDLE  ALT=\"%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
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document.write( "So after completing the square, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29x%5E2-3x%2B6 ALIGN=MIDDLE  ALT=\"%282%2F3%29x%5E2-3x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > transforms to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8 ALIGN=MIDDLE  ALT=\"%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29x%5E2-3x%2B6=%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8 ALIGN=MIDDLE  ALT=\"%282%2F3%29x%5E2-3x%2B6=%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x%5E2-3x%2B6 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x%5E2-3x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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=2%2F3 ALIGN=MIDDLE  ALT=\"a=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=9%2F4 ALIGN=MIDDLE  ALT=\"h=9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=21%2F8 ALIGN=MIDDLE  ALT=\"k=21%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Remember, the vertex of <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;\" > is (h,k).\r<BR>\n" );
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document.write( "So the vertex of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%28x-9%2F4%29%5E2%2B21%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(\frac{9}{4},\frac{21}{8}\right)\"> since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=9%2F4 ALIGN=MIDDLE  ALT=\"h=9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=21%2F8 ALIGN=MIDDLE  ALT=\"k=21%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also, remember that the axis of symmetry is simply <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 axis of symmetry is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9%2F4 ALIGN=MIDDLE  ALT=\"x=9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If you need more help, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
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document.write( "Also, feel free to check out my <a  rel=nofollow HREF=\"http://www.freewebs.com/jimthompson5910/home.html\">tutoring website</a>\r<BR>\n" );
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document.write( "Jim </SPAN>\n" );
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