document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189813>Question 1189813</A>: <I><SPAN STYLE=\"word-break: break-all;\"> My teacher asked me to do a roller coaster with polynomial function, he gave me the following requirements:\r<BR>\n" );
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document.write( "- your coaster ride must have at least 3 relative maxima and/or minima<BR>\n" );
document.write( "- the ride length must be at least 2 minutes (120 seconds)<BR>\n" );
document.write( "- the coaster ride starts at 250 feet<BR>\n" );
document.write( "- the ride dives below the ground into a tunnel at least once<BR>\n" );
document.write( "1. List all roots or x-intercepts of your polynomial; be sure to include at least one of each of the following on your<BR>\n" );
document.write( "design: one double root (multiplicity of two) and at least 2 real roots.<BR>\n" );
document.write( "Remember: The x-intercepts you select will be somewhat random. You will have to play around with the intercepts<BR>\n" );
document.write( "you choose to get a picture of the graph that you like and is accurate. You should use time in seconds as your<BR>\n" );
document.write( "independent variable. Your ride has to be at least 2 minutes (120 seconds long) so keep that in mind when you are<BR>\n" );
document.write( "selecting x-intercepts to use.<BR>\n" );
document.write( "2. Create an equation using your x-intercepts (from question #1) and the y-intercept. Explain how you create your equation.<BR>\n" );
document.write( "Avoid using digits 1 and 2 as x-intercepts </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849263 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849263\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's a possible roller coaster design using a polynomial function, addressing all the requirements:\r<BR>\n" );
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document.write( "**1. Roots/x-intercepts:**\r<BR>\n" );
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document.write( "I've chosen the following x-intercepts, keeping in mind the 120-second requirement and avoiding 1 and 2:\r<BR>\n" );
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document.write( "*   x = 0 (starting point)<BR>\n" );
document.write( "*   x = 10 (first hill)<BR>\n" );
document.write( "*   x = 40 (first valley/tunnel entrance)<BR>\n" );
document.write( "*   x = 40 (double root - tunnel bottom)<BR>\n" );
document.write( "*   x = 60 (tunnel exit)<BR>\n" );
document.write( "*   x = 90 (second hill)<BR>\n" );
document.write( "*   x = 110 (second valley)<BR>\n" );
document.write( "*   x = 120 (ride's end)\r<BR>\n" );
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document.write( "**2. Equation:**\r<BR>\n" );
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document.write( "To create the equation, we'll use the factored form of a polynomial.  Each root corresponds to a factor:\r<BR>\n" );
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document.write( "f(x) = A * x * (x - 10) * (x - 40)² * (x - 60) * (x - 90) * (x - 110) * (x - 120)\r<BR>\n" );
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document.write( "*   **A:**  This is a scaling factor. We'll determine it using the starting height of 250 feet.\r<BR>\n" );
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document.write( "Since f(0) = 250:\r<BR>\n" );
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document.write( "250 = A * 0 * (-10) * (-40)² * (-60) * (-90) * (-110) * (-120)\r<BR>\n" );
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document.write( "Notice that if we plug in zero we will get zero, so we need to change the formula to fit the initial condition, we will add 250 to the formula.\r<BR>\n" );
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document.write( "f(x) = A * x * (x - 10) * (x - 40)² * (x - 60) * (x - 90) * (x - 110) * (x - 120) + 250\r<BR>\n" );
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document.write( "Now plug in zero:\r<BR>\n" );
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document.write( "250 = A * 0 * (-10) * (-40)² * (-60) * (-90) * (-110) * (-120) + 250\r<BR>\n" );
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document.write( "250 = 250\r<BR>\n" );
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document.write( "We still need to find A, to do this we need to select a point in the graph and plug it in, for example f(5) = 300\r<BR>\n" );
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document.write( "300 = A * 5 * (-5) * (-35)² * (-55) * (-85) * (-105) * (-115) + 250\r<BR>\n" );
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document.write( "50 = A * 5 * (-5) * (-35)² * (-55) * (-85) * (-105) * (-115)\r<BR>\n" );
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document.write( "Solving for A:\r<BR>\n" );
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document.write( "A ≈ 1.357e-11\r<BR>\n" );
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document.write( "Therefore, the final equation is:\r<BR>\n" );
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document.write( "f(x) = 1.357e-11 * x * (x - 10) * (x - 40)² * (x - 60) * (x - 90) * (x - 110) * (x - 120) + 250\r<BR>\n" );
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document.write( "**Explanation:**\r<BR>\n" );
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document.write( "*   Each factor (x - root) makes the function equal to zero at that root, creating the x-intercepts.<BR>\n" );
document.write( "*   The double root at x = 40 makes the graph \"kiss\" the x-axis at that point, creating the tunnel.<BR>\n" );
document.write( "*   The scaling factor A controls the overall shape and height of the roller coaster. We solve for A using the initial condition f(0) = 250.\r<BR>\n" );
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document.write( "**Important Notes:**\r<BR>\n" );
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document.write( "*   This equation will likely produce very large numbers due to the high powers of x. Graphing software might struggle with it directly.  You might need to adjust the scaling factor *A* or use a different approach to get a visually pleasing graph.<BR>\n" );
document.write( "*   You can adjust the positions and multiplicities of the roots to fine-tune the shape of your roller coaster.<BR>\n" );
document.write( "*   Be sure to use graphing software (like Desmos or Wolfram Alpha) to visualize the function and make sure it meets all the requirements (3 relative extrema, below ground level, 120 seconds long).  You may have to iterate a few times to get the shape you want.<BR>\n" );
document.write( " </SPAN>\n" );
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