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 #575556 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=575556\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Parabolas are single curves that can open up or down.<BR>\n" );
document.write( "There is no \"or\" with hyperbolas;<BR>\n" );
document.write( "a hyperbola can open up AND down,<BR>\n" );
document.write( "or it could open left AND right,<BR>\n" );
document.write( "or it could open southeast and northwest,<BR>\n" );
document.write( "or .....<BR>\n" );
document.write( "Hyperbolas, like most people, have 2 arms (two separate curves, or \"branches\").<BR>\n" );
document.write( "Those arms are flirty curves that approach the center,<BR>\n" );
document.write( "only to turn around before getting to that center,<BR>\n" );
document.write( "and then go away along a symmetric path.<BR>\n" );
document.write( "A popular kind of hyperbola opens to the left and right,<BR>\n" );
document.write( "and (with its center) looks like parentheses drawn the wrong way around a dot/point:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%29%22+%2A+%22%28%22 ALIGN=MIDDLE  ALT=\"%22%29%22+%2A+%22%28%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Another popular shape is with the openings towards top and bottom:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C1%2C11%2C1%2C11%2C6%2Bsqrt%281%2B%28x-6%29%5E2%29%2C6-sqrt%281%2B%28x-6%29%5E2%29%2C6%2Bsqrt%280.0025-%28x-6%29%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There are tilted hyperbolas too, like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C1%2C11%2C1%2C11%2C6%2B1%2F%28x-6%29%2Cx%5E2%2B10%2C6%2Bsqrt%280.0016-%28x-6%29%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , but those make calculations too complicated,<BR>\n" );
document.write( "so you will (hopefully) not see that.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The curve represented by your equation is a hyperbola.<BR>\n" );
document.write( "We know that because the terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > have coefficients (8 and -10) with opposite signs.<BR>\n" );
document.write( "You may be able to figure out which ways the hyperbola opens through some formula given in class,<BR>\n" );
document.write( "but I have to work up what that hyperbola looks like to understand it.<BR>\n" );
document.write( "(Besides, I cannot memorize formulas, so I resort to understanding and thinking through the problem).<BR>\n" );
document.write( "To figure out what your hyperbola looks like, you can \"complete the squares\":<BR>\n" );
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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E2+-+48x+-+10y%5E2+-+100y+=+98 ALIGN=MIDDLE  ALT=\"8x%5E2+-+48x+-+10y%5E2+-+100y+=+98\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now I am going to add some terms to both sides of the equal sign to get an equivalent equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E2+-+48x+%2B+72+-+10y%5E2+-+100y+-+250+=+98+%2B+72+-+250 ALIGN=MIDDLE  ALT=\"8x%5E2+-+48x+%2B+72+-+10y%5E2+-+100y+-+250+=+98+%2B+72+-+250\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There is a method to mu madnes, as you will soon see:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%28x%5E2-6x%2B9%29+-+10%28y%5E2-10y-25%29+=+-80 ALIGN=MIDDLE  ALT=\"8%28x%5E2-6x%2B9%29+-+10%28y%5E2-10y-25%29+=+-80\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%28x-3%29%5E2+-+10%28y-5%29%5E2+=+-80 ALIGN=MIDDLE  ALT=\"8%28x-3%29%5E2+-+10%28y-5%29%5E2+=+-80\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Do you see what I meant by \"completing the squares\"?<BR>\n" );
document.write( "I was building up <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%28x-3%29%5E2 ALIGN=MIDDLE  ALT=\"8%28x-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E2+-+48x ALIGN=MIDDLE  ALT=\"8x%5E2+-+48x\"   DISABLED_style= \"max-width:90%; height:auto;\" > by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72 ALIGN=MIDDLE  ALT=\"72\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and I was building up <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-+10%28y-5%29%5E2 ALIGN=MIDDLE  ALT=\"-+10%28y-5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-+10y%5E2+-+100y ALIGN=MIDDLE  ALT=\"-+10y%5E2+-+100y\"   DISABLED_style= \"max-width:90%; height:auto;\" > by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-+250 ALIGN=MIDDLE  ALT=\"+-+250\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%28x-3%29%5E2+-+10%28y-5%29%5E2+=+-80 ALIGN=MIDDLE  ALT=\"8%28x-3%29%5E2+-+10%28y-5%29%5E2+=+-80\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "I divide both sides of the equal sign by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-80%29 ALIGN=MIDDLE  ALT=\"%28-80%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the right side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%28x-3%29%5E2%2F%28-80%29+-+10%28y-5%29%5E2%2F%28-80%29+=+-80%2F%28-80%29 ALIGN=MIDDLE  ALT=\"8%28x-3%29%5E2%2F%28-80%29+-+10%28y-5%29%5E2%2F%28-80%29+=+-80%2F%28-80%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28x-3%29%5E2%2F10+%2B+%28y-5%29%5E2%2F8+=+1 ALIGN=MIDDLE  ALT=\"-%28x-3%29%5E2%2F10+%2B+%28y-5%29%5E2%2F8+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can rearrange the equation to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-5%29%5E2%2F8-%28x-3%29%5E2%2F10+=+1 ALIGN=MIDDLE  ALT=\"%28y-5%29%5E2%2F8-%28x-3%29%5E2%2F10+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > if we want.<BR>\n" );
document.write( "The last equation looks more like the textbook<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a hyperbola opening up and down.<BR>\n" );
document.write( "(I just checked).<BR>\n" );
document.write( "Without studying the book, you can realize that the curve represented by the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-5%29%5E2%2F8-%28x-3%29%5E2%2F10+=+1 ALIGN=MIDDLE  ALT=\"%28y-5%29%5E2%2F8-%28x-3%29%5E2%2F10+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "1) is symmetrical with respect to the lines<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=0 ALIGN=MIDDLE  ALT=\"x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=0 ALIGN=MIDDLE  ALT=\"y-5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5 ALIGN=MIDDLE  ALT=\"y=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because that equation does not distinguish between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=1 ALIGN=MIDDLE  ALT=\"x-3=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=-1 ALIGN=MIDDLE  ALT=\"x-3=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=2 ALIGN=MIDDLE  ALT=\"x-3=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=-2 ALIGN=MIDDLE  ALT=\"x-3=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=1 ALIGN=MIDDLE  ALT=\"y-5=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=-1 ALIGN=MIDDLE  ALT=\"y-5=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=2 ALIGN=MIDDLE  ALT=\"y-5=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=-2 ALIGN=MIDDLE  ALT=\"y-5=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ;<BR>\n" );
document.write( "2) you also realize that as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%5E2%3E=0 ALIGN=MIDDLE  ALT=\"%28x-3%29%5E2%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-5%29%5E2%2F8=1 ALIGN=MIDDLE  ALT=\"%28y-5%29%5E2%2F8=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-5%29%5E2%3E=8 ALIGN=MIDDLE  ALT=\"%28y-5%29%5E2%3E=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y-5%3E=sqrt%288%29%2C%22or%22%2Cy-5%3C=-sqrt%288%29%29 ALIGN=MIDDLE  ALT=\"system%28y-5%3E=sqrt%288%29%2C%22or%22%2Cy-5%3C=-sqrt%288%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y%3E=5%2Bsqrt%288%29%2C%22or%22%2Cy%3C=5-sqrt%288%29%29 ALIGN=MIDDLE  ALT=\"system%28y%3E=5%2Bsqrt%288%29%2C%22or%22%2Cy%3C=5-sqrt%288%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the hyperbola opens up and down,<BR>\n" );
document.write( "with no points between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5-sqrt%288%29 ALIGN=MIDDLE  ALT=\"5-sqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Bsqrt%288%29 ALIGN=MIDDLE  ALT=\"5%2Bsqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "-\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );