document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208319>Question 1208319</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write an equation for a roller coaster that has at least 3 relative maxima and/or minima, travels below the surface of the ground at some point, and have at least one of the zeros in the function to have a multiplicity that is greater than 1. The equation should be written in h(x) in factored form. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846663 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=846663\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Pick any three numbers at random. <BR>\n" );
document.write( "I'll select 1,4,7.\r<BR>\n" );
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document.write( "These will be the roots, aka x intercepts, of the polynomial curve.<BR>\n" );
document.write( "This is where the curve either crosses the x axis or touches it.\r<BR>\n" );
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document.write( "If x = 1 is a root, then x-1 = 0, which leads to (x-1) being a factor.<BR>\n" );
document.write( "The other factors are (x-4) and (x-7)\r<BR>\n" );
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document.write( "Let's say that the root at x = 7 is a double root. <BR>\n" );
document.write( "Meaning that instead of crossing the x axis, it bounces off the x axis at this point.<BR>\n" );
document.write( "We'll update (x-7) to (x-7)^2\r<BR>\n" );
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document.write( "All together we have<BR>\n" );
document.write( "(x-1)(x-4)(x-7)^2<BR>\n" );
document.write( "I'll let the student expand this out to get a 4th degree polynomial. \r<BR>\n" );
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document.write( "Graph of (x-1)(x-4)(x-7)^2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-2%2C10%2C-60%2C60%2C-1000%2C%28x-1%29%28x-4%29%28x-7%29%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-2%2C10%2C-60%2C60%2C-1000%2C%28x-1%29%28x-4%29%28x-7%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "graphing window parameters are:<BR>\n" );
document.write( "xMin = -2, xMax = 10<BR>\n" );
document.write( "yMin = -60, yMax = 60<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> and <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> are two (of many) graphing tools you can use.<BR>\n" );
document.write( "If you are most familiar with something like a TI83, then it's best to stick with it.\r<BR>\n" );
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document.write( "This isn't the only possible answer. You can pick any other values to be your roots. Also, you can scale the graph up or down. Feel free to get creative. <BR>\n" );
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