Question

Finding the asymptotes of a rational function: Quadratic over linear

Answer 1

Okay, here we have this:

Considering the provided function, we are going to calculate the asymptotes, so we obtain the following:

Vertical asymptotes:

They correspond to the singularities of the functions or zeros of the denominator, in this case:

2x+1=0

2x=-1

x=-1/2

The vertical asymptote is x=-1/2.

Horizontal asymptotes:

Since the degree of the numerator is equal to the degree of the denominator plus 1, the asymptote is steep, and corresponds to the quotient of the polynomial division. Then:

`\begin{gathered} (6x^2+7x-9)/(2x+1) \\ =3x+(4x-9)/(2x+1) \\ =3x+2+(-11)/(2x+1) \end{gathered}`

Therefore the sloped asymptote is:

y=3x+2

Graphing the function and asymptotes: