document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1102814>Question 1102814</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Identify the type of graph of the following polar equations. How can the graphs best be described?\r<BR>\n" );
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document.write( "a.  rsin&#952; = -11\r<BR>\n" );
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document.write( "https://www.dropbox.com/s/xo8bahbd0r5qxfh/Screen%20Shot%202017-11-21%20at%209.18.13%20AM.png?dl=0\r<BR>\n" );
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document.write( "b.  r = 3-2sin&#952;\r<BR>\n" );
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document.write( "https://www.dropbox.com/s/i99qmpa4vmrir0u/Screen%20Shot%202017-11-21%20at%209.18.23%20AM.png?dl=0\r<BR>\n" );
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document.write( "c.  r^2 = 25cos2&#952;\r<BR>\n" );
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document.write( "https://www.dropbox.com/s/vboagd17yavux3s/Screen%20Shot%202017-11-21%20at%209.18.36%20AM.png?dl=0\r<BR>\n" );
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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 #717545 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=717545\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> c . You probably asked question 1102813,<BR>\n" );
document.write( "and from that question you should know that the answer to part c is<BR>\n" );
document.write( "a lemniscate symmetric about the origin and the x- and y-axes,<BR>\n" );
document.write( "or as your answer option says, in polar coordinates,<BR>\n" );
document.write( "\"symmetric about the pole and the lines <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=0 ALIGN=MIDDLE  ALT=\"theta=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the vertical line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F2 ALIGN=MIDDLE  ALT=\"theta=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\"<BR>\n" );
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document.write( "a. As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=r%2Asin%28theta%29 ALIGN=MIDDLE  ALT=\"y=r%2Asin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is part of the conversion from polar to cartesian coordinates,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2Asin%28theta%29=-11 ALIGN=MIDDLE  ALT=\"r%2Asin%28theta%29=-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > translates into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-11 ALIGN=MIDDLE  ALT=\"y=-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and represents a horizontal line.<BR>\n" );
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document.write( "b. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3-2sin%28theta%29 ALIGN=MIDDLE  ALT=\"r=3-2sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graphs ans a limacon with a dimple,<BR>\n" );
document.write( "because, in absolute value, the coefficient of the trigonometric function (2)<BR>\n" );
document.write( "is between 1/2 and 1 times the other term (3).<BR>\n" );
document.write( "The dimple is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F2 ALIGN=MIDDLE  ALT=\"theta=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > because that is where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > is minimum.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F6 ALIGN=MIDDLE  ALT=\"theta=pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3-2%2Asin%28pi%2F6%29=3%2A2%2A%281%2F2%29=3-1=2 ALIGN=MIDDLE  ALT=\"r=3-2%2Asin%28pi%2F6%29=3%2A2%2A%281%2F2%29=3-1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=r%2Asin%28pi%2F6%29=2%2A%281%2F2%29=1 ALIGN=MIDDLE  ALT=\"y=r%2Asin%28pi%2F6%29=2%2A%281%2F2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F4 ALIGN=MIDDLE  ALT=\"theta=pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3-2%2Asin%28pi%2F4%29=3%2A2%2A%28sqrt%282%29%2F2%29=3-sqrt%282%29 ALIGN=MIDDLE  ALT=\"r=3-2%2Asin%28pi%2F4%29=3%2A2%2A%28sqrt%282%29%2F2%29=3-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > has increased to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=r%2Asin%28pi%2F4%29=%283-sqrt%282%29%29%2A%28sqrt%282%29%2F2%29=%283sqrt%282%29-2%29%2F2=about+1.12 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but as we keep approaching <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F2 ALIGN=MIDDLE  ALT=\"theta=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > starts to decrease forming a dimple.<BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F2 ALIGN=MIDDLE  ALT=\"theta=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3-2%2Asin%28pi%2F2%29=3%2A2%2A1=3-2=1 ALIGN=MIDDLE  ALT=\"r=3-2%2Asin%28pi%2F2%29=3%2A2%2A1=3-2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=r%2Asin%28pi%2F2%29=1%2A1=1 ALIGN=MIDDLE  ALT=\"y=r%2Asin%28pi%2F2%29=1%2A1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > again. </SPAN>\n" );
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