document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1103734>Question 1103734</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am trying to help a student, but all of the online resources have x and y on opposite sides of the equation. Why is that? How do we change it?\r<BR>\n" );
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document.write( "Here is what he has been asked to solve: Identify the vertex, axis of symmetry, focus and directrix of the given parabola.<BR>\n" );
document.write( "x = 1/3(y+2)^2+3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #718434 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=718434\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "If a parabola is in the \"usual\" orientation -- opening up or down, with a vertical axis of symmetry, then the equation can be written in the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%281%2F4p%29%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y+=+%281%2F4p%29%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A parabola of this type opens up if p is positive or down if p is negative.<br><BR>\n" );
document.write( "If a parabola is \"lying on its side\" -- opening right or left, with a horizontal axis of symmetry, then the equation can be written in the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281%2F4p%29%28y-k%29%5E2%2Bh ALIGN=MIDDLE  ALT=\"x+=+%281%2F4p%29%28y-k%29%5E2%2Bh\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A parabola of this type opens to the right if p is positive or tothe left if p is negative.<br><BR>\n" );
document.write( "I think that is the difference you are talking about (\"...x and y on opposite sides\").<br><BR>\n" );
document.write( "The equation of your parabola is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281%2F3%29%28y%2B2%29%5E2%2B3 ALIGN=MIDDLE  ALT=\"x+=+%281%2F3%29%28y%2B2%29%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the parabola is lying on its side, with a horizontal axis of symmetry.<br><BR>\n" );
document.write( "From the standard form, we see that the vertex (h,k) is (3,-2).  And (1/4p) is equal to 1/3, so 4p=3, so p = 4/3.<br><BR>\n" );
document.write( "The parameter p in either of these forms for the equation of a parabola is the distance from the vertex to the focus and from the vertex to the directrix.<br><BR>\n" );
document.write( "With p being positive for this parabola, the graph opens to the right; so the focus is 4/3 units to the right of the vertex, and the directrix is 4/3 units to the left of the vertex.<br><BR>\n" );
document.write( "So now we have everything we need to answer the questions:<br><BR>\n" );
document.write( "vertex: (3,-2)<BR>\n" );
document.write( "axis of symmetry: y = -2  [the horizontal line passing through the vertex]<BR>\n" );
document.write( "focus: (13/3,-2)  [4/3 units to the right of the vertex]<BR>\n" );
document.write( "directrix: x = 5/3  [4/3 units to the left of the vertex]<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C200%2C-2%2C10%2C-10%2C5%2Csqrt%283%28x-3%29%29-2%2C-sqrt%283%28x-3%29%29-2%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C200%2C-2%2C10%2C-10%2C5%2Csqrt%283%28x-3%29%29-2%2C-sqrt%283%28x-3%29%29-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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