document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205368>Question 1205368</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Test the symmetry (with respect to the polar axis, to the pole, and to the vertical line θ=π/2) and sketch the graph of the polar equation r = 3+2sin(θ) (4 points)\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842140 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=842140\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  the polar equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+3%2B2sin%28theta%29 ALIGN=MIDDLE  ALT=\"r+=+3%2B2sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > -axis  -  called the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=polar ALIGN=MIDDLE  ALT=\"polar\"   DISABLED_style= \"max-width:90%; height:auto;\" > axis<BR>\n" );
document.write( "the 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;\" > (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >-axis)<BR>\n" );
document.write( "the pole is origin \r<BR>\n" );
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document.write( " test for symmetry about polar-axis: replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-theta ALIGN=MIDDLE  ALT=\"-theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > and see is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+ ALIGN=MIDDLE  ALT=\"r+\"   DISABLED_style= \"max-width:90%; height:auto;\" >unchanged \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+3%2B2sin%28-theta%29 ALIGN=MIDDLE  ALT=\"r+=+3%2B2sin%28-theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >....since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-theta%29=-sin%28theta%29 ALIGN=MIDDLE  ALT=\"sin%28-theta%29=-sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have\r<BR>\n" );
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document.write( "<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;\" > =>not same as what we started with, <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 changed; so, no symmetry about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >-axis or  the polar axis\r<BR>\n" );
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document.write( "test the 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;\" > ( <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >-axis): replace (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >) with (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi-theta ALIGN=MIDDLE  ALT=\"pi-theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and see is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > unchanged\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+3%2B2sin%28pi-theta%29 ALIGN=MIDDLE  ALT=\"r+=+3%2B2sin%28pi-theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >...since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi-theta%29=sin%28theta%29 ALIGN=MIDDLE  ALT=\"sin%28pi-theta%29=sin%28theta%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=r+=+3%2B2sin%28theta%29 ALIGN=MIDDLE  ALT=\"r+=+3%2B2sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> same as what we started with, <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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=not+ ALIGN=MIDDLE  ALT=\"not+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=changed ALIGN=MIDDLE  ALT=\"changed\"   DISABLED_style= \"max-width:90%; height:auto;\" >; so, there is symmetry about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >-axis or  the the 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;\" >\r<BR>\n" );
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document.write( "test the pole is origin (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180 ALIGN=MIDDLE  ALT=\"180\"   DISABLED_style= \"max-width:90%; height:auto;\" >°  rotation), replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-r ALIGN=MIDDLE  ALT=\"-r\"   DISABLED_style= \"max-width:90%; height:auto;\" > and see is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > unchanged\r<BR>\n" );
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document.write( ".<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-r+=+3%2B2sin%28theta%29 ALIGN=MIDDLE  ALT=\"-r+=+3%2B2sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = >  obviously <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 changed, so no symmetry abut pole (origin)\r<BR>\n" );
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document.write( "graph it:\r<BR>\n" );
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