1 OBJECTIVE TSW (1) determine vertical, horizontal, and oblique asymptotes, and (2) quiz over sec ASSIGNMENTS DUE WS Sec. 3.4 – Day 3  wire basket Sec. 3.4: pp (47-63 odd)  black tray TODAY’S ASSIGNMENTS − WS Sec. 3.5: Oblique Asymptotes − Due on Monday, 07 March − Sec. 3.5: p. 353 (37-46 all) − Due on Wednesday/Thursday, March QUIZ: Sec. 3.4 will be given after the lesson. TEST: Sec. 3.4 – 3.6 is on Wednesday/Thursday, 09/10 March College Algebra K/DC Friday, 04 March 2016

3-2 Due on Wednesday/Thursday, March Sec. 3.5: p. 353 (37-46 all) Due on Wednesday/Thursday, March Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. If there are none, write NONE.

3-3 Rational Functions: Graphs, Applications, and Models 3.5 Asymptotes

An asymptote is a line that a function gets very close to as x gets close to some specific value or as x  |∞|. There are three types of asymptotes that a rational function may have: (1) vertical, (2) horizontal, and (3) oblique (slanted). Not all rational functions will have each kind. Some rational functions may have vertical and horizontal, some may have vertical and oblique, some may have vertical only, some may not have any. 3-4

Asymptotes In order to find asymptotes, a rational function (fraction) must be in lowest terms. a 1)Vertical asymptote(s) Set the denominator equal to 0 and solve. If a is a zero of the denominator, then the line x = a is a vertical asymptote. 3-5

Asymptotes In order to find asymptotes, a rational function (fraction) must be in lowest terms. 2)Horizontal Asymptote(s) a)If the numerator has a lower degree than the denominator, then the horizontal line y = 0 is a horizontal asymptote. 3-6

Asymptotes In order to find asymptotes, a rational function (fraction) must be in lowest terms. 2)Horizontal Asymptote(s) a n b n b)If the numerator and denominator have the same degree, then the horizontal asymptote is where a n and b n are the leading coefficients of the numerator and denominator, respectively. 3-7

Asymptotes In order to find asymptotes, a rational function (fraction) must be in lowest terms. 3)Oblique Asymptotes If the numerator’s degree is exactly one more than the denominator’s, then there will be an oblique (slanted) asymptote. To find it, divide the numerator by the denominator and disregard the remainder. Set the rest of the quotient equal to y to get the equation of the asymptote. 3-8

Finding Asymptotes of Rational Functions Find all asymptotes of the function 3-9 To find the vertical asymptotes, set the denominator equal to 0 and solve. The equations of the vertical asymptotes are x = –4 and x = 4. The numerator has lower degree than the denominator, so the horizontal asymptote is y = 0. There is no oblique asymptote.

Finding Asymptotes of Rational Functions Find all asymptotes of the function 3-10 To find the vertical asymptotes, set the denominator equal to 0 and solve. The equation of the vertical asymptote is. The numerator and the denominator have the same degree, so the equation of the horizontal asymptote is The is no oblique asymptote.

Finding Asymptotes of Rational Functions Find all asymptotes of the function 3-11 To find the vertical asymptotes, set the denominator equal to 0 and solve. The equation of the vertical asymptote is x = 3. Since the degree of the numerator is exactly one more than the denominator, there is no horizontal asymptote, but there is an oblique asymptote.

Finding Asymptotes of Rational Functions Divide the numerator by the denominator and, disregarding the remainder; set the rest of the quotient equal to y to obtain the equation of the asymptote The equation of the oblique asymptote is y = 2x + 6.

3-13 Due on Wednesday/Thursday, March Sec. 3.5: p. 353 (37-46 all) Due on Wednesday/Thursday, March Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. If there are none, write NONE.