document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192190>Question 1192190</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what are the proper steps i would need to do in order to sketch a comprehensive graph of the polynomial function f(x)=0.1x^5-0.9x^3  by showing all the necessary steps of graphing a polynomial function. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824123 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52852)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824123\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "what are the proper steps i would need to do in order to sketch a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cross%28comprehensive%29%29 ALIGN=MIDDLE  ALT=\"highlight%28cross%28comprehensive%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph <BR>\n" );
document.write( "of the polynomial function f(x)=0.1x^5-0.9x^3 by showing all the necessary steps <BR>\n" );
document.write( "of graphing a polynomial function.<BR>\n" );
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document.write( "First what you need to understand is that a sketch is not a graph.\r<BR>\n" );
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document.write( "A graph must follow to a strict definition of a function and should reproduce the function literally and precisely.\r<BR>\n" );
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document.write( "A sketch should only reproduce the major behavior and the major tendencies of a function.\r<BR>\n" );
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document.write( "For a sketch, you should reproduce\r<BR>\n" );
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document.write( "    (a)  the zeroes of a function;\r\n" );
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document.write( "    (b)  the end behavior of a function (its behavior at x ---> oo  and  x ---> -oo);\r\n" );
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document.write( "    (c)  the asymptotical behavior of a function (horizontal and vertical asymptotes);\r\n" );
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document.write( "    (d)  local maximums and local minimums of a function;\r\n" );
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document.write( "    (e)  at the zero points (x-intercepts), you should show if the sketch (the plot) intersects x-axis and changes the sign or,\r\n" );
document.write( "         in opposite, remains the sign unchanged.\r\n" );
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document.write( "If you can reproduce all these elements and peculiar properties of a function, it is just enough for the sketch.\r<BR>\n" );
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document.write( "Concretely for the given function f(x) = 0.1x^5 - 0.9x^3\r<BR>\n" );
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document.write( "you start by re-writing it in the equivalent form  f(x) = 0.1x^3*(x^2 - 9) = 0.1x^3*(x-3)*(x+3).\r<BR>\n" );
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document.write( "From this form, you see\r<BR>\n" );
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document.write( "    (a)  end behavior:  it tends to -oo  at  x ---> -oo;\r\n" );
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document.write( "                        it tends to  oo  at  x --->  oo;\r\n" );
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document.write( "    (b)  it has zeroes  at  x= 0;  x= 3;  x= -3.\r\n" );
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document.write( "    (c)  the roots are of odd multiplicities:  multiplicity 3 at x= 0;\r\n" );
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document.write( "                                               multiplicity 1 at x= -3;\r\n" );
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document.write( "                                               multiplicity 1 at x = 3.\r\n" );
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document.write( "         Hence, at each zero, the plot (the sketch) intersects x-axis, changing the sign.\r\n" );
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document.write( "    (d)  the plot is in negative half-plane y < 0 at x < -3;  \r\n" );
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document.write( "                     in positive half-plane y > 0 in the interval  -3 < x < 0;\r\n" );
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document.write( "                     in negative half-plane y < 0 in the interval  0 < x <  3;\r\n" );
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document.write( "                     in positive half-plane y > 0 in the interval  3 < x < oo.\r\n" );
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document.write( "    (e)  There is positive local maximum in the interval -3 < x < 0;\r\n" );
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document.write( "         There is negative local minimum in the interval  0 < x < 3.\r\n" );
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document.write( "I'd say, it is enough to describe the sketch for this function.\r\n" );
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document.write( "What I described in this post, is not a universal prescription and is not a universal algorithm/methodology.\r<BR>\n" );
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document.write( "You will get more experience with the time, when you will make sketches for 10 - 20 functions on your own . . . \r<BR>\n" );
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document.write( "For periodic trigonometric functions, you should be able to describe and to reproduce their midlines; their amplitudes, periods; \r<BR>\n" );
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document.write( "horizontal and vertical shifts; horizontal and vertical compressing or stretching coefficients.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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