Calculus: Find Formula for Rational Function w/ Asymptotes & x-intercept

[Wierddemon]

I can't figure this question out, anyone have any ideas?

"Find a formula for a rational function whose horizontal asymptote is y = -8/3 and vertical asymptotes at x = -2 and x = 4 and whose ONLY x-intercept is at x = -5."

 

[courtrigrad]

What is a horizontal asymptote? It is defined as a line $$y = L$$ such that:

$$\lim_{x\rightarrow \infty} f(x) = L$$ or
$$\lim_{x\rightarrow -\infty} f(x) = L$$.

A vertical asymptote is defined as a line $$x=a$$ such that $$\lim_{x\rightarrow a} f(x) = \infty$$. (or $$-\infty$$).

So one function is $$f(x) = \frac{-8x^{2}+200}{3(x-4)(x+2)}$$

 

[Wierddemon]

that isn't right because it never passes through x = -5

 

[courtrigrad]

yes it does. graph it, and look at the value at $$x = -5$$.

 

[Wierddemon]

oh yeah, you're right, but it also has an x value of 5 and the only x-intercept should be -5