document.write( "Question 1113152: How can I create a \"Roller Coaster\" polynomial with the following requirements:
\n" ); document.write( "- at least 3 relative maxima and/or minima
\n" ); document.write( "- at least 2 real roots and one of the zero must have a multiplicity of 2
\n" ); document.write( "- the ride length must be at least 4.5 minutes
\n" ); document.write( "- the ride starts at 450 feet
\n" ); document.write( "- the ride dives below the ground into a tunnel at least once
\n" ); document.write( "Thanks!
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #728309 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
This is not a physics problem. and a realistic roller coaster design is not expected.
\n" ); document.write( "You just make \"x\"=time in minutes since the start of the ride,
\n" ); document.write( "and \"h%28x%29\" is the polynomial function representing height of the roller coaster car in feet as a function of time,
\n" ); document.write( "with a domain like [0, 4.5] for a 4.5-minute ride, or [0, 5] for a 5-minute ride.
\n" ); document.write( "A graphing calculator would help immensely.
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\n" ); document.write( "The graphs of higher degree polynomials look somewhat like roller coasters.
\n" ); document.write( "For example, a degree 4 polynomial with a positive leading coefficient could look like this, with 2 relative minima and one relative maximum.
\n" ); document.write( "\"graph%28300%2C300%2C-1%2C9%2C-5%2C5%2C%28x-1%29%28x-2%29%28x-3%29%28x-4.5%29%29\"
\n" ); document.write( "That would satisfy the first condition \"at least 3 relative maxima and/or minima\".
\n" ); document.write( "A higher degree could give you more relative maxima and minima.
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\n" ); document.write( "The real roots (zeros) are the points where the graph crosses the x-axis,
\n" ); document.write( "and if that happens for a value \"x=a\" ,
\n" ); document.write( "it means that \"%28x-a%29\" is a factor of the polynomial.
\n" ); document.write( "The graph above shows the function \"f%28x%29=%28x-1%29%28x-2%29%28x-3%29%28x-4.5%29\" ,
\n" ); document.write( "and that satisfies some of the roller coaster problem requirements.
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\n" ); document.write( "If \"x=b\" is a zero of multiplicity 2, it means \"%28x-b%29%5E2\" is a factor,
\n" ); document.write( "and the function is tangent to the axis at that point, but does not cross the axis.
\n" ); document.write( "To have a zero of multiplicity 2, you could use something like
\n" ); document.write( "\"f%28x%29=%28x-1%29%5E2%28x-3%29%28x-5%29\" .
\n" ); document.write( "That would give you a graph that dips below the x-axis between \"x=3\" and \"x=5\"
\n" ); document.write( "(the roller coaster \"dives below the ground into a tunnel\") once between 3 and 5 minutes),
\n" ); document.write( "has real roots/zeros at \"x=1\" , \"x=3\" , and \"x=5\" ,
\n" ); document.write( "and the \"x=1\" zero has multiplicity 2.
\n" ); document.write( "The graph for that is
\n" ); document.write( "\"graph%28300%2C300%2C-1%2C9%2C-10%2C10%2C%28x-1%29%5E2%28x-3%29%28x-5%29%29\"
\n" ); document.write( "In that case, the ride could end at ground level at x=5,
\n" ); document.write( "but maybe you do not want to end the ride at a point when the roller coaster is going up so sharply.
\n" ); document.write( "To end at height zero with the roller coaster going neither up nor down,
\n" ); document.write( "you can give the last zero multiplicity 2, as in
\n" ); document.write( "\"f%28x%29=-%28x-1%29%5E2%28x-3%29%28x-5%29%5E2\" : \"graph%28300%2C300%2C-1%2C9%2C-15%2C35%2C-%28x-1%29%5E2%28x-3%29%28x-5%29%5E2%29\" ,
\n" ); document.write( "
\n" ); document.write( "or \"f%28x%29=-%28x-1%29%5E2%28x-3%29%28x-4.5%29%5E2\" : \"graph%28300%2C300%2C-1%2C9%2C-15%2C35%2C-%28x-1%29%5E2%28x-3%29%28x-4.5%29%5E2%29\" .
\n" ); document.write( "With a graphing calculator, you can try different functions to find a graph shape you like.
\n" ); document.write( "Maybe you would prefer \"f%28x%29=-%28x-1%29%5E2%28x-4%29%28x-6%29%5E2\" : \"graph%28300%2C300%2C-1%2C9%2C-20%2C80%2C-%28x-1%29%5E2%28x-4%29%28x-6%29%5E2%29\" for a 6-minute ride.
\n" ); document.write( "Maybe you want a higher degree polynomial for a complicated ride with more ups and downs.
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\n" ); document.write( "The last requirement you have to meet is \"the ride starts at 450 feet\".
\n" ); document.write( "For that, you need \"h%280%29=450\" .
\n" ); document.write( "
\n" ); document.write( "For \"f%28x%29=-%28x-1%29%5E2%28x-3%29%28x-5%29%5E2\" , \"f%280%29=-%28-1%29%5E2%28-3%29%28-5%29%5E2=1%2A3%2A25=75\"
\n" ); document.write( "If you multiply that function times \"450%2F75=6\" , you \"stretch\" the graph vertically, with the same general shape,
\n" ); document.write( "and have \"h%28x%29=-6%28x-1%29%5E2%28x-3%29%28x-5%29%5E2\" , with \"h%280%29=450\" .
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\n" ); document.write( "From \"f%28x%29=-%28x-1%29%5E2%28x-4%29%28x-6%29%5E2\" , with \"f%280%29=1%2A4%2A36=144\" ,
\n" ); document.write( "multiplying times \"450%2F144=3.125\" you would get
\n" ); document.write( "\"h%28x%29=-3.125%28x-1%29%5E2%28x-4%29%28x-6%29%5E2\" . \n" ); document.write( "
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