Question 149236
First, let's find the asymptotes of the equation {{{y=(x)/(x^2-4)}}}





<b> Horizontal Asymptote: </b>


Since the degree of the numerator (which is {{{1}}}) is less than the degree of the denominator (which is {{{2}}}), the horizontal asymptote is always {{{y=0}}}


So the horizontal asymptote is {{{y=0}}}




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<b> Vertical Asymptote: </b>

To find the vertical asymptote, just set the denominator equal to zero and solve for x


{{{x^2-4=0}}} Set the denominator equal to zero



{{{x^2=0+4}}}Add 4 to both sides



{{{x^2=4}}} Combine like terms on the right side



{{{x=0+-sqrt(4)}}} Take the square root of both sides



{{{x=2}}} or {{{x=-2}}} Break up the expression and simplify.



So the vertical asymptotes are {{{x=-2}}} or {{{x=2}}} 




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Now we need to test each region to see if it lies above or below the x-axis


Region 1:


This region is to the left of the vertical asymptote {{{x=-2}}}


So let's plug in {{{x=-3}}}


{{{y=(x)/(x^2-4)}}} Start with the given equation.



{{{y=(-3)/((-3)^2-4)}}} Plug in {{{x=-3}}}.



{{{y=-3/5}}} Simplify.


Since the y-value is negative, this means that <font size=4><b>every</b></font> point in the interval *[Tex \LARGE \left(-\infty,-2\right)] is below the x-axis.


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Region 2:


This region lies between the vertical asymptote {{{x=-2}}} and  the x-axis {{{x=0}}}


So let's plug in {{{x=-1}}}


{{{y=(x)/(x^2-4)}}} Start with the given equation.



{{{y=(-1)/((-1)^2-4)}}} Plug in {{{x=-1}}}.



{{{y=1/3}}} Simplify.


Since the y-value is positive, this means that <font size=4><b>every</b></font> point in the interval *[Tex \LARGE \left(-2,0\right)] is above the x-axis.



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Region 3:


This region lies between the x-axis {{{x=0}}} and the vertical asymptote {{{x=2}}} 


So let's plug in {{{x=1}}}


{{{y=(x)/(x^2-4)}}} Start with the given equation.



{{{y=(1)/((1)^2-4)}}} Plug in {{{x=1}}}.



{{{y=-1/3}}} Simplify.


Since the y-value is negative, this means that <font size=4><b>every</b></font> point in the interval *[Tex \LARGE \left(0,2\right)] is below the x-axis.


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Region 3: 


This region lies to the right of the vertical asymptote {{{x=2}}} 




So let's plug in {{{x=3}}}


{{{y=(x)/(x^2-4)}}} Start with the given equation.



{{{y=(3)/((3)^2-4)}}} Plug in {{{x=3}}}.



{{{y=3/5}}} Simplify.


Since the y-value is positive, this means that <font size=4><b>every</b></font> point in the interval *[Tex \LARGE \left(2,\infty\right)] is above the x-axis.



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So with all of this information, we can now graph the function {{{y=(x)/(x^2-4)}}} 


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