# Math Help - Rational Function question

1. ## Rational Function question

Given the rational function f(x) = 2x^3 - 7x + 3 / x^2 - 4

a) Determine all of the asymptotes of f(x). Explain your work clearly.
b) Make a neat sketch of f(x). Label all important parts.

I determined that the vertical asymptotes are x = -2, 2 (hopefully I'm right), and there are no horizontal asymptotes. I'm wondering if there is a slant asymptote? How can I tell? If there is one how do I find what the slant asymptote is?

How would the graph look like?

Thanks for your time.

2. Originally Posted by zuuberbat
Given the rational function f(x) = 2x^3 - 7x + 3 / x^2 - 4

...
I'm wondering if there is a slant asymptote? How can I tell? If there is one how do I find what the slant asymptote is?

How would the graph look like?

Thanks for your time.
I assume that you mean:

$f(x) = \dfrac{2x^3-7x+3}{x^2-4}$

If so: Do the long division to rearrange the term of the function:

$(2x^3-7x+3) \div (x^2-4) = 2x+\dfrac{x+3}{x^2-4}$

If |x| approaches infinity the value of the fraction is going to be zero. Therefore the asymptote has the equation:

$y= 2x$