document.write( "Question 669204: 13- Show work, graph the function using transformations, state domain, range and asymptotes\r
\n" ); document.write( "\n" ); document.write( "F(x)=5- 1/(x+4)²
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Algebra.Com's Answer #416276 by KMST(5328) $\"\"$ $\"About$

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$\"F%28x%29=5-1%2F%28x%2B4%29%5E2\"$ is related to the functions $\"f%28x%29=1%2Fx\"$ and $\"g%28x%29=1%2Fx%5E2\"$ .
\n" ); document.write( "You know that the function $\"f%28x%29=1%2Fx\"$ has a graph like this:
\n" ); document.write( " $\"graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C1%2Fx%29\"$ .
\n" ); document.write( "The x-axis ( $\"y=0\"$ ) is a horizontal asymptote that the function $\"f%28x%29=1%2Fx\"$ approaches as $\"x\"$ and $\"-x\"$ become larger and larger,
\n" ); document.write( "and you know that the function
\n" ); document.write( "does not exist for $\"x=0\"$ because you cannot divide by zero.
\n" ); document.write( "Instead, you have a vertical asymptote at $\"x=0\"$ ,
\n" ); document.write( "and there is no limit to the values $\"f%28x%29\"$ can take.
\n" ); document.write( "It is never zero, but can take any other value.
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\n" ); document.write( "Something similar happens with the function $\"g%28x%29=1%2Fx%5E2\"$ ,
\n" ); document.write( "which does not exist for $\"x=0\"$ and graphs as $\"graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C1%2Fx%5E2%29\"$
\n" ); document.write( "The only difference is that $\"g%28x%29=1%2Fx%5E2\"$ is never negative.
\n" ); document.write( "The domain is all real numbers except zero; the range is all $\"y%3E0\"$ , and the asymptotes are the same.
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\n" ); document.write( "The function $\"h%28x%29=1%2F%28x%2B4%29%5E2\"$ is like $\"g%28x%29=1%2Fx%5E2\"$ ,
\n" ); document.write( "but everything is shifted 4 units to the left.
\n" ); document.write( "What happened to $\"g%28x%29=1%2Fx%5E2\"$ for $\"x=0\"$ , $\"x=1\"$ , and $\"x\"$
\n" ); document.write( "now happens for $\"h%28x%29=1%2F%28x%2B4%29%5E2\"$ at $\"x=-4\"$ , $\"x=-3\"$ , and $\"x-4\"$ .
\n" ); document.write( "The graph for $\"h%28x%29=1%2F%28x%2B4%29%5E2\"$ is $\"graph%28300%2C300%2C-9%2C1%2C-5%2C5%2C1%2F%28x%2B4%29%5E2%29\"$ .
\n" ); document.write( "The vertical asymptote is $\"x=-4%29%2C%0D%0Aand+the+function+does+not+exist+for+%7B%7B%7Bx=-4\"$ .
\n" ); document.write( "(All other real numbers are in the domain of $\"f%28x%29\"$ ).
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\n" ); document.write( "Adding a minus sign in front just flips the graph.
\n" ); document.write( "Everything becomes the reflection under the x-axis:
\n" ); document.write( " $\"graph%28300%2C300%2C-9%2C1%2C-5%2C5%2C-1%2F%28x%2B4%29%5E2%29\"$
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\n" ); document.write( "Adding $\"5\"$ to that, just moves the graph (and asymptotes) up by 5 units.
\n" ); document.write( " $\"graph%28300%2C300%2C-9%2C1%2C-3%2C7%2C5-1%2F%28x%2B4%29%5E2%2C5%29\"$
\n" ); document.write( "The domain is all the real numbers except $\"x=-4\"$ .
\n" ); document.write( "The vertical asymptote is $\"x=-4\"$ .
\n" ); document.write( "The range is all $\"y%3C5\"$ .
\n" ); document.write( "The horizontal asymptote (shown in green) is $\"y=5\"$ . \n" ); document.write( "

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