document.write( "Question 1036217: How can i write a rational function with a horizontal asymptote at y=3, a vertical asymptote at x=2, y-intercept at y=4, and is never negative. \n" ); document.write( "
Algebra.Com's Answer #650849 by Edwin McCravy(20055)\"\" \"About 
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How can i write a rational function
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document.write( "That means:\r\n" );
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document.write( "It must be a fractional equation with polynomial numrator and denominator:\r\n" );
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document.write( "\"%22f%28x%29%22=POLYNOMIAL%2FPOLYNOMIAL\"\r\n" );
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document.write( " with a horizontal asymptote at y=3, \r\n" );
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document.write( "That means: \r\n" );
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document.write( "1.  The degree of the numerator and denominator are the same, and\r\n" );
document.write( "    The leading coefficient of the numerator is 3 times the leading\r\n" );
document.write( "    coefficient of the denominator.\r\n" );
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\n" ); document.write( "...is never negative.
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document.write( "That means the numerator and denominator must be the same even \r\n" );
document.write( "power of an expression.  We'll choose the smallest even power 2,\r\n" );
document.write( "of the simplest forms which is linear form with leading coefficient\r\n" );
document.write( "1 and multiply the numerator by 3 so that the leading coefficient\r\n" );
document.write( "of the numerator will be 3 times the leading coefficient of the\r\n" );
document.write( "denominator.  So it is of the form:\r\n" );
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document.write( "\"%22f%28x%29%22=3%28x%2Ba%29%5E2%2F%28x%2Bb%29%5E2\"\r\n" );
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\n" ); document.write( "...a vertical asymptote at x=2,
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document.write( "That means x-2 must be a factor of the denominator. So we make b=-2:\r\n" );
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document.write( "\"%22f%28x%29%22=3%28x%2Ba%29%5E2%2F%28x-2%29%5E2\"\r\n" );
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\n" ); document.write( "...y-intercept at y=4,
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document.write( "This means that it goes through (0,4), so we substitute\r\n" );
document.write( "(x,y) = (0,4)\r\n" );
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document.write( "and since y = f(x) we set f(x)=4 and x=0:\r\n" );
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document.write( "\"%22f%28x%29%22=3%28x%2Ba%29%5E2%2F%28x-2%29%5E2\"\r\n" );
document.write( "\"4=3%280%2Ba%29%5E2%2F%280-2%29%5E2\"\r\n" );
document.write( "\"4=3a%5E2%2F2%5E2%29\"\r\n" );
document.write( "\"4=3a%5E2%2F4\"\r\n" );
document.write( "\"16=3a%5E2\"\r\n" );
document.write( "\"16%2F3=a%5E2\"\r\n" );
document.write( "\"sqrt%2816%2F3%29=a\" <--take positive square root\r\n" );
document.write( "\"4%2Fsqrt%283%29=a\"\r\n" );
document.write( "\"4sqrt%283%29%2F3=a\" <--after rationalizing denominator\r\n" );
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document.write( "Substitute for \"a\":\r\n" );
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document.write( "\"%22f%28x%29%22=3%28x%2B4sqrt%283%29%2F3%29%5E2%2F%28x-2%29%5E2\"\r\n" );
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document.write( "That is a form of the equation.  We can simplify that:\r\n" );
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document.write( "\"%22f%28x%29%22=3%28x%5E2%2Bexpr%288sqrt%283%29%2F3%29x%2B16%2A3%2F9%29%2F%28x-2%29%5E2\"\r\n" );
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document.write( "\"%22f%28x%29%22=3%28x%5E2%2Bexpr%288sqrt%283%29%2F3%29x%2B16%2F3%29%2F%28x-2%29%5E2\"\r\n" );
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document.write( "\"%22f%28x%29%22=%283x%5E2%2B8sqrt%283%29x%2B16%29%2F%28x-2%29%5E2\"\r\n" );
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document.write( "Graph:\r\n" );
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document.write( "Edwin
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