Rational Fcn f(x) = 2x^3-7x+3/x^2-4, find asympt., graph

[zuuberbat]

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 after I found all the asymptotes?

Thanks for your time.

 

[mmm4444bot]

Re: Rational Function Question

... I determined that the vertical asymptotes are x = -2, 2 ...YUP, X = 2 AND X = -2 ARE EQUATIONS FOR THESE ASYMPTOTES

and there are no horizontal asymptotes. YUP, YUP ... (don't forget to explain your methods)

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? See below.

How would the graph look like after I found all the asymptotes? You'll know after you sketch it ...

Hello Über Bat:

Either you have a textbook that does not list the steps for sketching graphs of rational functions, or you did not read it.

Same goes for determining slant asymptotes.

I'm too tired to type up a precise listing, so I'm going to paraphrase.

N = degree of numerator

D = degree of denominator

If N < D, then y = 0 is the horizontal asymptote equation

If N = D, then y = [ratio of leading coefficients] is the horizontal asymptote equation

If N > D, then there are no horizontal asymptotes

If N = D + 1, then there is a slant asymptote.

Use polynomial division (either longhand or synthetic) to find the quotient and remainder. Since the remainder vanishes as x blows up (positively or negatively), the quotient defines the slant asymptote's line.

TO SKETCH:

1) Find the y-intercept

2) Find the x-intercept(s)

3) Test values of x on either side of each vertical asymptote to determine if the function is going to positive or negative infinity

4) Determine any horiztonal or slant asymptote

5) Use all of this information to determine the general shape of the graph, and calculate any additional coordinates needed to draw a "nice" sketch

Cheers,

~ Mark