document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=943929>Question 943929</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am confused on how hyperbolas open left or right, can you please explain that to me. and I have to solve these and not sure if left or right \r<BR>\n" );
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document.write( "1. 8x^2 - 10y^2 - 48x - 100y - 98 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #575540 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=575540\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hyperbolas consist of two vaguely parabola shaped pieces that open either up and down or right and left.  Also, just like parabolas each of the pieces has a vertex.  Note that they aren’t really parabolas, they just resemble parabolas.\r<BR>\n" );
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document.write( "There are also two lines on each graph.  These lines are called asymptotes and as the graphs show as we make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > large (in both the positive and negative sense) the graph of the hyperbola gets closer and closer to the asymptotes.  \r<BR>\n" );
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document.write( "The asymptotes are not officially part of the graph of the hyperbola.  However, they are usually included so that we can make sure and get the sketch correct.  The point where the two asymptotes cross is called the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=center ALIGN=MIDDLE  ALT=\"center\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the hyperbola.\r<BR>\n" );
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document.write( " There are two <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=standard ALIGN=MIDDLE  ALT=\"standard\"   DISABLED_style= \"max-width:90%; height:auto;\" > forms of the hyperbola:\r<BR>\n" );
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document.write( "form: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2+-%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2+-%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > => hyperbola opens left and right,has  vertices at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2Ba ALIGN=MIDDLE  ALT=\"h%2Ba\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h-a ALIGN=MIDDLE  ALT=\"h-a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >) , \r<BR>\n" );
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document.write( "and form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fb%5E2+-%28x-h%29%5E2%2Fa%5E2+=1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fb%5E2+-%28x-h%29%5E2%2Fa%5E2+=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > =>hyperbola opens up and down, has vertices (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%2Bb ALIGN=MIDDLE  ALT=\"k%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k-b ALIGN=MIDDLE  ALT=\"k-b\"   DISABLED_style= \"max-width:90%; height:auto;\" >) \r<BR>\n" );
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document.write( "both have: <BR>\n" );
document.write( "Center at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >),  Slope of Asymptotes ± <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fa ALIGN=MIDDLE  ALT=\"b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, Equations of Asymptotes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > ±<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28+b%2Fa%29%28x-h%29 ALIGN=MIDDLE  ALT=\"%28+b%2Fa%29%28x-h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "1.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+8x%5E2+-+10y%5E2+-+48x+-+100y+-+98+=+0 ALIGN=MIDDLE  ALT=\"+8x%5E2+-+10y%5E2+-+48x+-+100y+-+98+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > first write it in standard form\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%288x%5E2++-+48x%29+-+%2810y%5E2%2B+100y%29++=+98 ALIGN=MIDDLE  ALT=\"+%288x%5E2++-+48x%29+-+%2810y%5E2%2B+100y%29++=+98\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...complete square\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+8%28x%5E2++-+6x%29+-+10%28y%5E2%2B+10y%29++=+98 ALIGN=MIDDLE  ALT=\"+8%28x%5E2++-+6x%29+-+10%28y%5E2%2B+10y%29++=+98\"   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=+8%28x%5E2++-+6x%2B_%29+-_-+10%28y%5E2%2B+10y%2B_%29+-_+=+98 ALIGN=MIDDLE  ALT=\"+8%28x%5E2++-+6x%2B_%29+-_-+10%28y%5E2%2B+10y%2B_%29+-_+=+98\"   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=+8%28x%5E2++-+6x%2B3%5E2%29+-3%5E2%2A8-+10%28y%5E2%2B+10y%2B5%5E2%29+-5%5E2%28-10%29+=+98 ALIGN=MIDDLE  ALT=\"+8%28x%5E2++-+6x%2B3%5E2%29+-3%5E2%2A8-+10%28y%5E2%2B+10y%2B5%5E2%29+-5%5E2%28-10%29+=+98\"   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=8+%28x-3%29%5E2-72+-10+%28y%2B5%29%5E2+%2B250=98 ALIGN=MIDDLE  ALT=\"8+%28x-3%29%5E2-72+-10+%28y%2B5%29%5E2+%2B250=98\"   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=8+%28x-3%29%5E2-10+%28y%2B5%29%5E2+=98%2B72-250 ALIGN=MIDDLE  ALT=\"8+%28x-3%29%5E2-10+%28y%2B5%29%5E2+=98%2B72-250\"   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=8+%28x-3%29%5E2-10+%28y%2B5%29%5E2+=-80 ALIGN=MIDDLE  ALT=\"8+%28x-3%29%5E2-10+%28y%2B5%29%5E2+=-80\"   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=8+%28x-3%29%5E2%2F-80-10+%28y%2B5%29%5E2%2F-80+=-80%2F-80 ALIGN=MIDDLE  ALT=\"8+%28x-3%29%5E2%2F-80-10+%28y%2B5%29%5E2%2F-80+=-80%2F-80\"   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=-%28x-3%29%5E2%2F10%2B+%28y%2B5%29%5E2+%2F8=1 ALIGN=MIDDLE  ALT=\"-%28x-3%29%5E2%2F10%2B+%28y%2B5%29%5E2+%2F8=1\"   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=%28y%2B5%29%5E2+%2F8-%28x-3%29%5E2%2F10=1 ALIGN=MIDDLE  ALT=\"%28y%2B5%29%5E2+%2F8-%28x-3%29%5E2%2F10=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "as you can see, you have hyperbola in a form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28y-k%29%5E2%2Fb%5E2+-%28x-h%29%5E2%2Fa%5E2+=1+ ALIGN=MIDDLE  ALT=\"+%28y-k%29%5E2%2Fb%5E2+-%28x-h%29%5E2%2Fa%5E2+=1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> Opens up and down, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-5 ALIGN=MIDDLE  ALT=\"k=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=3 ALIGN=MIDDLE  ALT=\"h=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, => center at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   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=b=sqrt%288%29=2sqrt%282%29 ALIGN=MIDDLE  ALT=\"b=sqrt%288%29=2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=sqrt%2810%29 ALIGN=MIDDLE  ALT=\"a=sqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
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document.write( "vertices (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%2Bb ALIGN=MIDDLE  ALT=\"k%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >) =(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2B2sqrt%282%29 ALIGN=MIDDLE  ALT=\"-5%2B2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5-2sqrt%282%29 ALIGN=MIDDLE  ALT=\"-5-2sqrt%282%29\"   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=+graph%28+600%2C600%2C+-10%2C+15%2C+-15%2C+10%2C-sqrt%288%28%28x-3%29%5E2%2F10%2B1%29%29-5+%2Csqrt%288%28%28x-3%29%5E2%2F10%2B1%29%29-5%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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