document.write( "Question 844729: Describe the vertical asymptote(s) and hole(s) for the graph of y=(x+2)(x+4)/(x+4)(x+1). \n" ); document.write( "

Algebra.Com's Answer #508914 by KMST(5328) $\"\"$ $\"About$

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For $\"x=-4\"$ , $\"y=%28x%2B2%29%28x%2B4%29%2F%28%28x%2B4%29%28x%2B1%29%29\"$ does not exist, because the denominator is zero.
\n" ); document.write( "However, for any $\"x%3C%3E-4\"$ ,
\n" ); document.write( " $\"y=%28x%2B2%29%28x%2B4%29%2F%28%28x%2B4%29%28x%2B1%29%29=%28x%2B2%29%2F%28x%2B1%29\"$ which is continuous at $\"x=-4\"$ .
\n" ); document.write( "So, at $\"x=-4\"$ we have a hole in the graph.
\n" ); document.write( "The function $\"y=%28x%2B2%29%2F%28x%2B1%29\"$ is what results from \"plugging\" that hole.
\n" ); document.write( "For $\"x=-1\"$ , $\"y=%28x%2B2%29%28x%2B4%29%2F%28%28x%2B4%29%28x%2B1%29%29\"$ does not exist, and there is no equivalent continuous function. (No plugging possible).
\n" ); document.write( "At $\"x=-1\"$ , the functions $\"y=%28x%2B2%29%28x%2B4%29%2F%28%28x%2B4%29%28x%2B1%29%29\"$ and $\"y=%28x%2B2%29%2F%28x%2B1%29\"$ have a vertical asymptote.
\n" ); document.write( "The function changes sign at that point.
\n" ); document.write( " $\"y=%28x%2B2%29%2F%28x%2B1%29\"$ is positive for $\"x%3E-1\"$ , where $\"x%2B1%3E0\"$ and $\"x%2B2%3E0\"$ .
\n" ); document.write( "It is negative for $\"-2%3Cx%3C-1\"$ , where $\"x%2B1%3C0\"$ and $\"x%2B2%3E0\"$ .
\n" ); document.write( " $\"graph%28300%2C300%2C-6%2C4%2C-10%2C10%2C%28x%2B2%29%2F%28x%2B1%29%2C200%28x%2B1%29%29\"$ \n" ); document.write( "

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