(4, 2) (2, 0) Vertical shift two units downward (2, 4) (0, 2) (1, 2) ( 1, 0) Horizontal shrink each x-value is multiplied by 1 2

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1 Section. Rational Functions 9. i i i i 9i. i i i. i 7i i i i i. 9 i9 i i. g f. g f. g f, ), ), ), ), ), ), ), ), ), ), ), ) Horizontal shift two units to the right Vertical shift two units downward Vertical stretch each -value is multiplied b ). g f 7. g f. g f, ), ), ), ) Reflection in the -ais, ), ), ), ) Horizontal shrink each -value is multiplied b, ), ), ), ) Horizontal stretch each -value is multiplied b Section. Rational Functions You should know the following basic facts about rational functions. a) A function of the form f ND, D, where N and D are polnomials, is called a rational function. b) The domain of a rational function is the set of all real numbers ecept those which make the denominator zero. c) If f ND is in reduced form, and a is a value such that Da, then the line a is a vertical asmptote of the graph of f. f or f as a. The line b is a horizontal asmptote of the graph of f if f b as or. e) Let f N where N and D have no common factors. D a n n a n n... a a b m m b m m... b b. If n < m, then the -ais is a horizontal asmptote.. If then a n n m, is a horizontal asmptote. b m. If n > m, then there are no horizontal asmptotes. Vocabular Check. rational functions. vertical asmptote. horizontal asmptote. slant asmptote

2 Chapter Polnomial and Rational Functions. f a) f... f f... b) The zero of the denominator is, so is a vertical asmptote. The degree of the numerator is less than the degree of the denominator so the -ais, or, is a horizontal asmptote. c) The domain is all real numbers ecept f a) f f.. f... b) The zero of the denominator is, so is a vertical asmptote. The degree of the numerator is equal to the degree of the denominator, so the line is a horizontal asmptote. c) The domain is all real numbers ecept f a)..9 f.79.. f. 7.9 f.. b) The zeros of the denominator are ± so both and are vertical asmptotes. Since the degree of the numerator equals the degree of the denominator, is a horizontal asmptote c) The domain is all real numbers ecept ± f a). f..9.9 f....9 f.. b) The zeros of the denominator are ± so both and are vertical asmptotes. Because the degree of the numerator is less than the degree of the denominator, the -ais or is a horizontal asmptote c) The domain is all real numbers ecept ± f. Domain: all real numbers ecept Vertical asmptote: Horizontal asmptote: Degree of N < degree of D f Domain: all real numbers ecept Vertical asmptote: Horizontal asmptote: Degree of N < degree of D

3 Section. Rational Functions 7 7. f. f Domain: all real numbers ecept Vertical asmptote: Domain: all real numbers ecept Horizontal asmptote: Vertical asmptote: Degree of N degree of D Horizontal asmptote: Degree of N degree of D 9. f. f Domain: all real numbers ecept ± Domain: all real numbers ecept Vertical asmptotes: ± Vertical asmptote: Horizontal asmptote: None Horizontal asmptote: None Degree of N > degree of D Degree of N > degree of D. f 9. f Domain: All real numbers. The denominator has no real zeros. [Tr the Quadratic Formula on the denominator.] Domain: All real numbers. The denominator has no real zeros. [Tr the Quadratic Formula on the denominator.] Vertical asmptote: None Vertical asmptote: None Horizontal asmptote: Horizontal asmptote: Degree of N degree of D Degree of N degree of D. f. f. f Vertical asmptote: Vertical asmptote: Vertical asmptote: Horizontal asmptote: Horizontal asmptote: Horizontal asmptote: Matches graph. Matches graph a). Matches graph c).. f 7. g. h Vertical asmptote: Horizontal asmptote: Matches graph b). The onl zero of g is. makes g undefined. No real solution, h has no real zeros.

4 Chapter Polnomial and Rational Functions 9. f is a zero of f.. g is a real zero of g.. f,. f 9, Domain: all real numbers ecept ± Domain: all real numbers ecept ± Horizontal asmptote: Degree of N < degree of D Vertical asmptote: Since is a common factor of N and D, is not a vertical asmptote of f. The degree of the numerator is less than the degree of the denominator, so the graph has the line as a horizontal asmptote. Vertical asmptote: Since is a common factor of N and D, is not a vertical asmptote of f.. f Domain: all real numbers ecept and Horizontal asmptote: Degree of N degree of D,. Vertical asmptote: Since is a common factor of N and D, is not a vertical asmptote of f. f, Domain: all real numbers ecept and Horizontal asmptote: Degree of N degree of D Vertical asmptote: Since is a common factor of N and D, is not a vertical asmptote of f.. f. f 7,, Domain: all real numbers ecept and Domain: all real numbers ecept or Horizontal asmptote: Degree of N degree of D Vertical asmptote: Since is a common factor of N and D, is not a vertical asmptote of f. Horizontal asmptote: Degree of N degree of D Vertical asmptote: Since is a common factor of N and D, is not a vertical asmptote of f.

5 Section. Rational Functions 9 7. f. f a) Domain: all real numbers ecept a) Domain: all real numbers ecept b) -intercept:, b) -intercept:, c) Vertical asmptote: Horizontal asmptote: c) Vertical asmptote: Horizontal asmptote:,, 9. h. g a) Domain: all real numbers ecept a) Domain: all real numbers ecept b) -intercept:, b) -intercept:, c) Vertical asmptote: Horizontal asmptote: c) Vertical asmptote: Horizontal asmptote:,, Note: This is the graph of f Eercise 7) reflected about the -ais. Note: This is the graph of f Eercise ) reflected about the -ais.

6 Chapter Polnomial and Rational Functions. C. P a) Domain: all real numbers ecept a) Domain: all real numbers ecept b) -intercept:, b) -intercept:, -intercept:, -intercept:, c) Vertical asmptote: Horizontal asmptote: c) Vertical asmptote: Horizontal asmptote: C 7,, ) ), ),. f 9. ft t t t t a) Domain: all real numbers a) Domain: all real numbers t ecept t b) Intercept: c) Horizontal asmptote: ±, ± ± b) t-intercept:, c) Vertical asmptote: t Horizontal asmptote: t, ) t, )

7 Section. Rational Functions. gs s s. f a) Domain: all real numbers s a) Domain: all real numbers ecept b) Intercept: c) Horizontal asmptote:, s b) -intercept:, c) Vertical asmptote: Horizontal asmptote: gs 9 7 9, ) s, 7. h. g 9 a) Domain: all real numbers ecept ± a) Domain: all real numbers ecept ± b) -intercepts:,,, -intercept:, c) Vertical asmptotes:, Horizontal asmptote: b) -intercept: -intercepts:,,, c) Vertical asmptotes: ± Horizontal asmptote: 7, ,.), ), ), ), )

8 Chapter Polnomial and Rational Functions 9. f. f a) Domain: all real numbers ecept, ± b) -intercepts: -intercept:,,,, a) Domain: all real numbers ecept,, or b) -intercepts: -intercept:,,,, c) Vertical asmptotes:, and Horizontal asmptotes: c) Vertical asmptotes:,, Horizontal asmptote:. f 9 9,, 9, ), ),, ). f,. f, a) Domain: all real numbers ecept and a) Domain: all real numbers ecept or b) Intercept: c) Vertical asmptote: Horizontal asmptote:, ), b) -intercept: -intercept: none c) Vertical asmptote: Horizontal asmptote:,.), 7

9 Section. Rational Functions. f. f,, a) Domain: all real numbers ecept and a) Domain: all real numbers ecept or b) -intercept: -intercept:,, b) -intercept:, -intercept:, c) Vertical asmptote: c) Vertical asmptote: Horizontal asmptote: 7 Horizontal asmptote:, ), ) ) ),, ). f t t t a) Domain: all real numbers t ecept t b) t-intercept:, -intercept:, c) Vertical asmptote: none Horizontal asmptote: none t t t t, t. f a) Domain: all real numbers b) -intercept:, -intercept:, c) Vertical asmptote: none Horizontal asmptote: none t 9,, ), ) t, ), )

10 Chapter Polnomial and Rational Functions 7. f, g a) Domain of f : all real numbers ecept Domain of g: all real numbers b) Because is a factor of both the numerator and the denominator of f, is not a vertical asmptote. f has no vertical asmptotes. c).. f. Undef... f, g a) Domain of f : all real numbers ecept and Domain of g: all real numbers b) Since is a factor of both the numerator and the denominator of f, neither nor is a vertical asmptote of f. Thus, f has no vertical asmptotes. c).. g.. f Undef.. Undef.. g).. e) Because there are onl a finite number of piels, the utilit ma not attempt to evaluate the function where it does not eist. e) Because there are onl a finite number of piels, the utilit ma not attempt to evaluate the function where it does not eist. 9. f, g. f, g 7 a) Domain of f : all real numbers ecept and Domain of g: all real numbers ecept b) Because is a factor of both the numerator and the denominator of f, is not a vertical asmptote. The onl vertical asmptote of f is. c)... f g Undef. Undef. Undef. a) Domain of f : all real numbers ecept and Domain of g: all real numbers ecept b) Since is a factor of both the numerator and the denominator of f, is not a vertical asmptote of f. Thus, f has as its onl vertical asmptote. c) f Undef. Undef. g) Undef. e) Because there are onl a finite number of piels, the utilit ma not attempt to evaluate the function where it does not eist. e) Because there are onl a finite number of piels, the utilit ma not attempt to evaluate the function where it does not eist.

11 Section. Rational Functions. h a) Domain: all real numbers ecept b) Intercepts: c) Vertical asmptote: Slant asmptote:,,,. a) Domain: all real numbers ecept b) No intercepts c) Vertical asmptote: Slant asmptote: g 9 9 =, ), ) =. f a) Domain: all real numbers ecept b) No intercepts c) Vertical asmptote: Slant asmptote: = f f a) Domain: all real numbers ecept b) -intercepts:,,, c) Vertical asmptote: Slant asmptote: f =, ), ). g a) Domain: all real numbers ecept b) No intercepts c) Vertical asmptote: Slant asmptote: g =

12 Chapter Polnomial and Rational Functions. h 7. ft t t t t a) Domain: all real numbers ecept a) Domain: all real numbers t ecept t b) Intercept: c) Vertical asmptote: Slant asmptote: h, b) Intercept: c) Vertical asmptote: t Slant asmptote: t t 7, 7 7 = +, ) = t,.) t. f 9 9 a) Domain: all real numbers ecept 9. f a) Domain: all real numbers ecept ± b) Intercept:, b) Intercept:, c) Vertical asmptote: c) Vertical asmptotes: ± Slant asmptote: Slant asmptote: 9 f 9 7 =, ) = 9, )

13 Section. Rational Functions 7. g. f a) Domain: all real numbers ecept ± b) Intercept: c) Vertical asmptotes: Slant asmptote: g, 7 ± 7 a) Domain: all real numbers ecept b) -intercept: c) Vertical asmptote: Slant asmptote:, f 7 =, ), ) =. f. f a) Domain: all real numbers ecept b) -intercept:, c) Vertical asmptote: Slant asmptote: , 9 = 7 7 a) Domain: all real numbers ecept and b) intercept:, - - intercepts: c) Vertical asmptote: Slant asmptote: 7, 7,,,,,.), )., ) = 7

14 Chapter Polnomial and Rational Functions. f. f 7 9, a) Domain: all real numbers ecept or b) -intercept:, -intercepts:,,, Domain: all real numbers ecept -intercept:, Vertical asmptote: Slant asmptote: c) Vertical asmptote: Slant asmptote: 7 Line:, ), = + 7, ). f 7. g Domain: all real numbers ecept Domain: all real numbers ecept Vertical asmptote: Vertical asmptote: Slant asmptote: Slant asmptote: Line: Line:. h 9. Domain: all real numbers ecept a) -intercept:, Vertical asmptote: Slant asmptote: b) Line: 7. a) -intercept:, 7. b) a) -intercepts: ±, b) ±

15 Section. Rational Functions 9 7. a) -intercepts:,,, 7. b), C a) b) p p, p <, C. million dollars C 7 million dollars C7 7 7 million dollars 7 c) C as. No, it would not be possible to remove % of the pollutants. 7. C,p p, p < 7. t N.t, t a), a) N deer b) C, The cost would be $.7. C, The cost would be $,. C,9 9.7,, The cost would be $,. c) C as. No. The model is undefined for p. N deer N deer b) The herd is limited b the horizontal asmptote: N deer. 7. a)..7 C C C b) Domain:..7 and Thus, 9. Using interval notation, the domain is, 9. c)..... C As the tank is filled, the concentration increases more slowl. It approaches the horizontal asmptote of C.7.

16 Chapter Polnomial and Rational Functions 77. a) A and Thus, b) Domain: Since the margins on the left and right are each inches, >. In interval notation, the domain is,. c) A. 7 9 Area) The area is minimum when.7 inches and.7 inches. The area is minimum when is approimatel. 7. A and Thus, A 9, >. B graphing the area function, we see that A is minimum when. inches and. inches a) Let t time from Akron to Columbus and t time from Columbus back to Akron. t t t t t t Thus, t t. b) Vertical asmptote: c) Horizontal asmptote: e) Yes. You would epect the average speed for the round trip to be the average of the average speeds for the two parts of the trip. f) No. At miles per hour ou would use more time in one direction than is required for the round trip at an average speed of miles per hour.

17 Section. Rational Functions. a) b) S The sales in is estimated to be $7,,. c) Probabl not. The graph has a horizontal asmptote at Future sales ma eceed this limiting value. S. million dollars... False. Polnomial functions do not have vertical asmptotes.. False. The graph of f crosses, which is a horizontal asmptote.. Vertical asmptote: None The denominator is not zero for an value of unless the numerator is also zero there). Horizontal asmptote: The degree of the numerator equals the degree of the denominator. f is one possible function. There are man correct answers.. Vertical asmptotes:, are factors of the denominator. Horizontal asmptotes: None The degree of the numerator is greater than the degree of the denominator. f is one possible function. There are man correct answers i i > > 7 > < 9. < 9. < < < < < < 7 7 or Answers will var.