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

Question 149236: Graph
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
First, let's find the asymptotes of the equation

Horizontal Asymptote:

Since the degree of the numerator (which is ) is less than the degree of the denominator (which is ), the horizontal asymptote is always

So the horizontal asymptote is

--------------------------------------------------

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

Set the denominator equal to zero

Add 4 to both sides

Combine like terms on the right side

Take the square root of both sides

or Break up the expression and simplify.

So the vertical asymptotes are or

------------------

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

So let's plug in

Start with the given equation.

Plug in .

Simplify.

Since the y-value is negative, this means that every point in the interval

is below the x-axis.

------

Region 2:

This region lies between the vertical asymptote and the x-axis

So let's plug in

Start with the given equation.

Plug in .

Simplify.

Since the y-value is positive, this means that every point in the interval

is above the x-axis.

------

Region 3:

This region lies between the x-axis and the vertical asymptote

So let's plug in

Start with the given equation.

Plug in .

Simplify.

Since the y-value is negative, this means that every point in the interval

is below the x-axis.

-----------------
Region 3:

This region lies to the right of the vertical asymptote

So let's plug in

Start with the given equation.

Plug in .

Simplify.

Since the y-value is positive, this means that every point in the interval

is above the x-axis.

----------------

So with all of this information, we can now graph the function

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