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 -00 1.054-100 1.0515 100 0.95146 y = 1 00 0.97537 ) Zeros o n : = 1 Zeros o d : = {-1, } = -1, = y -00-0.015-100 -0.03 100 0.03001 00 0.015 3) 4 y -00 0.01985-100 0.0394 100-0.0406 00-0.00 Zeros o n : = 1.5 Zeros o d : = {0, 3} = 0, = 3 01, TESCCC 04/7/13 page 1 o 6
4) Zeros o n : = Zeros o d : = {-3, } = -3 y -00-0.015-100 -0.0309 100 0.0913 00 0.01478 Hole or removable discontinuity) at = 5) 1 4 y -00-1.0001-100 -1.0003 100-1.0003 00-1.0001 Zeros o n : = {-1, 1} Zeros o d : = {-, } = -, = y = -1 6) 5 1 Zeros o n : No real zeros Zeros o d : = {-, } = -, = y -00 1.6669-100 1.6675 100 1.6675 y = 5/3 = 1.6666 00 1.6669 01, TESCCC 04/7/13 page o 6
7) 5 10 3 9 Zeros o n : = - Zeros o d : = {-3, 0, 3} = -3, = 0, = 3 y -00 1. 10-4 -100 4.9 10-4 100 5.1 10-4 00 1.3 10-4 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
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 -00 0.9755 y = 1-100 0.95098 100 1.051 00 1.053 14) Zeros o n : = {-1, } Zeros o d : = 1 = 1 y -00-66.66-100 -33.33 100 33.37 00 66.663 Slant linear) asymptote 15) 4 Zeros o n : = {0, 3} Zeros o d : = 1.5 = 1.5 y -00 50.37-100 5.369 100-4.6 00-49.6 Slant linear) asymptote 01, TESCCC 04/7/13 page 4 o 6
16) Zeros o n : = {-3, } Zeros o d : = y -00-65.67-100 -3.33 100 34.333 00 67.667 Linear, with removable disc. at = 17) 4 1 Zeros o n : = {-, } Zeros o d : = {-1, 1} = -1, = 1 y -00-0.9999-100 -0.9997 100-0.9997 y = -1 00-0.9999 18) 5 1 Zeros o n : = {-, } Zeros o d : No real zeros y -00 0.5999-100 0.59969 100 0.59969 y = 3/5 = 0.6 00 0.5999 01, TESCCC 04/7/13 page 5 o 6
19) 3 9 5 10 Zeros o n : = {-3, 0, 3} Zeros o d : = - = - y -00 8079-100 039 100 1959 00 7919 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