16 Rational Functions Worksheet

Transcription

1 16 Rational Functions Worksheet Concepts: The Definition of a Rational Function Identifying Rational Functions Finding the Domain of a Rational Function The Big-Little Principle The Graphs of Rational Functions Vertical and Horizontal Asymptotes Holes in the Graphs of Rational Functions (Section 4.) 1. Describe the end behavior of the following rational functions. (a) f() = degree same so divide out top degree terms. If is large, f() 2 behaves like 3 = 3. So answer is: as f() 3 (b) g() = 2 - degree same so divide out top degree coefficients, to get 2 = 2. So for similar reason as above, (c) h() = (d) k() = as g() 2 as h() 0 as k() 0 1

2 (e) l() = denominator does not dominate in degree here, so divide out by top degree terms. So if is large, l() behaves like 2 =. Therefore (f) m() = as as l() l() - denominator does not dominate in degree here, so divide out by top degree terms. So if is large, m() behaves like 2 (g) n() = as as is large enough, n() behaves like m() m() =. Therefore - degree same so divide out top degree terms. So if 23 3 = 2 as n() 2 (h) o() = (2 + )4 (6 ) 3 - degree same so divide out top degree terms. So if (3 1)( 2) 6 is large enough, o() behaves like (2)4 ( ) 3 = 16 (3) 6 3. Therefore as o() 16 3 (i) p() = (2 + )4 (6 ) 3 (3 1)( 2) 7 as p() 0 2. Find all vertical, horizontal, oblique asymptotes, holes, -intercepts, and y-intercepts for the following rational functions. Show the algebra that justifies your answer. Graph these functions. (a) f() = The roots of denominator: 2 only 2

3 The roots of denominator that have multiplicity more than in numerator: 2 only Therefore the vertical line = 2 is a vertical asymptote. As seen earlier the end behaviour of this function is that as, f() 3. This itself gives rise to a horizontal asymptote. That is, The horizontal line y = 3 is a horizontal asymptote. The above means that the end behaviour is that as There are no oblique asymptotes because that would give the end behaviour: f() ± as which is not the case here. (b) g() = 2 The roots of denominator: 7 only The roots of denominator that have multiplicity more than in numerator: 7 only Therefore the vertical line = 7 is a vertical asymptote. The degrees of the numerator and the denominator match. So the end behaviour is like 2 so as, f() 2. The end behaviour itself gives the horizontal asymptote. That is, The horizontal line y = 2 is a horizontal asymptote. There are no oblique asymptotes as that would reflect the following end behaviour: f() ± as which is not the case here. (c) h() = The roots of denominator: 2 and 4 The roots of denominator that have multiplicity more than in numerator: 2 and 4 (because each of 2 and 4 occurs with multiplicity 1 in denominator and 0 in numerator) Therefore the vertical lines = 2 and = 4 are vertical asymptotes. 3

4 Denominator dominates so the end behaviour is that as, f() 0. Therefore this itself gives a horizontal asymptote. That is, The horizontal line y = 0 is a horizontal asymptote. There are no oblique asymptotes because it gives the end behaviour that as, f() ± which is not the case here. (d) k() = The roots of denominator: 1 and 7 The roots of denominator that have multiplicity more than in numerator: 1 (because it occurs with multiplicity 1 in denominator and 0 in numerator) Therefore the vertical line = 1 is a vertical asymptote. 7 Therefore there is a hole at = 7. Denominator dominates so the end behaviour is that as, f() 0. Therefore this itself gives a horizontal asymptote. That is, The horizontal line y = 0 is a horizontal asymptote. There are no oblique asymptotes because it gives the end behaviour that as, f() ± which is not the case here. (e) l() = The roots of denominator: 7 only The roots of denominator that have multiplicity more than in numerator: 7 only Therefore the vertical line = 7 is a vertical asymptote. Therefore there are no holes. The numerator dominates, so the end behaviour is that as, f() and as, f(). There are no horizontal asymptotes because if y = a were a horizontal asymptote then the end behaviour it gives is that as, f() a which is not the case here. It might therefore have an oblique asymptote. So performing long division of gives quotient Q() = 13 and remainder R is not zero. Also, Q is a polynomial of degree 1. Therefore, y = 13 is an oblique asymptote. (f) m() =

5 The roots of denominator: 7 only The roots of denominator that have multiplicity more than in numerator: Therefore there are no vertical asymptotes. 7 only (because it occurs with multiplicity 1 in denominator and also in numerator) Therefore there is a hole at = 7. The numerator of the rational function dominates so the end behaviour is that as, f() and as, f(). Therefore there is no horizontal asymptote for reasons as seen before. It might have an oblique asymptote. So perform long division of gives quotient is Q() = 1 and remainder R = 0. So if is large enough, p = Q which is a polynomial and therefore, there are no horizontal or oblique q asymptotes.