SOLUTION: Graph {{{y=x/(x^2-4)}}}

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Question 149236: Graph y=x%2F%28x%5E2-4%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, let's find the asymptotes of the equation y=%28x%29%2F%28x%5E2-4%29




Horizontal Asymptote:

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

x%5E2-4=0 Set the denominator equal to zero


x%5E2=0%2B4Add 4 to both sides


x%5E2=4 Combine like terms on the right side


x=0%2B-sqrt%284%29 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=%28x%29%2F%28x%5E2-4%29 Start with the given equation.


y=%28-3%29%2F%28%28-3%29%5E2-4%29 Plug in x=-3.


y=-3%2F5 Simplify.

Since the y-value is negative, this means that every point in the interval 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=%28x%29%2F%28x%5E2-4%29 Start with the given equation.


y=%28-1%29%2F%28%28-1%29%5E2-4%29 Plug in x=-1.


y=1%2F3 Simplify.

Since the y-value is positive, this means that every point in the interval 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=%28x%29%2F%28x%5E2-4%29 Start with the given equation.


y=%281%29%2F%28%281%29%5E2-4%29 Plug in x=1.


y=-1%2F3 Simplify.

Since the y-value is negative, this means that every point in the interval 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=%28x%29%2F%28x%5E2-4%29 Start with the given equation.


y=%283%29%2F%28%283%29%5E2-4%29 Plug in x=3.


y=3%2F5 Simplify.

Since the y-value is positive, this means that every point in the interval is above the x-axis.


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So with all of this information, we can now graph the function y=%28x%29%2F%28x%5E2-4%29

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