Rational Functions Horizontal Asymptotes A horizontal asymptote of

Rational Functions: Horizontal Asymptotes: A horizontal asymptote of a rational function is a horizontal line (equation: y = number) such that as values of the independent variable, x, decrease without bound or increase without bound, the function values (y-values) approach (get closer and closer to) the number (from either above or below). The next two slides illustrate the definition. Table of Contents

Rational Functions: Horizontal Asymptotes The rational function shown graphed has horizontal asymptote: y = 2 (dashed line), since as x-values decrease without bound, y-values approach 2 from "below". -2 30 20 10 0 2 4 6 -10 -20 -30 x -1 - 1000 y 0. 5 1. 54 1. 994 Table of Contents x decreasing without bound y approaches 2 from "below" Slide 2

Rational Functions: Horizontal Asymptotes Also note as x-values increase without bound, y-values approach 2 from "above". 30 20 10 -2 0 2 4 6 -10 -20 -30 x 6 10 1000 y 4 2. 86 2. 006 Table of Contents x increasing without bound y approaches 2 from "above" Slide 3

Rational Functions: Horizontal Asymptotes Example 1: Algebraically find the horizontal asymptote of, First, compare the degree of the polynomial in the numerator with the degree of the polynomial in the denominator. There are 3 possibilities to consider. (1) If degree of numerator < degree of denominator, the horizontal asymptote is y = 0 (the x-axis). (2) If degree of numerator > degree of denominator, there is no horizontal asymptote. Table of Contents Slide 4

Rational Functions: Horizontal Asymptotes (3) If degree of numerator = degree of denominator, the horizontal asymptote is In the function, the degrees of the polynomials in both the numerator and denominator are 1, so the horizontal asymptote is found using (3) above. This function’s horizontal asymptote is y = 2. (This was the function shown graphed in the preceding slides!) Table of Contents Slide 5

Rational Functions: Horizontal Asymptotes Try: Algebraically find the horizontal asymptote of, This horizontal asymptote is y = 0. Try: Algebraically find the horizontal asymptote of, This horizontal asymptote is y = 1/4. Table of Contents Slide 6

Rational Functions: Horizontal Asymptotes Table of Contents