Chapter 4.5, Problem 2AYU

To determine

To find: If the graph of a rational function $R$ has the horizontal asymptote $y=2$, the degree of the numerator of $R$ equals the degree of the denominator of $R$. Explain why.

Answer to Problem 2AYU

Therefore a rational function $R$ has the horizontal asymptote $y=2$, the degree of the numerator of $R$ equals the degree of the denominator of $R$.

Explanation of Solution

A horizontal line is an asymptote only to the far left and the far right of the graph. "Far" left or "far" right is defined as anything past the vertical asymptotes or $x\text{-intercepts}$. Horizontal asymptotes are not asymptotic in the middle. It is okay to cross a horizontal asymptote in the middle.

The location of the horizontal asymptote is determined by looking at the degrees of the numerator $\left(\text{n}\right)$ and denominator $\left(\text{m}\right)$.

• If $n<m$, the $x\text{-axis}$, $y=0$ is the horizontal asymptote.
• If $n=m$, then $y=\frac{a_n }{b_m}$ is the horizontal asymptote. That is, the ratio of the leading coefficients.

Therefore a rational function $R$ has the horizontal asymptote $y=2$, the degree of the numerator of $R$ equals the degree of the denominator of $R$.