Question 631227
<font face="Times New Roman" size="+2">

WARNING! Danger, Will Robinson.  Terminology Error!  *[tex \LARGE y\ =\ 2x\ -\ 4] is NOT a horizontal asymptote.  Horizontal asymptotes are horizontal, hence the name.  Linear functions with a non-zero (2 is not zero) slope are NOT horizontal.  Asymptotes that are slanted or oblique are called Slant Asymptotes or Oblique Asymptotes.

The fact that the function has zeros at -2 and 3 tells us that the factors of the numerator polynomial are *[tex \LARGE (x\ +\ 2)], *[tex \LARGE (x\ -\ 3)], and some constant *[tex \LARGE k] (because all polynomial equations *[tex \LARGE k\left(p(x)\right)\ =\ 0] where *[tex \LARGE p(x)] is a polynomial with degree *[tex \LARGE n] and *[tex \LARGE k\ \neq\ 0]have identical solution sets)

Hence, the numerator is *[tex \LARGE kx^2\ -\ kx\ -\ 6k]

The fact that the function has a vertical asymptote of *[tex \LARGE -1] means that the denominator polynomial has a zero at *[tex \LARGE -1], therefore the denominator polynomial must be *[tex \LARGE x\ +\ 1].

If a rational function has a numerator that is one degree greater than the degree of the denominator, then the function will have a slant asymptote equal to the quotient of a polynomial long division of the numerator by the denominator.

Perform the polynomial long division of *[tex \LARGE kx^2\ -\ kx\ -\ 6k\ \div\ x\ +\ 1].

Your quotient will have a factor of *[tex \LARGE k] in it, but if you set the quotient equal to the given slant asymptote *[tex \LARGE 2x\ -\ 4], you will very quickly see the value of *[tex \LARGE k].

Then it is simply a matter of constructing your function from the derived numerator and denominator.

Go to <a href="http://www.purplemath.com/modules/polydiv2.htm">Purple Math Polynomial Long Division</a> if you need a refresher on polynomial long division.

John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>