How to write an equation for a rational function with: Horizontal asymptote at y = 5?

Aug 12, 2016

Beware!!! Many answers possible.

$\frac{5 {x}^{2}}{{x}^{2} + 4}$

This example will have a horizontal asymptote at $y = 5$ (since the ratio between the highest degrees $=$ 5) and no vertical asymptote (since if ${x}^{2} + 4 = 0 \to {x}^{2} = - 4 \to x = \emptyset$).

You will have a horizontal asymptote at $y = 5$ anytime that the degree of the denominator equals that of the numerator and the ratio between the numerator and the denominator equals $5$. If that is the only asymptote, the denominator when set to $0$ like in the example above needs to have no solution; otherwise there will be vertical asymptotes.

Practice exercises:

$1$. Determine an equation for a rational function with a horizontal asymptote at $y = - 2$

$2.$ Determine an equation for a rational function with vertical asymptotes at $x = - 3$ and $x = 5$ and a horizontal asymptote at $y = 7$.

Hopefully this helps, and good luck!