document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=967250>Question 967250</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can someone pretty please help me figure this out?<BR>\n" );
document.write( "For the following equation of a parabola determine the vertex, focus, and directix, then sketch it's graph.<BR>\n" );
document.write( "x^2+4x-3y+4=0 <BR>\n" );
document.write( "Thank you so much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #591181 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=591181\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "try this reference.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/parabola.htm\" target = \"_blank\">http://www.purplemath.com/modules/parabola.htm</a>\r<BR>\n" );
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document.write( "the information pertinent to your problem is as follows:\r<BR>\n" );
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document.write( "you want to know the vertex, the focus and the directrix.\r<BR>\n" );
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document.write( "there are 2 basic forms of the equation of a parabola that you will need.\r<BR>\n" );
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document.write( "the first form is the vertex form.\r<BR>\n" );
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document.write( "that form is y = a * (x-h)^2 + k\r<BR>\n" );
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document.write( "(h,k) is the vertex.<BR>\n" );
document.write( "a is the coefficient of the (x-h)^2 term.\r<BR>\n" );
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document.write( "you need to transform your equation to this form.\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "x^2+4x-3y+4=0\r<BR>\n" );
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document.write( "associate the first two terms together and then complete the squares on them.\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "(x^2 + 4x) - 3y + 4 = 0.\r<BR>\n" );
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document.write( "complete the squares on the x terms to get:\r<BR>\n" );
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document.write( "(x + 2)^2 - 4 - 3y + 4 = 0\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "(x + 2)^2 - 3y = 0\r<BR>\n" );
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document.write( "add 3y to both sides of the equation to get:\r<BR>\n" );
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document.write( "(x + 2)^2 = 3y\r<BR>\n" );
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document.write( "divide both sides of the equation by 3 to get:\r<BR>\n" );
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document.write( "(x + 2)^2 / 3 = y\r<BR>\n" );
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document.write( "this is the same as y = 1/3 * (x + 2)^2 \r<BR>\n" );
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document.write( "this is now in the vertex form of y = a * (x-h)^2 + k\r<BR>\n" );
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document.write( "a = 1/3<BR>\n" );
document.write( "h = -2<BR>\n" );
document.write( "k = 0\r<BR>\n" );
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document.write( "the vertex is at (h,k) = (-2,0)\r<BR>\n" );
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document.write( "the second form you will need is the conics form of the equation of a parabola.\r<BR>\n" );
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document.write( "that form is 4p(y – k) = (x – h)^2\r<BR>\n" );
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document.write( "start with the vertex form of y = 1/3 * (x + 2)^2 \r<BR>\n" );
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document.write( "multiply both sides of the equation by 3 to get:\r<BR>\n" );
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document.write( "3y = (x + 2)^2\r<BR>\n" );
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document.write( "the equation is now in the conics form of 4p * (y - k) = (x - h)^2\r<BR>\n" );
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document.write( "this is because 3y is the same as 3 * (y - 0).<BR>\n" );
document.write( "that's where the k of 0 comes from.\r<BR>\n" );
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document.write( "in this equation:\r<BR>\n" );
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document.write( "4p = 3<BR>\n" );
document.write( "k = 0<BR>\n" );
document.write( "h = -2\r<BR>\n" );
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document.write( "since 4p = 3, then divide by 4 to get p = 3/4.\r<BR>\n" );
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document.write( "p is the distance from the focus to the vertex and from the directrix to the vertex.\r<BR>\n" );
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document.write( "the vertex is in between the focus and the directrix.\r<BR>\n" );
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document.write( "the focus is inside the parabola.\r<BR>\n" );
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document.write( "the directrix is outside the partabola.\r<BR>\n" );
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document.write( "the vertex is at (-2,0)<BR>\n" );
document.write( "the focus is at (-2,3/4)<BR>\n" );
document.write( "the directrix is at y = -3/4\r<BR>\n" );
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document.write( "the directrix is a line perpendicular to the axis of symmetry of the parabola.\r<BR>\n" );
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document.write( "there is also the standard form of the equation of a parabola.\r<BR>\n" );
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document.write( "that form is y = ax^2 + bx + c.\r<BR>\n" );
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document.write( "you can convert your equation to this form by solving for y.\r<BR>\n" );
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document.write( "it's not necessary, however, to solve this problem.\r<BR>\n" );
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document.write( "solving this problem required the vertex form and the conics form.\r<BR>\n" );
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document.write( "the relationship between p in the conics form and a in the vertex form is:\r<BR>\n" );
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document.write( "4p = 1 / a\r<BR>\n" );
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document.write( "a = 1 / 4p\r<BR>\n" );
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document.write( "the vertex form of your equation was y = 1/3 * (x + 2)^2\r<BR>\n" );
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document.write( "the conics form of your equation was 3y = (x + 2)^2\r<BR>\n" );
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document.write( "4p is equal to 1/a which made 4p equal to 1 / (1/3) = 3.\r<BR>\n" );
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document.write( "a is equal to 1/(4p) which made a equal to 1/3.\r<BR>\n" );
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document.write( "the graph of your equation looks like this:\r<BR>\n" );
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document.write( "all 3 forms of the equation were graphed.\r<BR>\n" );
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document.write( "you can see that they all point to the same graph.\r<BR>\n" );
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document.write( "this means the different forms of the equation are equivalent to each other.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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