document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=312310>Question 312310</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I can't seem to get these right, so thanks for helping if you can. <BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "1.) Change the following equation into vertex form: \r<BR>\n" );
document.write( "\n" );
document.write( "y = 12x^2 - 96x + 200\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2.) Write an equation for the ellipse that has foci (0,-15) and (0,15) and focal constant 34. (And what exactly is a focal constant?) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #223289 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=223289\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 12x^2\ -\ 96x\ +\ 200\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factor out the lead coefficient on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\">, divide the resulting coefficient on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> by 2, square the result, add that result inside of the parentheses, and compensate by subtracting the added term times the lead coefficient.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 12(x^2\ -\ 8x\ +\ 16)\ +\ 200\ -\ 192\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factor the trinomial\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 12(x\ -\ 4)^2\ +\ 8\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So, parabola, vertical symmetry, vertex at (4,8), focus at (4,11), directrix <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ 5\">.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=========================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First things first.  The focal constant is the sum of the distances of the foci to the ellipse at any point -- such being the definition of an ellipse.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your foci are at (0,-15) and (0,15).  This tells us a couple of things.  First, the major axis is on the x-axis and the center is at the origin.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The distance from one of the focal points to a vertex plus the distance of the other focal point to the same vertex is 34...just like the sum of the distances from the foci to any other point on the ellipse.  But considering the distances along the x-axis, you can use the fact that the foci are 30 units apart.  So, the distance from the near focus to a vertex is 34 - 30 divided by 2, or 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now we know that the two vertices are at (0,-17) and (0,17) and we have a semi-minor axis that measures 17.  But we knew that already because the focal constant is always equal to the measure of the major axis.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Next there is another relationship that also holds for every ellipse.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\"> is the distance from the center of the ellipse to either focus, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> is the measure of the semi-major axis (one-half of the major axis or the distance from the center of the ellipse to either vertex.) and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\"> is the measure of the semi-minor axis (one half of the minor axis or the measure of the segment from the center of the ellipse to either endpoint of the minor axis), then the following relationship holds:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ c^2\ =\ a^2\ -\ b^2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In your case, you have found <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ 17\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\ =\ 15\">, so a little arithmetic tells us <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ 8\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now we know enough to create the desired equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "An equation of an ellipse with center at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (h, k)\">, horizontal semi-major axis <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> and vertical semi-minor axis <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\"> is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{(x\,-\,h)^2}{a^2}\ +\ \frac{(y\ -\ k)^2}{b^2}\ =\ 1\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Tossing in the numbers we know:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Center: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (0, 0)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Semi-major axis: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ 17\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Semi-minor axis: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ 8\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{x^2}{17^2}\ +\ \frac{y^2}{8^2}\ =\ 1\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Which can also be expressed as:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{x^2}{289}\ +\ \frac{y^2}{64}\ =\ 1\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "or\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 64x^2\ +\ 289y^2\ =\ 18496\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );