document.write( "Question 1158026: What function has the following characteristics?\r
\n" ); document.write( "\n" ); document.write( "A zero at x = 3
\n" ); document.write( "A hole when x = 5
\n" ); document.write( "A vertical asymptote at x = -1
\n" ); document.write( "A horizontal asymptote at y = 3
\n" ); document.write( "A y-intercept at y = -2 \n" ); document.write( "
Algebra.Com's Answer #854559 by KMST(5385)\"\" \"About 
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There could be other ways to get a function that does that, but we could easily find a ratio of polynomials (a rational function) that could do that.\r
\n" ); document.write( "\n" ); document.write( "Characteristic #1: A zero at x = 3
\n" ); document.write( "Having \"%28x-3%29\" as a factor would do that.
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\n" ); document.write( "Characteristic #2: A hole when x = 5
\n" ); document.write( "Having \"%28x-5%29%2F%28x-5%29\" as a factor (having \"%28x-5%29\" as a factor in numerator and denominator) would do that.
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\n" ); document.write( "Characteristic #3: A vertical asymptote at x = -1
\n" ); document.write( "Having \"%28x-%28-1%29%29=%28x%2B1%29%29\" as a factor only in the denominator would do that.
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\n" ); document.write( "So far we have the building blocks for a function that has the 3 characteristics above,
\n" ); document.write( "\"f%5B1%5D\"\"%28x%29\"\"%22=%22\"\"%28x-3%29%28x-5%29%2F%28x-5%29%2F%28x%2B1%29\"\"%22=%22\"\"%28x%5E2-8x%2B15%29%2F%28x%5E2-4x-5%29\" ,
\n" ); document.write( "with a graph that looks like this:
\n" ); document.write( "\"graph%28300%2C300%2C-14%2C6%2C-10%2C10%2C%28x%5E2-8x%2B15%29%2F%28x%5E2-4x-5%29%29\" .
\n" ); document.write( "It even has a horizontal asymptote, but at \"y=1\" ,
\n" ); document.write( "and its y-intercept is \"f%5B1%5D%280%29=%28-3%29%28-5%29%2F%28-5%29%2F%28%2B1%29=-3\" .
\n" ); document.write( "The asymptote and intercept are not yet what we want.
\n" ); document.write( "If we add \"2\" , the horizontal asymptote becomes \"y=3\", but then \"x=3\" is not a zero.
\n" ); document.write( "If we include \"3\" as another factor, the he horizontal asymptote becomes \"y=3\" , but then the y- intercept becomes \"-9\" .
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\n" ); document.write( "We need to meet the required characteristics #4 and #5 without loosing what we already have achieved.
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\n" ); document.write( "Characteristic #4: A horizontal asymptote at y = 3
\n" ); document.write( "Characteristic #5: A y-intercept at y = -2
\n" ); document.write( "\"f%5B1%5D\"\"%28x%29\" has a \"y=1\" horizontal asymptote because it is the ratio of polynomials of the same degree, and when divide numerator \"x%5E2-8x%2B15\" by denominator \"x%5E2-4x-5\" the quotient is \"1\", and the remainder polynomial degree is less,
\n" ); document.write( "so that we could re-write \"f%5B1%5D\"\"%28x%29\" as \"quotient%2Bremainder%2Fdivisor\" as
\n" ); document.write( "\"f%5B1%5D\"\"%28x%29\"\"%22=%22\"\"%28%28x%5E2-4x-5%29%2B%28-4x%2B20%29%29%2F%28x%5E2-4x-5%29\"\"%22=%22\"\"1%2B%28-4x%2B20%29%2F%28x%5E2-4x-5%29\" ,
\n" ); document.write( "and we know that
\n" ); document.write( "Of course the reason the quotient is \"1\" is that the leading coefficients of \"x%5E2-8x%2B15\" are \"x%5E2-4x-5\" both \"1\", and their ratio is \"1\" .
\n" ); document.write( "The leading coefficient of \"x%5E2-8x%2B15\" is the product of the leading coefficients of the factors \"%28x-3%29\" and \"%28x-5%29\" that we had to include to get characteristics #1 and 2.
\n" ); document.write( "The leading coefficient of \"x%5E2-4x-5\" is the product of the leading coefficients of the factors \"%28x-3%29\" and \"%28x%2B1%29\" that we had to include to get characteristics #2 and #3.
\n" ); document.write( "The value of the y-intercept \"%28-3%29%28-5%29%2F%28-5%29%2F%28%2B1%29=-3\" depended only on the independent terms of factors \"%28x-3%29\" , \"%28x-5%29\" , and \"%28x%2B1%29\" .
\n" ); document.write( "We could include an extra factor \"%28x%2B1%29\" in the denominator,
\n" ); document.write( "and an extra factor \"%283x%2Bb%29\" in the numerator
\n" ); document.write( "to try to achieve characteristics #4 and #5.
\n" ); document.write( "Horizontal asymptote \"y=3\" is achieved by \"f%28x%29=%283x%2Bb%29%28x%5E2-8x%2B15%29%2F%28x%2B1%29%2F%28x%5E2-4x-5%29\"
\n" ); document.write( "because the ratio of leading coefficients would be \"3%2F1=3\" .
\n" ); document.write( "To get \"A y-intercept at y = -2\"we need \"f%280%29=b%2A15%2F%281%28-5%29%29=15b%2F%28-5%29%0D%0A=-3b=-2\"-->\"b=2%2F3%29%29%29%0D%0APutting+it+all+together+we+get%0D%0A%7B%7B%7Bf%28x%29\"\"%22=%22\"\"%283x%2B2%2F3%29%28x-3%29%28x-5%29%2F%28x%2B1%29%2F%28x-5%29%2F%28x%2B1%29\"\"%22=%22\"
\n" ); document.write( "\"%289x%2B2%29%28x-3%29%28x-5%29%2F%28x%2B1%29%5E2%2F%283x-15%29\"\"%22=%22\"\"%289x%5E3-70x%5E2-151x-30%29%2F%283x%5E3-9x%5E2-27x-15%29\"
\n" ); document.write( "The graph would then be
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