document.write( "Question 987330: State the vertical, horizontal asymptotes and zeros of the rational function, f(x) = (x^2 + 3x + 2)/(x^2 + 5x + 4)
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Algebra.Com's Answer #608074 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "Instead of doing your problem for you, I'll do one exactly in every detail\r\n" );
document.write( "like yours so that you can do yours yourself using this as a model.  You\r\n" );
document.write( "won't learn it as well if I do it for you. I'll do this one instead:\r\n" );
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\n" ); document.write( "State the vertical, horizontal asymptotes and zeros of the rational function,
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document.write( "First we'll get the horizontal asymptote's equation:\r\n" );
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document.write( "Since the numerator and denominator have the same degree, the horizontal\r\n" );
document.write( "asymptote's equation is \r\n" );
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document.write( "That's\r\n" );
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document.write( "Now let's answer the question \"Why is there no zero at x = –3?\"\r\n" );
document.write( "Let's substitute -2 for x and see:\r\n" );
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document.write( "One of the impossible things in mathematics is division\r\n" );
document.write( "by zero.  So that's why I crossed this out, and it's why f(-3)\r\n" );
document.write( "is not a zero.\r\n" );
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document.write( "Now we'll go back and find the vertical asymptotes\r\n" );
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document.write( "We factor numerator and denominator:\r\n" );
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document.write( "Now since the (x+3)'s will cancel, that will give us another function g(x)\r\n" );
document.write( "which is exactly like f(x) except it WILL have a zero at x=-3 whereas f(x)\r\n" );
document.write( "does not have a zero there:\r\n" );
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document.write( "So we get\r\n" );
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document.write( "And we find the vertical asymptote's equation by setting the denominator = 0:\r\n" );
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document.write( "x+4 = 0\r\n" );
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document.write( "So the vertical asymptote's equation is x = -4\r\n" );
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document.write( "Now we graph g(x) by graphing the horizontal and vertical asymptotes (in green)\r\n" );
document.write( "and making a table of values:\r\n" );
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document.write( " x   y\r\n" );
document.write( "-14  2.5\r\n" );
document.write( "-9   3\r\n" );
document.write( "-5   7\r\n" );
document.write( "-3  -3\r\n" );
document.write( " 0   1\r\n" );
document.write( " 6  1.5    \r\n" );
document.write( "  \r\n" );
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document.write( "That's the graph of g(x) but the graph of f(x) must\r\n" );
document.write( "have a hole at (-3,-3) because that's not part of the\r\n" );
document.write( "graph:\r\n" );
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document.write( "Edwin

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