document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=672886>Question 672886</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine whether the graph of y=4x^2+1 is symmetric with respect to the x-axis, y-axis, and/or origin. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #418401 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=418401\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4x%5E2%2B1+ ALIGN=MIDDLE  ALT=\"y=4x%5E2%2B1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we need to check if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29+=+-f%28x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29+=+-f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, if yes than a function is symmetric with respect to the origin\r<BR>\n" );
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document.write( "first find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y ALIGN=MIDDLE  ALT=\"-y\"   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=-y=-%284x%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"-y=-%284x%5E2%2B1%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=-y=-4x%5E2-1 ALIGN=MIDDLE  ALT=\"-y=-4x%5E2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%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=y=4%28-x%29%5E2%2B1+ ALIGN=MIDDLE  ALT=\"y=4%28-x%29%5E2%2B1+\"   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=y=4x%5E2%2B1+ ALIGN=MIDDLE  ALT=\"y=4x%5E2%2B1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29+%3C%3E+-f%28x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29+%3C%3E+-f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...a function is not symmetric with respect to the origin\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C600%2C-10%2C10%2C-10%2C10%2Cgrid%281%29%2Ccircle%280%2C0%2C0.2%29%2Cgraph%28600%2C600%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "A graph is said to be symmetric about the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-axis ALIGN=MIDDLE  ALT=\"x-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" > if whenever (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is on the graph then so is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b ALIGN=MIDDLE  ALT=\"-b\"   DISABLED_style= \"max-width:90%; height:auto;\" >) \r<BR>\n" );
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document.write( "as you can see from a graph, there is no point such as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2C-b ALIGN=MIDDLE  ALT=\"a%2C-b\"   DISABLED_style= \"max-width:90%; height:auto;\" > because the graph is above <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-axis ALIGN=MIDDLE  ALT=\"x-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "see it on a graph<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C600%2C-10%2C10%2C-10%2C10%2Cgrid%281%29%2Ccircle%281%2C5%2C0.2%29%2Ccircle%281%2C-5%2C0.2%29%2Clocate%281%2C5-.2%2C%22%281%2C5%29%22%29%2Clocate%281%2C-5-.2%2C%22%281%2C-5%29%22%29%2Cgraph%28600%2C600%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "A graph is said to be symmetric about the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-axis ALIGN=MIDDLE  ALT=\"y-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" > if whenever (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is on the graph then so is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a ALIGN=MIDDLE  ALT=\"-a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   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=-a ALIGN=MIDDLE  ALT=\"-a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   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;\" >), there is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   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=drawing%28600%2C600%2C-10%2C10%2C-10%2C10%2Cgrid%281%29%2Clocate%28-1%2C5-.2%2C%22%28-1%2C5%29%22%29%2Clocate%281%2C5-.2%2C%22%281%2C5%29%22%29%2Ccircle%281%2C5%2C0.2%29%2Ccircle%28-1%2C5%2C0.2%29%2Cgraph%28600%2C600%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4x%5E2%2B1 ALIGN=MIDDLE  ALT=\"y=4x%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is symmetric with respect to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-axis ALIGN=MIDDLE  ALT=\"y-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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