document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204299>Question 1204299</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What two parts of a polynomial function indicate the end behaviours of the graph? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840639 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=840639\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The two things we look for are: the <font color=red>leading coefficient</font> and the <font color=red>degree of the polynomial</font>.<BR>\n" );
document.write( "Both of those help form the leading term.<BR>\n" );
document.write( "The other terms won't affect the end behavior.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Something like y = 6x^3 + 10x^2 + 5x + 7 has the same end behavior as y = 6x^3.<BR>\n" );
document.write( "This is because as x gets really really large, the leading term has the most effect compared to the other terms. <BR>\n" );
document.write( "Cubing a large number makes it much bigger compared to squaring it for example.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Consider the monomial y = a*x^n<BR>\n" );
document.write( "If a > 0 and n is even, then the two endpoints rise up together.<BR>\n" );
document.write( "Example: y = 3x^2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C-1000%2C3x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C-1000%2C3x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If a < 0 and n is even, then the endpoints fall together<BR>\n" );
document.write( "Example: y = -3x^2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C-1000%2C-3x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C-1000%2C-3x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If n is odd, then the endpoints point in opposite directions.<BR>\n" );
document.write( "a > 0 has the end behavior \"fall to the left, rise to the right\"<BR>\n" );
document.write( "Example: y = 2x^5<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C-1000%2C2x%5E5%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C-1000%2C2x%5E5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a < 0 has the end behavior \"rise to the left, fall to the right\" when n is odd.<BR>\n" );
document.write( "Example: y = -5x^3<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C-1000%2C-5x%5E3%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C-1000%2C-5x%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I encourage you to try out other examples. Use Desmos or GeoGebra as a graphing tool.\r<BR>\n" );
document.write( "\n" );
document.write( "---------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Summary chart<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >n is even</TD><TD >n is odd</TD></tr><tr><TD >a > 0</TD><TD >rise together</TD><TD >fall to the left, rise to the right</TD></tr><tr><TD >a < 0</TD><TD >fall together</TD><TD >rise to the left, fall to the right</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );