Problem 112 - Algebra and Trigonometry

Question

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if there is one, of the function's graph.

Step-by-Step Solution

Answer

If degrees of the numerator and the denominator polynomials are equal, the horizontal asymptote would be the ratio of the leading coefficients. If the degree of denominator is greater, the y = 0 is the horizontal asymptote. If the degree of the numerator is greater, there is no horizontal asymptote but there may be a slant asymptote.

Step 1: Identify degrees of polynomials

Take a look at the provided rational function and identify the degrees of the numerator and the denominator polynomials. The degree of a polynomial is the highest exponent of its variable.

Step 2: Case when the degree of denominator is greater

If the degree of the denominator polynomial is greater than that of the numerator, the x-axis (y = 0) is the horizontal asymptote of the function's graph. This is because as x approaches positive or negative infinity, the denominator's high degree causes the function to approach 0.

Step 3: Case when degrees are equal

If the degrees of the numerator and the denominator are equal, the horizontal asymptote is the ratio of the coefficients of the highest degree terms. Given a function in the form \(\frac{ax^n}{bx^n}\), the horizontal asymptote would be at y = a/b.

Step 4: Case when degree of numerator is greater

If the degree of the numerator is greater than the denominator, there is no horizontal asymptote. However, there may be a slant (oblique) asymptote which is found by using polynomial division.