4.2 Properties of Rational Functions. 188 CHAPTER 4 Polynomial and Rational Functions. Are You Prepared? Answers

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1 88 CHAPTER 4 Polnomial and Rational Functions 5. Obtain a graph of the function for the values of a, b, and c in the following table. Conjecture a relation between the degree of a polnomial and the number of turning points after completing the table. In the table, a can be, 2, or ; b can be, 2, or ; and c can be, 2,, or 4. Values of a, b, and c Degree of Polnomial Number of Turning Points a, b, c a, b, c 2 4 a, b, c 5 a, b, c 4 a, b 2, c a, b 2, c 2 a, b 2, c a, b 2, c 4 a, b, c a, b, c 2 a, b, c a, b, c 4 a 2, b, c a 2, b, c 2 a 2, b, c a 2, b, c 4 a 2, b 2, c a 2, b 2, c 2 a, b, c 4 Are You Prepared? Answers. - 2, 02, 2, 02, 0, Yes;. Down; 4 4. Local maimum value 6.48 at = ; local minimum value - at = 2 5. True 6. 5, 02;5 4.2 Properties of Rational Functions PREPARING FOR THIS SECTION Before getting started, review the following: Rational Epressions (Appendi A, Section A.5, pp. A6 A42) Polnomial Division (Appendi A, Section A., pp. A25 A28) Now Work the Are You Prepared? problems on page 96. Graph of f2 = (Section.2, Eample 2, p. 6) Graphing Techniques: Transformations (Section 2.5, pp ) OBJECTIVES Find the Domain of a Rational Function (p. 89) 2 Find the Vertical Asmptotes of a Rational Function (p. 92) Find the Horizontal or Oblique Asmptote of a Rational Function (p. 9) Ratios of integers are called rational numbers. Similarl, ratios of polnomial functions are called rational functions. Eamples of rational functions are R2 = F2 = 2-4 G2 = 2 4 -

2 SECTION 4.2 Properties of Rational Functions 89 DEFINITION A rational function is a function of the form R2 = p2 q2 where p and q are polnomial functions and q is not the zero polnomial. The domain of a rational function is the set of all real numbers ecept those for which the denominator q is 0. Find the Domain of a Rational Function EXAMPLE Finding the Domain of a Rational Function (a) The domain of R2 = 22-4 is the set of all real numbers ecept - 5; that + 5 is, the domain is 5 ƒ Z (b) The domain of R2 = is the set of all real numbers ecept - 2 and 2; 2-4 that is, the domain is 5 ƒ Z - 2, Z 26. (c) The domain of R2 = is the set of all real numbers. 2 + (d) The domain of R2 = 2 - is the set of all real numbers ecept ; that is, - the domain is 5 ƒ Z Although reduces to +, it is important to observe that the functions - R2 = and f2 = + are not equal, since the domain of R is 5 ƒ Z 6 and the domain of f is the set of all real numbers. Now Work PROBLEM 5 If R2 = p2 is a rational function and if p and q have no common factors, q2 then the rational function R is said to be in lowest terms. For a rational function R2 = p2 in lowest terms, the real zeros, if an, of the numerator in the domain q2 of R are the -intercepts of the graph of R and so will pla a major role in the graph of R.The real zeros of the denominator of R [that is, the numbers,if an,for which q2 = 0], although not in the domain of R, also pla a major role in the graph of R. We have alread discussed the properties of the rational function =. (Refer to Eample 2, page 6). The net rational function that we take up is H2 = 2.

3 90 CHAPTER 4 Polnomial and Rational Functions EXAMPLE 2 Graphing 2 Table ,000 0,000 00,000, , ,000 00,000,000 Figure 28 H() = 2 Analze the graph of H2 =. 2 Solution The domain of H2 = is the set of all real numbers ecept 0.The graph has no 2 -intercept, because can never equal 0. The graph has no -intercept because the H() equation H2 = 0 has no solution. Therefore, the graph of H will not cross or 2 touch either of the coordinate aes. Because 4 H- 2 = = 2 = H2 H is an even function, so its graph is smmetric with respect to the -ais. Table 9 shows the behavior of H2 = for selected positive numbers. (We 2 will use smmetr to obtain the graph of H when 6 0.) From the first three rows of Table 9, we see that, as the values of approach (get closer to) 0, the values of H2 become larger and larger positive numbers, so H is unbounded in the positive direction. We use limit notation, lim H2 = q, read the limit of H2 as :0 approaches zero equals infinit, to mean that H2 : q as : 0. Look at the last four rows of Table 9. As : q, the values of H2 approach 0 (the end behavior of the graph). In calculus, this is smbolized b writing lim H2 = 0. Figure 28 shows the graph. Notice the use of red dashed lines to : q conve the ideas discussed above. 0 5 (, 4) (, 4) 2 2 (2, 4 (, ) (, ) ) ( ) 2, EXAMPLE (, ) Using Transformations to Graph a Rational Function Graph the rational function: R2 = (, ) (, ) Solution The domain of R is the set of all real numbers ecept. To graph R, start with the graph of = = 2. See Figure 29 for the steps. 2 2 Figure (, ) 5 (, 2) (, 2) 5 Replace b 2; Add ; shift right shift up 2 units unit (a) (b) 2 ( 2) 2 (c) ( 2) 2 Now Work PROBLEM

4 SECTION 4.2 Properties of Rational Functions 9 Table 0 Asmptotes Let s investigate the roles of the vertical line = 2 and the horizontal line = in Figure 29(c). First, we look at the end behavior of R2 = Table 0(a) shows the values of R at = 0, 00, 000, 0,000. Notice that, as becomes unbounded in the positive direction, the values of R approach, so lim. From Table 0(b) : q we see that, as becomes unbounded in the negative direction, the values of R also approach, so lim. : - q Even though = 2 is not in the domain of R, the behavior of the graph of R near = 2 is important. Table 0(c) shows the values of R at =.5,.9,.99,.999, and We see that, as approaches 2 for 6 2, denoted : 2 -, the values of R are increasing without bound, so lim R2 = q. :2 - From Table 0(d), we see that, as approaches 2 for 7 2, denoted : 2 +, the values of R are also increasing without bound, so lim. R2 = q :2 + R() R() R() R() , (a) , (b) ,00.999,000, ,000,00 (c) , ,000, ,000,00 (d) The vertical line = 2 and the horizontal line = are called asmptotes of the graph of R. DEFINITION Let R denote a function. If, as : - q or as : q, the values of R2 approach some fied number L, then the line = L is a horizontal asmptote of the graph of R. [Refer to Figures 0(a) and (b).] If, as approaches some number c, the values ƒ R2 ƒ : q[r2 : - q or R2 : q],then the line = c is a vertical asmptote of the graph of R.[Refer to Figures 0(c) and (d).] Figure 0 c c R( ) L L R( ) (a) End behavior: As, the values of R( ) approach L [ lim R() L]. That is, the points on the graph of R are getting closer to the line L; L is a horizontal asmptote. (b) End behavior: As, the values of R() approach L [ lim R() L]. That is, the points on the graph of R are getting closer to the line L; L is a horizontal asmptote. (c) As approaches c, the values of R() [ lim c R() ; lim R() ]. That is, c the points on the graph of R are getting closer to the line c; c is a vertical asmptote. (d) As approaches c, the values of R() [ lim c R() ; lim R() ]. That is, c the points on the graph of R are getting closer to the line c; c is a vertical asmptote.

5 92 CHAPTER 4 Polnomial and Rational Functions Figure A horizontal asmptote, when it occurs, describes the end behavior of the graph as : q or as : - q. The graph of a function ma intersect a horizontal asmptote. A vertical asmptote, when it occurs, describes the behavior of the graph when is close to some number c. The graph of a rational function will never intersect a vertical asmptote. There is a third possibilit. If, as : - q or as : q, the value of a rational function R2 approaches a linear epression a + b, a Z 0, then the line = a + b, a Z 0, is an oblique asmptote of R. Figure shows an oblique asmptote. An oblique asmptote, when it occurs, describes the end behavior of the graph. The graph of a function ma intersect an oblique asmptote. Now Work PROBLEM 25 2 Find the Vertical Asmptotes of a Rational Function The vertical asmptotes of a rational function R2 = p2, in lowest terms, are q2 located at the real zeros of the denominator of q2. Suppose that r is a real zero of q, so - r is a factor of q. As approaches r, smbolized as : r, the values of - r approach 0, causing the ratio to become unbounded, that is, ƒ R2 ƒ : q. Based on the definition, we conclude that the line = r is a vertical asmptote. THEOREM WARNING If a rational function is not in lowest terms, an application of this theorem ma result in an incorrect listing of vertical asmptotes. Locating Vertical Asmptotes A rational function R2 = p2, in lowest terms,will have a vertical asmptote q2 = r if r is a real zero of the denominator q. That is, if - r is a factor of the denominator q of a rational function R2 = p2, in lowest terms, R will have the vertical asmptote = r. q2 EXAMPLE 4 Finding Vertical Asmptotes Find the vertical asmptotes, if an, of the graph of each rational function. (a) F2 = + (b) R2 = (c) H2 = (d) G2 = Solution (a) F is in lowest terms and the onl zero of the denominator is. The line = is the vertical asmptote of the graph of F. WARNING In Eample 4(a), the vertical (b) R is in lowest terms and the zeros of the denominator 2-4 are - 2 and 2. The asmptote is. Do not sa that the lines = - 2 and = 2 are the vertical asmptotes of the graph of R. vertical asmptote is. (c) H is in lowest terms and the denominator has no real zeros, because the equation 2 + = 0 has no real solutions. The graph of H has no vertical asmptotes. (d) Factor the numerator and denominator of G2 to determine if it is in lowest terms. G2 = = = Z The onl zero of the denominator of G2 in lowest terms is - 7. The line = - 7 is the onl vertical asmptote of the graph of G. As Eample 4 points out, rational functions can have no vertical asmptotes, one vertical asmptote, or more than one vertical asmptote.

6 SECTION 4.2 Properties of Rational Functions 9 Eploration Graph each of the following rational functions: R() = - R() = ( - ) 2 R() = ( - ) R() = ( - ) 4 Each has the vertical asmptote =. What happens to the value of R() as approaches from the right side of the vertical asmptote; that is, what is lim R()? What happens to the value of R() as : + approaches from the left side of the vertical asmptote; that is, what is lim R()? How does the : multiplicit of the zero in the denominator affect the graph of R? - Now Work PROBLEM 47 (FIND THE VERTICAL ASYMPTOTES, IF ANY.) Find the Horizontal or Oblique Asmptote of a Rational Function The procedure for finding horizontal and oblique asmptotes is somewhat more involved. To find such asmptotes, we need to know how the values of a function behave as : - q or as : q. That is, we need to find the end behavior of the rational function. If a rational function R2 is proper, that is, if the degree of the numerator is less than the degree of the denominator, then as : - q or as : q the value of R2 approaches 0. Consequentl, the line = 0 (the -ais) is a horizontal asmptote of the graph. THEOREM If a rational function is proper, the line = 0 is a horizontal asmptote of its graph. EXAMPLE 5 Finding a Horizontal Asmptote Find the horizontal asmptote, if one eists, of the graph of R2 = Solution Since the degree of the numerator,, is less than the degree of the denominator, 2, the rational function R is proper. The line = 0 is a horizontal asmptote of the graph of R. To see wh = 0 is a horizontal asmptote of the function R in Eample 5, we investigate the behavior of R as : - q and : q. When ƒ ƒ is ver large, the numerator of R, which is - 2, can be approimated b the power function =, while the denominator of R, which is , can be approimated b the power function = 4 2. Appling these ideas to R2, we find R2 = L 4 2 = c 4 : c 0 For ƒ ƒ ver large As : - q or : q This shows that the line = 0 is a horizontal asmptote of the graph of R. If a rational function R2 = p2 is improper, that is, if the degree of the q2 numerator is greater than or equal to the degree of the denominator, we use long division to write the rational function as the sum of a polnomial f2 (the quotient) r2 plus a proper rational function ( r2 is the remainder). That is, we write q2 R2 = p2 q2 = f2 + r2 q2

7 94 CHAPTER 4 Polnomial and Rational Functions r2 r2 where f2 is a polnomial and is a proper rational function. Since is r2 q2 q2 proper, as : - q or as : q. As a result, q2 : 0 The possibilities are listed net. R2 = p2 : f2 as : - q or as : q q2. If f2 = b, a constant, the line = b is a horizontal asmptote of the graph of R. 2. If f2 = a + b, a Z 0, the line = a + b is an oblique asmptote of the graph of R.. In all other cases, the graph of R approaches the graph of f, and there are no horizontal or oblique asmptotes. We illustrate each of the possibilities in Eamples 6, 7, and 8. EXAMPLE 6 Solution Finding a Horizontal or Oblique Asmptote Find the horizontal or oblique asmptote, if one eists, of the graph of H2 = Since the degree of the numerator, 4, is greater than the degree of the denominator,, the rational function H is improper. To find a horizontal or oblique asmptote, we use long division As a result, H2 = As : - q or as : q, = L 22 = 2 : 0 As : - q or as : q, we have H2 : +. We conclude that the graph of the rational function H has an oblique asmptote = +. EXAMPLE 7 Finding a Horizontal or Oblique Asmptote Find the horizontal or oblique asmptote, if one eists, of the graph of R2 =

8 SECTION 4.2 Properties of Rational Functions 95 Solution Since the degree of the numerator, 2, equals the degree of the denominator, 2, the rational function R is improper. To find a horizontal or oblique asmptote, we use long division. As a result, Then, as : - q or as : q, R2 = = L = - 4 : 0 As : - q or as : q, we have R2 : 2. We conclude that = 2 is a horizontal asmptote of the graph. In Eample 7, notice that the quotient 2 obtained b long division is the quotient of the leading coefficients of the numerator polnomial and the denominator polnomial a 8 This means that we can avoid the long division process for rational 4 b. functions where the numerator and denominator are of the same degree and conclude that the quotient of the leading coefficients will give us the horizontal asmptote. Now Work PROBLEMS 4 AND 45 EXAMPLE 8 Solution Finding a Horizontal or Oblique Asmptote Find the horizontal or oblique asmptote, if one eists, of the graph of G2 = Since the degree of the numerator, 5, is greater than the degree of the denominator,, the rational function G is improper. To find a horizontal or oblique asmptote, we use long division As a result, G2 = Then, as : - q or as : q, = L 22 = 2 : 0

9 96 CHAPTER 4 Polnomial and Rational Functions As : - q or as : q, we have G2 : We conclude that, for large values of ƒ ƒ, the graph of G approaches the graph of = That is, the graph of G will look like the graph of = as : - q or : q. Since = is not a linear function, G has no horizontal or oblique asmptote. SUMMARY Consider the rational function Finding a Horizontal or Oblique Asmptote of a Rational Function in which the degree of the numerator is n and the degree of the denominator is m.. If n 6 m (the degree of the numerator is less than the degree of the denominator), then R is a proper rational function, and the graph of R will have the horizontal asmptote = 0 (the -ais). 2. If n Ú m (the degree of the numerator is greater than or equal to the degree of the denominator), then R is improper. Here long division is used. (a) If n = m (the degree of the numerator equals the degree of the denominator), the quotient obtained will be a n the number, and the line = a n is a horizontal asmptote. b m R2 = p2 q2 = a n n + a n - n - + Á + a + a 0 b m m + b m - m - + Á + b + b 0 b m (b) If n = m + (the degree of the numerator is one more than the degree of the denominator), the quotient obtained is of the form a + b (a polnomial of degree ), and the line = a + b is an oblique asmptote. (c) If n Ú m + 2 (the degree of the numerator is two or more greater than the degree of the denominator), the quotient obtained is a polnomial of degree 2 or higher, and R has neither a horizontal nor an oblique asmptote. In this case, for ver large values of ƒ ƒ, the graph of R will behave like the graph of the quotient. Note: The graph of a rational function either has one horizontal or one oblique asmptote or else has no horizontal and no oblique asmptote. 4.2 Assess Your Understanding Are You Prepared? Answers are given at the end of these eercises. If ou get a wrong answer, read the pages listed in red.. True or False The quotient of two polnomial epressions is a rational epression. (pp. A6 A42) 2. What are the quotient and remainder when 4-2 is divided b (pp. A25 A28). Graph = (p. 6). 4. Graph = using transformations. (pp ) Concepts and Vocabular 5. True or False The domain of ever rational function is the set of all real numbers. 6. If, as : - q or as : q, the values of R2 approach some fied number L, then the line = L is a of the graph of R. 7. If, as approaches some number c, the values of ƒ R2 ƒ : q, then the line = c is a of the graph of R. 8. For a rational function R, if the degree of the numerator is less than the degree of the denominator, then R is. 9. True or False The graph of a rational function ma intersect a horizontal asmptote. 0. True or False The graph of a rational function ma intersect a vertical asmptote.. If a rational function is proper, then is a horizontal asmptote. 2. True or False If the degree of the numerator of a rational function equals the degree of the denominator, then the ratio of the leading coefficients gives rise to the horizontal asmptote.

10 SECTION 4.2 Properties of Rational Functions 97 Skill Building In Problems 24, find the domain of each rational function.. R2 = 4-4. R2 = H2 = G2 = F2 = Q2 = R2 = R2 = 4-2. H2 = G2 = R2 = F2 = In Problems 25 0, use the graph shown to find (a) The domain and range of each function (b) The intercepts, if an (c) Horizontal asmptotes, if an (d) Vertical asmptotes, if an (e) Oblique asmptotes, if an (0, 2) (, 2) 4 4 (, 0) (, 0) (, 2) 0. (, ) (, 2) In Problems 42, graph each rational function using transformations.. F2 = Q2 = + 2. R2 = R2 = 5. H2 = G2 = R2 = R2 = G2 = F2 = R2 = R2 = - 4 2

11 98 CHAPTER 4 Polnomial and Rational Functions In Problems 4 54, find the vertical, horizontal, and oblique asmptotes, if an, of each rational function. 4. R2 = 47. T2 = R2 = R2 = P2 = R2 = H2 = Q2 = G2 = G2 = F2 = F2 = Applications and Etensions (a) Let ohms, and graph as a function of. 55. Gravit In phsics, it is established that the acceleration R = 0 R tot R 2 due to gravit, g (in m/sec 2 ), at a height h meters above sea level is given b (b) Find and interpret an asmptotes of the graph obtained in part (a)..99 * 0 4 (c) If R 2 = 2R, what value of R will ield an R tot of gh2 = 7 ohms? 6.74 * h2 2 Source: en.wikipedia.org/wiki/series_and_parallel_circuits where 6.74 * 0 6 is the radius of Earth in meters. (a) What is the acceleration due to gravit at sea level? (b) The Willis Tower in Chicago, Illinois, is 44 meters tall. What is the acceleration due to gravit at the top of the Willis Tower? (c) The peak of Mount Everest is 8848 meters above sea level. What is the acceleration due to gravit on the peak of Mount Everest? (d) Find the horizontal asmptote of gh2. (e) Solve gh2 = 0. How do ou interpret our answer? 56. Population Model A rare species of insect was discovered in the Amazon Rain Forest. To protect the species, environmentalists declared the insect endangered and transplanted the insect into a protected area. The population P of the insect t months after being transplanted is t2 Pt2 = t (a) How man insects were discovered? In other words, what was the population when t = 0? (b) What will the population be after 5 ears? (c) Determine the horizontal asmptote of Pt2. What is the largest population that the protected area can sustain? 57. Resistance in Parallel Circuits From Ohm s law for circuits, it follows that the total resistance R tot of two components hooked in parallel is given b the equation R R 2 R tot = R + R 2 where R and R 2 are the individual resistances. 58. Newton s Method In calculus ou will learn that, if p2 = a n n + a n- n- + Á + a + a 0 is a polnomial function, then the derivative of p2 is p 2 = na n n- + n - 2a n- n- 2 + Á + 2a 2 + a Newton s Method is an efficient method for approimating the -intercepts (or real zeros) of a function, such as p2. The following steps outline Newton s Method. STEP : Select an initial value that is somewhat close to the -intercept being sought. STEP 2: Find values for using the relation n + = n - p n 2 p n 2 n =, 2, Á until ou get two consecutive values n and n + that agree to whatever decimal place accurac ou desire. n + STEP : The approimate zero will be. Consider the polnomial p2 = (a) Evaluate p52 and p- 2. (b) What might we conclude about a zero of p? Eplain. (c) Use Newton s Method to approimate an -intercept, r, - 6 r 6 5,of p2 to four decimal places. (d) Use a graphing utilit to graph p2 and verif our answer in part (c). (e) Using a graphing utilit, evaluate pr2 to verif our result. 0 Eplaining Concepts: Discussion and Writing 59. If the graph of a rational function R has the vertical asmptote = 4, the factor - 4 must be present in the denominator of R. Eplain wh. 60. If the graph of a rational function R has the horizontal asmptote = 2, the degree of the numerator of R equals the degree of the denominator of R. Eplain wh. 6. Can the graph of a rational function have both a horizontal and an oblique asmptote? Eplain. 62. Make up a rational function that has = 2 + as an oblique asmptote. Eplain the methodolog that ou used.