document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1073333>Question 1073333</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the following equation, determine if the following symmetry exists. <BR>\n" );
document.write( "f(x)=-2x^3+4x^2-2x+1 <BR>\n" );
document.write( " <BR>\n" );
document.write( "1. x- axis <BR>\n" );
document.write( "2. y=x <BR>\n" );
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 #688190 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=688190\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Neither<BR>\n" );
document.write( "1. For the graph of a  relation to be symmetrical with respect to the x-axis,for every point (x,y), its reflection over the x-axis, point (X,-Y) must also be in the graph.<BR>\n" );
document.write( "That means that changing y to -y, you get the same equation.<BR>\n" );
document.write( "That cannot happen with a function,<BR>\n" );
document.write( "because by definition a function assigns just one y value to every x.<BR>\n" );
document.write( "It cannot assign y and -y for the same x.<BR>\n" );
document.write( "2. I assume that means symmetry about the y=x line.<BR>\n" );
document.write( "For that symmetry, if a point (x,y) is part of the graph, (y,x) must be part of the graph too.<BR>\n" );
document.write( "That means that swapping the x and y variables you get the same equation.<BR>\n" );
document.write( "Some functions could do that, but the only polynomial that can do that is the degree 1 polynomial f(x)=x or y=x.<BR>\n" );
document.write( "If one of the variables has an exponent that the other does not have,<BR>\n" );
document.write( "swapping variables cannot possibly yield the same equation.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-2x%5E3%2B4x%5E2-2x%2B1 ALIGN=MIDDLE  ALT=\"f%28x%29=-2x%5E3%2B4x%5E2-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a polynomial in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Polynomials in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > have graphs like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C0.05%28x%2B5%29%28x-1%29%28x-3%29%2B1%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C0.05%28x%2B5%29%28x-1%29%28x-3%29%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C0.05%28x%2B5%29%28x%2B1%29%28x-2%29%28x-6%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C0.05%28x%2B5%29%28x%2B1%29%28x-2%29%28x-6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With any luck,<BR>\n" );
document.write( "a polynomial of even degree may be symmetrical with respect to the y-axis,<BR>\n" );
document.write( "or a polynomial of odd degree may be symmetrical about the origin. </SPAN>\n" );
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