How do you graph this rational function?

Question

(x)/(x^2-4)

i found the x and y intercept to be 0.

there are vertical asymptotes at -2 and +2 cause i made the denominator 0.

i dont know how to do the horizontal asymptote (i think it might be a slant asymptote?) maybe synthetic divison?

can someone tell me what the equation of the line would be? oh and the graphing calculator doesnt help

Answer 1

In order to make sense of this graph we have to find the limits of the critical numbers of the equation.

Very good finding the vertical asymptotes!  Now just find out how the graph goes near them.  For instance as x/(x^2-4) approaches -2 from the left (using values the get closer to x=-2, such as -3,-2.9 etc.), the numerator will always be negative, and the denominator always positive, but as it approaches -2 from the left the denominator shrinks and shrinks, making the value enormously negative.

Now you do this as x approaches -2 from the right, 2 from the left, and 2 from the right.

In order to find the horizontal asymptote, find what happens when x approaches + and - infinity.  As it approaches infinity the equation looks more and more like x/(x^2), (because the -4 is not going to make much of a difference with such a big number as infinity) which simplifies to 1/x. Therefore, as x approaches - infinity the line will approach 0 from the bottom (because 1/-infinity is a little negative and almost 0), and likewise as x approach + infinity the line will approach 0 from the top.

Hope this helps! (Wish I could just draw it out for you, makes it way less wordy)

Answer 2

The horizontal asymptote is the x-axis.

When degree of numerator is < degree of denominator, the

horizontal asymptote is the x-axis.

Answer 3

To find horizontal asymptote, you look at the highest degree of x's in the numerator and denominator:

(x) / (x^2)

Now, since the degree of the polynomial on the bottom is larger than on the top, the horizontal asymptote is y=0. if it were reverse, there is no asymptote and if they are the same you just take the leading coefficient of the top and bottom and divide them together.

Now just plot points to figure out where the graph goes. there are 3 sections: x < -2, -2 < x < 2, & x > 2.

x | y

-4 -1/3

-3 -3/5

---------

-1 1/4

0 0

1 -1/4

---------

3 3/5

4 1/3

So in the first section it should come fromt he left really close but below to y=0, then go down to negative infinity at x = -2. Then it should look like a x^3 function in the middle section, then it should come from infinity at x= 2 and drop down next to y=0 and continue right.

I hope that helps!