Chapter 2 Polynomial and Rational Functions

2.6 Rational Functions and Asymptotes Objectives:  Find the domains of rational functions.  Find the horizontal and vertical asymptotes of graphs of rational functions.  Use rational functions to model and solve real-life problems. 2

Rational Functions  A rational function can be written in the form where N(x) and D(x) are polynomials.  A rational function is not defined at values of x for which D(x) = 0. 3

Reciprocal Function 4

Asymptotes  An asymptote is a boundary line that the graph of a function approaches, but never touches or crosses.  The line x = a is a vertical asymptote of the graph of f if, as x approaches a from either the left or the right, f ( x ) approaches ∞ or –∞.  The line y = b is a horizontal asymptote of the graph of f if, as x approaches ∞ or –∞, f ( x ) approaches b. 5

Examples  The following graphs show horizontal and vertical asymptotes of two rational functions. 6

Finding Asymptotes  Let f be a rational function:  Vertical Asymptotes:  Occur when the denominator equals zero.  Simplify the function if possible.  Set D(x) = 0 and solve for x. 7

Horizontal Asymptotes  The graph of f has at most one horizontal asymptote determined by comparing the degrees of N(x) and D ( x ).  Let n be the degree of the numerator and m be the degree of the denominator.  Let a n be the leading coefficient of the numerator and b n be the leading coefficient of the denominator. 8 If n < mHA: y = 0 If n = mHA: y = a n /b n If n > mNo HA

Examples  Find all HA and VA of each rational function. 9

Example  Find all HA and VA of the rational function. 10

 For a person with sensitive skin, the amount of time T, in hours, the person can be exposed to the sun with a minimal burning can be modeled by where s is the Sunsor Scale reading (based on the level of intensity of UVB rays). a. Find the amount of time a person with sensitive skin can be exposed to the sun with minimal burning when s = 10, s = 25, and s = 100. b. If the model were valid for all s > 0, what would be the horizontal asymptote of this function, and what would it represent? 11

Homework 2.6  Worksheet 2.6 # 7 – 12 (matching), 15, 17, 19, 35, 39 12