document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=133152>Question 133152</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the vertices of a hyperbola with the equation (x + 3)^2 - 4(y - 2)^2 = 4.\r<BR>\n" );
document.write( "\n" );
document.write( "a. (-4, 2) and (-2, 2)<BR>\n" );
document.write( " b. (-5, 2) and (-1, 2)<BR>\n" );
document.write( " c. (4, -3) and (0, -3)<BR>\n" );
document.write( " d. (3, -3) and (1, -3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #97345 by </B><A HREF=/tutors/aboutme.mpl?userid=nycsharkman><B>nycsharkman(136)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nycsharkman><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=97345\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the vertices of a hyperbola with the equation (x + 3)^2 - 4(y - 2)^2 = 4.\r<BR>\n" );
document.write( "\n" );
document.write( "a. (-4, 2) and (-2, 2)<BR>\n" );
document.write( "b. (-5, 2) and (-1, 2)<BR>\n" );
document.write( "c. (4, -3) and (0, -3)<BR>\n" );
document.write( "d. (3, -3) and (1, -3)\r<BR>\n" );
document.write( "\n" );
document.write( "The first thing you want to do is to write the equation in standard form.\r<BR>\n" );
document.write( "\n" );
document.write( "All hyperbola standard form equations = 1.\r<BR>\n" );
document.write( "\n" );
document.write( "So, this means you must divide everthing by 4 to get 1 on the right side.\r<BR>\n" );
document.write( "\n" );
document.write( "Why 4?  Because your equation equals 4 and we want it to equal 1. As you know, anything divided by itself = 1.\r<BR>\n" );
document.write( "\n" );
document.write( "Dividing everything by 4 and the result is:\r<BR>\n" );
document.write( "\n" );
document.write( "[(x + 3)^2]/4 - [4(y - 2)^2]/4 = 1\r<BR>\n" );
document.write( "\n" );
document.write( "Now this looks like the general form:\r<BR>\n" );
document.write( "\n" );
document.write( "[(x - h)^2]/a^2 - [(y - k)^2]/b^2\r<BR>\n" );
document.write( "\n" );
document.write( "The vertices are:\r<BR>\n" );
document.write( "\n" );
document.write( "(h + a, k) and (h - a, k)\r<BR>\n" );
document.write( "\n" );
document.write( "We know the value of h and k as given in the equation above.\r<BR>\n" );
document.write( "\n" );
document.write( "h = -3 and k = 2...The opposite of the original signs must be used.  In other words, 3 became -3 and -2 became 2 as the new values for h and k.\r<BR>\n" );
document.write( "\n" );
document.write( "What is a?\r<BR>\n" );
document.write( "\n" );
document.write( "Do you see the denominator 4 in the fraction [(x + 3)^2]/4 above?\r<BR>\n" );
document.write( "\n" );
document.write( "If I equate 4 to a^2 and take the square root, I will get 2 as the value of a.\r<BR>\n" );
document.write( "\n" );
document.write( "So, a = 2.  We reject -2 because vertices represent distance and distance CANNOT be negative.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I will now plug the values of h, k and a into (h + a, k) and (h - a, k) and find the vertices.\r<BR>\n" );
document.write( "\n" );
document.write( "(-3 + 2, 2) = (-1, 2)\r<BR>\n" );
document.write( "\n" );
document.write( "(-3 -2, 2) = (-5, 2)\r<BR>\n" );
document.write( "\n" );
document.write( "Answer: Choice B\r<BR>\n" );
document.write( "\n" );
document.write( "Did you follow?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );