2 6 Introduction to Rational Functions A rational

2. 6 Introduction to Rational Functions A rational function is a function of the form f(x) = N(x)/D(x), where N and D are both polynomials and D(x) 0. The domain of f is all real x’s except x values that give 0 in the denominator. N(x) and D(x) should have no common factors. Example 1 Domain is all reals except x=0 Plug in x value to find y value. This table shows x approaching 0(the excluded x value) from the left. x f(x) -1 -1/2 -1/1000 -1 -2 -1000 0 This table shows x approaching 0 from the right. x f(x) 0 1/100 1/10 1/2 1 1000 10 2 1

x f(x) -1 -1/2 -1/1000 -1 -2 -1000 x 0 1/100 1/10 1/2 1 1000 10 2 1 f(x) 0 Plot the sets of ordered pairs and this is the graph that you get. Note that as x approaches 0 from the left, f(x) decreases without bound. In contrast, as x approaches 0 from the right, f(x) increases without bound. Remember that the domain is all reals except x=0. What do you notice about the line x=0 and the graph of f ? ? ? The graph never touches the line. This line is a vertical asymptote. f(x) x

Vertical and Horizontal Asymptotes • 1. The line x = a is a vertical asymptote of the graph of f if f(x) as x a, either from the right or left. • 2. The line y = b is a horizontal asymptote of the graph of f if f(x) b as x

Asymptotes of Rational Functions Rules: where N(x) and D(x) have no common factors. 1. The graph of f has vertical asymptotes at the zeros of D(x). 2. The graph of f has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x). 3. 4. 5. a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote. b. If n=m, the line y=an/bm is a horizontal asymptote. c. If n>m , the graph of f has no horizontal

Asymptotes of Rational Functions Rules: • Slant Asymptotes – Only occur when the degree of the top is 1 more than the degree of the bottom. – The S. A. is derived by dividing the top by the bottom (long division or synthetic) and ignoring the remainder.

a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote. b. If n=m, the line y=an/bm is a horizontal asymptote. c. If n>m , the graph of f has no horizontal asymptote. Ex. 2 Find the Horizontal Asymptotes for each of the following functions. n<m therefore the horizontal asymptote is y=0.

a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote. b. If n=m, the line y=an/bm is a horizontal asymptote. c. If n>m , the graph of f has no horizontal asymptote. Ex. 2 Find the Horizontal Asymptotes for each of the following functions. n=m therefore, y=2/3 is the horizontal asymptote.

a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote. b. If n=m, the line y=an/bm is a horizontal asymptote. c. If n>m , the graph of f has no horizontal asymptote. Ex. 2 Find the Horizontal Asymptotes for each of the following functions. n>m, therefore there is no horizontal asymptote. Although this graph does not have a horizontal asymptote it does have a slant or oblique asymptote – the line y=2/3 x.

Ex. 3 For the function f, find a) the domain of f, b) the vertical asymptotes of f, and c) the horizontal asymptote of f. a) Set denominator =0 and solve. b) The graph of f has vertical asymptotes at the zeros of D(x). Therefore the vertical asymptote of f is c) If n=m, the line y=an/bm is a horizontal asymptote. Therefore, the horizontal asymptote is

Ex. 4 A Graph with Two Horizontal Asymptotes A function that is not rational can have two horizontal asymptotes-one to the left and one to the right. For instance, the graph of HA y = -1 HA y = 1

Find the following if possible: domain, vertical asymptote, horizontal asymptote, slant asymptote.

Find the following if possible: domain, vertical asymptote, horizontal asymptote, slant asymptote.

Ultraviolent Radiation • For a person with sensitive skin, the amount of time T (in hours) the person can be exposed to the sun with mininal burning can be modeled by where s is the Sunsor Scale Reading. The Sunsor Scale is based on the level of intensity of UVB rays.

Homework • Page 152 -155 1 -21 odd, 43