Rational Function

Question

Write a rational function F that has the specified characteristics. Vertical asymptote: None Horizontal asymptote: y=2

Answer 1

Hi Ashleigh!

Basically with this you want to pick a function such that the numerator and denominator have the same degree (giving us a nonzero but finite horizontal asymptote) and the denominator is never zero (which eliminates vertical asymptotes). I would pick the denominator first. Typically to get it to never be zero, you use a quadratic expression like x²+1, something where it is the so-called "sum of squares" that does not factor. Notice with this expression the leading coefficient is 1. If I want a horizontal asymptote of 2, then for the numerator I pick any quadratic expression such that the leading coefficient is 2, say 2x²-4x+3 (it doesn't matter if this factors or not). Then by the rules for finding horizontal asymptotes, here it would be y=2/1 or y=2 as desired. So F(x)=(2x²-4x+3)/(x²+1) is an example of a rational function with the desired characteristics. There are many others as long as you follow the same ideas.

I hope this helps!