Flip-Flop Functions KEY

Transcription

1 For each rational unction, list the zeros o the polynomials in the numerator and denominator. Then, using a calculator, sketch the graph in a window o [-5.75, 6] by [-5, 5], and provide an end behavior table or the unction. Finally, use these to describe the asymptotes o the unction. 1) Zeros o n : = Zeros o d : = -3 = -3 y y = ) Zeros o n : = 1 Zeros o d : = {-1, } = -1, = y ) 4 y Zeros o n : = 1.5 Zeros o d : = {0, 3} = 0, = 3 01, TESCCC 04/7/13 page 1 o 6

2 4) Zeros o n : = Zeros o d : = {-3, } = -3 y Hole or removable discontinuity) at = 5) 1 4 y Zeros o n : = {-1, 1} Zeros o d : = {-, } = -, = y = -1 6) 5 1 Zeros o n : No real zeros Zeros o d : = {-, } = -, = y y = 5/3 = , TESCCC 04/7/13 page o 6

3 7) Zeros o n : = - Zeros o d : = {-3, 0, 3} = -3, = 0, = 3 y Questions: 8) Fill in the blank: In most cases, the zeros o the polynomial in the denominator ) determine the locations o the vertical asymptotes. 9) Among the seven unctions provided, there is one case where the statement in #8 is not true. Which unction is it, and why doesn t the rule eactly work? Function #4 has a denominator with a zeros at = -3 and = ; however, there is no vertical asymptote at =. Instead there is a removable discontinuity there, because = is also a zero o the numerator ). 10) What graphical eature is determined by the zeros on the polynomial ) in the numerator o each unction? In most cases, the zeros o the numerator determine the -intercepts o the unction. Again, an eception occurs when a removable discontinuity is present.) 11) Many o the unctions provided have a horizontal asymptote at. Which ones? What do these unctions have in common? Functions #, 3, 4, and 7 each have a horizontal asymptote at. In each, the denominator has a polynomial o degree larger than the numerator. 1) Look back at the unctions which do NOT have a horizontal asymptote at. What do these unctions have in common? Functions # 1, 5, and 6 all have a horizontal asymptote, but NOT at. In each, the numerator and denominator have polynomials o equal degree. The horizontal asymptote ends up being y = leading coeicient o the numerator/leading coeicient o the denominator. 01, TESCCC 04/7/13 page 3 o 6

4 The ollowing unctions are reciprocals o the original seven rational unctions. In other words, their numerators and denominators are switched.) As beore, graph and provide inormation about each. 13) Zeros o n : = -3 Zeros o d : = = y y = ) Zeros o n : = {-1, } Zeros o d : = 1 = 1 y Slant linear) asymptote 15) 4 Zeros o n : = {0, 3} Zeros o d : = 1.5 = 1.5 y Slant linear) asymptote 01, TESCCC 04/7/13 page 4 o 6

5 16) Zeros o n : = {-3, } Zeros o d : = y Linear, with removable disc. at = 17) 4 1 Zeros o n : = {-, } Zeros o d : = {-1, 1} = -1, = 1 y y = ) 5 1 Zeros o n : = {-, } Zeros o d : No real zeros y y = 3/5 = , TESCCC 04/7/13 page 5 o 6

6 19) Zeros o n : = {-3, 0, 3} Zeros o d : = - = - y Questions: 0) Many o these reciprocal unctions #13 19) do NOT have horizontal asymptotes. Which ones? What do these unctions have in common? Functions # 14, 15, 16, and 19 do NOT have horizontal asymptotes. In each, the numerator is a polynomial with a degree that is larger than the denominator. Decide whether each statement is true T) o alse F), and circle the correct letter. When a statement is alse, provide a unction or problem number that can serve as a counter-eample. T F 1) A rational unction o the orm is undeined where = 0. d T F ) A rational unction o the orm has a vertical asymptote where = 0. d See #4, 16 T F 3) A rational unction o the orm will only have a horizontal asymptote d See #, 3, 4, 7 when the degrees o and are the same. a b T F 4) The graphs o, and its reciprocal unction g, are inverses b a o one another. See pairs: #1&13, &14, 3&15, etc. a b T F 5) The graphs o, and its reciprocal unction g, are relections b a o one another over the -ais. See pairs: #1&13, &14, 3&15, etc. 01, TESCCC 04/7/13 page 6 o 6