Problem 114 - College Algebra

Question

If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does, how do you find its equation?

Step-by-Step Solution

Answer

To tell if a rational function has a slant asymptote, compare the degrees of the numerator and denominator polynomials. If the numerator's degree is exactly one more than the denominator's, a slant asymptote exists. The equation of the slant asymptote is found by applying polynomial division to the numerator and denominator and writing the quotient in \(y=mx+b\) form.

Step 1: Identify whether the denominator's degree is less than the numerator's

Analyse the degrees of the numerator and denominator. If the degree of the numerator is exactly one more than the degree of the denominator, then a slant asymptote exists.

Step 2: Apply Polynomial Division

If a slant asymptote is possible, compute the division of the numerator's polynomial by the denominator's polynomial. The result is the equation of the slant asymptote.

Step 3: Write the Equations of the Slant Asymptote

Write the result of the division you computed in Step 2 as the equation of the slant asymptote. This equation will be of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.