Polnomial and Rational Functions. Polnomial Functions and Modeling. Graphing Polnomial Functions. Polnomial Division; The Remainder and Factor Theorems. Theorems about Zeros of Polnomial Functions. Rational Functions.6 Polnomial and Rational Inequalities.7 Variation and Applications SUMMARY AND REVIEW TEST A P P L I C A T I O N Adog s life span is tpicall much shorter than that of a human. We can use a graphing calculator to fit a polnomial function to data relating the age of a dog to a human s age. This function can then be used to estimate the equivalent human age for a dog of a given age. This problem appears as Eercise 76 in Section..
8 Chapter Polnomial and Rational Functions.. Polnomial Functions and Modeling Determine the behavior of the graph of a polnomial function using the leading-term test. Factor polnomial functions and find the zeros and their multiplicities. Use a graphing calculator to graph a polnomial function and find its real-number zeros, relative maimum and minimum values, and domain and range. Solve applied problems using polnomial models; fit linear, quadratic, power, cubic, and quartic polnomial functions to data. There are man kinds of functions. The constant, linear, and quadratic functions that we studied in Chapters and are part of a larger group of functions called polnomial functions. Polnomial Function A polnomial function P is given b P a n n a n n a n n a a 0, where the coefficients a n, a n,..., a, a 0 are real numbers and the eponents are whole numbers. The first nonzero coefficient,, is called the leading coefficient. The term a n n is called the leading term.the degree of the polnomial function is n.some eamples of polnomial functions are as follows. POLYNOMIAL FUNCTION DEGREE EXAMPLE Constant 0 Linear Quadratic Cubic a n f f f f Quartic f. 0..8.7
Section. Polnomial Functions and Modeling 9 The function f 0 can be described in man was: f 0 0 0 0 8, and so on. For this reason, we sa that the constant function f 0 has no degree. From our stud of functions in Chapters and, we know how to find or at least estimate man characteristics of a polnomial function. Let s consider two eamples for review. QUADRATIC FUNCTION (, 0) (, 0) (0, ) (, ) f() ( )( ) Function: f() ( ) ( ) Zeros:, -intercepts: (,0), (,0) -intercept: (0, ) Minimum: at Maimum: None Domain: All real numbers, (, ) Range: [, ) CUBIC FUNCTION 0 0 0 (.7,.6) 0 (, 0) (, 0) 0 (, 0) 0 (0, ) 0 (., 0.7) 0 0 0 g() ( )( )( ) Function: g() ( ) ( ) ( ) Zeros:,, -intercepts: (,0), (,0), (,0) -intercept: (0, ) Relative minimum: 0.7 at. Relative maimum:.6 at.7 Domain: All real numbers, (, ) Range: All real numbers, (, )
60 Chapter Polnomial and Rational Functions Polnomial Functions All graphs of polnomial functions have some characteristics in common. Compare the following graphs. Observe how the graphs of polnomial functions differ from the graphs of nonpolnomial functions. Also observe some characteristics of the graphs of polnomial functions. f() f() f() f() Nonpolnomial Functions h() h() h() / h() You probabl noted that the graph of a polnomial function is continuous, that is, it has no holes or breaks. It is also smooth; there are no sharp corners. Furthermore, the domain of a polnomial function is the set of all real numbers,,. f() g() h() A continuous function A discontinuous function A discontinuous function
Section. Polnomial Functions and Modeling 6 The Leading-Term Test The behavior of the graph of a polnomial function as becomes ver large l or ver small l is referred to as the end behavior of the graph. The leading term of a polnomial function determines its end behavior. becomes ver small becomes ver large Using the graphs shown below, let s see if we can discover some general patterns b comparing the end behavior of even- and odd-degree functions. We also observe the effect of positive and negative leading coefficients. Even Degree g() g() 6 0 8 6 6 8 0 g() g() 0 Odd Degree f () 0 8 6 6 8 0 0 0 0 0 0 0 0 0 0 0 f() f() 7 f() 9 0
6 Chapter Polnomial and Rational Functions We can summarize our observations as follows. The Leading-Term Test If a n n is the leading term of a polnomial function, then the behavior of the graph as l or as l can be described in one of the four following was. If n is even, and a n 0: If n is even, and a n 0: If n is odd, and a n 0: If n is odd, and a n 0: The portion of the graph is not determined b this test. EXAMPLE Using the leading-term test, match each of the following functions with one of the graphs A D, which follow. a) f b) f c) f d) f 6 A. B. C. D. Solution DEGREE OF SIGN OF LEADING TERM LEADING TERM LEADING COEFFICIENT GRAPH a) Even Positive D b) Odd Negative B c) Odd Positive A d) 6 Even Negative C
Section. Polnomial Functions and Modeling 6 Finding Zeros of Factored Polnomial Functions Let s review the meaning of the real zeros of a function and their connection to the -intercepts of the function s graph. C ONNECTING THE CONCEPTS ZEROS, SOLUTIONS, AND INTERCEPTS FUNCTION Quadratic Polnomial g 8, or REAL ZEROS OF THE FUNCTION; SOLUTIONS OF THE EQUATION To find the zeros of g, we solve g 0: 8 0 0 0 or 0 or. The solutions of 8 0 are and.the are the zeros of the function g.that is, g 0 and g 0. X-INTERCEPTS OF THE GRAPH The zeros of g are the -coordinates of the -intercepts of the graph of g. (, 0) (, 0) -intercepts 8 g() 8 Cubic Polnomial h 6, or To find the zeros of h, we solve h 0: The zeros of h are the -coordinates of the 6 0 -intercepts of the graph 0 of h. 0 or 0 or 0 or or. The solutions of 6 0 are,, and.the are the zeros of the function h. That is, h 0, h 0, and h 0. 8 (, 0) (, 0) 8 (, 0) h() 6 -intercepts
6 Chapter Polnomial and Rational Functions The connection between the real-number zeros of a function and the -intercepts of the graph of the function is easil seen in the preceding eamples. If c is a real zero of a function (that is, f c 0), then c,0 is an -intercept of the graph of the function. 7 60 0 0 60 principle of zero products review section.. ( ) ( ) 0 6 6 0 Figure ( ) ( ) 6 Figure EXAMPLE Consider P 7. Determine whether each of the numbers and is a zero of P. Solution We have P 7 7. Substituting into the polnomial Since P 0, we know that is not a zero of the polnomial. We also have P 7 0. Substituting into the polnomial Since P 0, we know that is a zero of P. Let s take a closer look at the polnomial function (see Connecting the Concepts on page 6). Is there a connection between the factors of the polnomial and the zeros of the function? The factors of h are,, and, and the zeros are,, and. We note that when the polnomial is epressed as a product of linear factors, each factor determines a zero of the function. Thus if we know the linear factors of a polnomial function f, we can easil find the zeros of f b solving the equation f 0 using the principle of zero products. EXAMPLE Find the zeros of f. Solution To solve the equation f 0, we use the principle of zero products, solving 0 and 0. The zeros of f are and. (See Fig..) EXAMPLE h 6 Find the zeros of g. Solution To solve the equation g 0, we use the principle of zero products, solving 0 and 0. The zeros of g are and. (See Fig..)
Section. Polnomial Functions and Modeling 6 Let s consider the occurrences of the zeros in the functions in Eamples and and their relationship to the graphs of those functions. In Eample, the factor occurs three times. In a case like this, we sa that the zero we obtain from this factor,, has a multiplicit of. The factor occurs one time. The zero we obtain from this factor,, has a multiplicit of. In Eample, the factors and each occur two times. Thus both zeros, and, have a multiplicit of. Note, in Eample, that the zeros have odd multiplicities and the graph crosses the -ais at both and. But in Eample, the zeros have even multiplicities and the graph is tangent to (touches but does not cross) the -ais at and. This leads us to the following generalization. Even and Odd Multiplicit If c k, k,is a factor of a polnomial function P and c k is not a factor and: k is odd, then the graph crosses the -ais at c,0 ; k is even, then the graph is tangent to the -ais at c,0. Some polnomials can be factored b grouping. Then we can use the principle of zero products to find their zeros. 9 8 EXAMPLE Find the zeros of f 9 8. Solution We factor b grouping, as follows: f 9 8 9 Grouping with and 9 with 8 and factoring each group 9 Factoring out. Factoring 9 Then, b the principle of zero products, the solutions of the equation f 0 are,, and. These are the zeros of f. Other factoring techniques can also be used. EXAMPLE 6 Find the zeros of f. Solution We factor as follows: f 9.
66 Chapter Polnomial and Rational Functions 60 0 0 60 We now solve the equation f 0 to determine the zeros. We use the principle of zero products: 9 0 0 or 9 0 or 9 or 9 i. The solutions are and i.these are the zeros of f. Onl the real-number zeros of a function correspond to the -intercepts of its graph. For instance, the real-number zeros of the function in Eample 6, and, can be seen on the graph of the function at left, but the nonreal zeros, i and i, cannot. Ever polnomial function of degree n, with n, has at least one zero and at most n zeros. This is often stated as follows: Ever polnomial function of degree n, with n, has eactl n zeros. This statement is compatible with the preceding statement, if one takes multiplicities into account. Finding Real Zeros on a Calculator Finding eact values of the real zeros of a function can be difficult. We can find approimations using a graphing calculator. EXAMPLE 7 Find the real zeros of the function f given b f 0. 0.6 0.. Approimate the zeros to three decimal places. Solution We use a graphing calculator, tring to create a graph that clearl shows the curvature. Then we look for points where the graph crosses the -ais. It appears that there are three zeros, one near, one near, and one near 6. We use the ZERO feature to find them. 0. 0.6 0. 8 7 8.680, 0., 0.6, 0 The zeros are approimatel.680,., and.6.
Section. Polnomial Functions and Modeling 67 Polnomial Models Polnomial functions have man uses as models in science, engineering, and business. The simplest use of polnomial functions in applied problems occurs when we merel evaluate a polnomial function. In such cases, a model has alread been developed. X 0... X Y 0 0. 8.6. 8.6 06.9 0.. 96.6 7.7 00 0 6 Maimum X.80 Y.766 00 EXAMPLE 8 Ibuprofen in the Bloodstream. The polnomial function M t 0.t.t 96.6t 7.7t can be used to estimate the number of milligrams of the pain relief medication ibuprofen in the bloodstream t hours after 00 mg of the medication has been taken. a) Find the number of milligrams in the bloodstream at t 0, 0.,,., and so on, up to 6 hr. Round the function values to the nearest tenth. b) Find the domain, the relative maimum and where it occurs, and the range. Solution a) We can evaluate the function with the TABLE feature of a graphing calculator set in AUTO mode. We start at 0 and use a step-value of 0.. M 0 0, M..9, M 0. 0., M 9., M, M. 6.9, M. 8., M 66, M., M. 0., M. 8.6, M 6 0. M 06.9, b) Recall that the domain of a polnomial function, unless restricted b a statement of the function, is,. The implications of the application restrict the domain of the function. If we assume that a patient had not taken an of the medication before, it seems reasonable that M 0 0; that is, at time 0, there is 0 mg of the medication in the bloodstream. After the medication has been taken, M t will be positive for a period of time and eventuall decrease back to 0 when t 6 and not increase again (unless another dose is taken). Thus the restricted domain is 0, 6. To determine the range, we find the relative maimum value of the function using the MAXIMUM feature. The maimum is about.76 mg. It occurs approimatel. hr, or hr 9 min, after the initial dose has been taken. The range is about 0,.76. In Chapter, we used regression to model data with linear functions. We now epand that procedure to include quadratic, cubic, and quartic models.
68 Chapter Polnomial and Rational Functions GCM EXAMPLE 9 Declining Number of Farms in the United States. Toda U.S. farm acreage is about the same as it was in the earl part of the twentieth centur, but the number of farms has decreased. Declining Number of Farms in the United States Number of farms (in millions) 7 6 0 900 90 90.70 6.66 6. 6.9 6.0 90 90 90.88.96 960 970.9.0.6.96.7.8 980 990 99 000 00 Year Sources: U.S. Department of Agriculture; U.S. Bureau of the Census; Statistical Abstract of the United States, 00 Looking at the graph above, we note that the data could be modeled with a cubic or a quartic function. a) Model the data with both cubic and quartic functions. Let the first coordinate of each data point be the number of ears after 900. That is, enter the data as 0,.70, 0, 6.66, 0, 6., and so on. Then using R, the coefficient of determination,decide which function is the better fit. The R -value gives an indication of how well the function fits the data. The closer R is to, the better the fit. b) Graph the function with the scatterplot of the data. c) Use the answer to part (a) to estimate the number of farms in 9, 97, 00, and 00. Solution a) Using the REGRESSION feature with DIAGNOSTIC turned on, we get the following. CubicReg a b c d a.80e b.0070977 c.960807 d.609609 R.98997 QuarticReg a b... e a.6697e 7 b.70868e 6 c.0007 d.0879 e.70867 QuarticReg a b... e b.70868e 6 c.0007 d.0879 e.70867 R.99890 Since the -value for the quartic function is closer to than that for the cubic function, the quartic function is the better fit. Note that a and R
Section. Polnomial Functions and Modeling 69 7 0 0 0 b are given in scientific notation on the graphing calculator, but we convert to decimal notation when we write the function. f 0.0000006697 0.0000070868 0.0007 0.0879.70867 b) The scatterplot and graph are shown at left. c) We evaluate the function found in part (a). X 7 0 0 Y.9.799.9.0 X With this function, we can estimate that there were about. million farms in 9 and about.79 million in 97. Looking at the bar graph shown on the preceding page, we see that these estimates appear to be fairl accurate. If we use the function to estimate the number of farms in 00 and in 00, we get about.9 million and.0 million, respectivel. These estimates are probabl not realistic since it is not reasonable to epect the number of forms to increase in the future. The quartic model has a high value for R, approimatel 0.99, over the domain of the data,but this number does not reflect the degree of accurac for etended values. It is alwas important when using regression to evaluate predictions with common sense and knowledge of current trends.
Section. Polnomial Functions and Modeling 69. Eercise Set Classif the polnomial as constant, linear, quadratic, cubic, or quartic and determine the leading term, the leading coefficient, and the degree of the polnomial.. g 0 8 Cubic; ; ;. f 0 0. 7 Quartic; 0. ; 0.;. h 0.9 0. Linear; 0.9; 0.9;. f 6 Constant; 6; 6; 0. g 0 0 Quartic; 0 ; 0; 6. h. 7 8 Cubic;. ;.; 7. h 7 Quartic; ; ; 8. f Quadratic; ; ; 9. g Cubic; 8 ; ; 0. f Linear; ; ;
70 Chapter Polnomial and Rational Functions In Eercises 8, select one of the following four sketches to describe the end behavior of the graph of the function. a) b) c) d). f (d). f 6 (a). f 6 (b). f (c). f. 6 0. 7 (c) 6. f 0. 6 (b) 7. f 0 0 (a) 8. f (d) 9. Use substitution to determine whether,, and are zeros of Yes; no; no f 9. 0. Use substitution to determine whether,, and are zeros of No; no; es f 6.. Use substitution to determine whether,, and are zeros of No; es; es g 6 8 6 9.. Use substitution to determine whether,, and are zeros of Yes; es; no g. Find the zeros of the polnomial function and state the multiplicit of each.. f, multiplicit ;, multiplicit. f, multiplicit ;, multiplicit ;, multiplicit. f 6, multiplicit ; 6, multiplicit 6. f 7 7, multiplicit ; 7, multiplicit ;, multiplicit 7. f 9, each has multiplicit 8. f, each has multiplicit 9. f 0, multiplicit ;, multiplicit ;, multiplicit 0. f. f 8, multiplicit ;, multiplicit ; 0, multiplicit. f 6, multiplicit ;, multiplicit. f,, each has multiplicit. f 0 9,, each has multiplicit. f,,, each has multiplicit 6. f,,each has multiplicit 7. f 8,, each has multiplicit 8. f 8 6,, each has multiplicit Using a graphing calculator, find the real zeros of the function. 9. f 0... f.6. f.. f 9.78,.67,. f., 0,., 0, f 0.8, 0.86,.88 f 8 8 7 8 6 6 70 6 8 8 0.,.87, 0.8, 0.0, 0.098, 0.,.9,.97 6. f 6 0.8, 9.87 Using a graphing calculator, estimate the real zeros, the relative maima and minima, and the range of the polnomial function. 7. g. 8. h 6 9. f 6.8 0. h 0. f 0. f.6 0., 0.7,.879.0,.0 Answers to Eercises 0 and 7 can be found on pp. IA- and IA-6.
Section. Polnomial Functions and Modeling 7 In Eercises 6, answer True or False to each statement.. If P, then the graph of the polnomial function P crosses the -ais at, 0. False. If P, then the graph of the polnomial function crosses the -ais at True P,0.. If P 6, then the graph of P is tangent to the -ais at, 0. True 6. If P, then the graph of P is tangent to the -ais at, 0. False 7. Tobacco Acreage. Tobacco was America s first eport, but tobacco fields are now disappearing due to lawsuits, a decrease in smoking, and competition from imports. 9. Vertical Leap. A formula relating an athlete s vertical leap V, in inches, to hang time T, in seconds, is V 8T. Anfernee Hardawa of the Phoeni Suns has a vertical leap of 6 in. (Source: National Basketball Association). What is his hang time? 0.866 sec 60. Investments in China. Total foreign direct investment in China, in billions of dollars, over the ears 99 to 00 is modeled b the cubic function f 0.6 7.7. 7.6, where is the number of ears since 99 (Source: Ministr of Foreign Trade and Economic Cooperation). Find the foreign investment in China in 99, 996, 000, and 00. Then use this model to estimate the investment in 00. $86 billion; $7 billion; $8 billion; $9 billion; $00 billion The quartic function f 0.8 6. 86.769 9. 7.66, where is the number of ears since 99, can be used to estimate the acreage (in thousands) of tobacco harvested from 99 to 00 (Source: National Agriculture Statistics Source, USDA). Estimate the average acreage harvested in 99 and in 000. 689,000; 6,000 8. Projectile Motion. A stone thrown downward with an initial velocit of. m sec will travel a distance of s meters, where s t.9t.t and t is in seconds. If a stone is thrown downward at. m sec from a height of 9 m, how long will it take the stone to reach the ground? sec 6. Games in a Sports League. If there are teams in a sports league and all the teams pla each other twice, a total of N games are plaed, where N. A softball league has 9 teams, each of which plas the others twice. If the league pas $ per game for the field and umpires, how much will it cost to pla the entire schedule? $0 6. Windmill Power. Under certain conditions, the power P, in watts per hour, generated b a windmill with winds blowing v miles per hour is given b P v 0.0v. 0.6 watts per hour a) Find the power generated b -mph winds. b) How fast must the wind blow in order to generate 0 watts of power in hr? 0 mph
7 Chapter Polnomial and Rational Functions 6. Bald Eagles. During winter, bald eagles travel south to find open water in states that have milder winter temperatures. The number of bald eagles spotted in Indiana during the winters from 996 to 00 can be modeled b the quartic function f.9.88 98.9.90 7.97, where is the number of ears since 996 (Source: Indiana Department of Natural Resources). Find the number of bald eagles in Indiana in the winters of 997, 999, 00, and 00. 0; 0; 69; 07 66. Threshold Weight. In a stud performed b Alvin Shemesh, it was found that the threshold weight W, defined as the weight above which the risk of death rises dramaticall, is given b W h h,. where W is in pounds and h is a person s height, in inches. Find the threshold weight of a person who is ft, 7 in. tall. 6 lb For the scatterplots and graphs in Eercises 67 7, determine which, if an, of the following functions might be used as a model for the data. a) Linear, f m b b) Quadratic, f a b c, a 0 c) Quadratic, f a b c, a 0 d) Polnomial, not quadratic or linear 67. (b) 68. (a) Sales (in millions) 8 0 0 0 Year Sales (in millions) 8 0 0 0 Year 6. French Language Registrations. Higher education French language course registrations in the United States from 970 to 998 can be modeled with the quartic function f 0.00 0.68.86 6. 60.06, 69. (c) Sales (in millions) 8 0 0 6 8 Year 70. (d) Sales (in millions) 0 0 0 0 Year 6 where f is in thousands and is the number of ears since 970. Use this model to find the number of higher education French language registrations in 970, 980, 990, and 998. Then use this function to estimate the French language registrations in 000. Round answers to the nearest thousand. 60,000;,000; 68,000; 0,000;,000 6. Interest Compounded Annuall. When P dollars is invested at interest rate r, compounded annuall, for t ears, the investment grows to A dollars, where 7. (a) Sales (in millions) 8 0 0 0 Year 7. Sales (in millions) 0 0 0 Year 6 (b) might fit. It is possible that there is no polnomial function that fits the data well. A P r t. a) Find the interest rate r if $000 grows to $68.0 in r. % b) Find the interest rate r if $0,000 grows to $,0 in r. 0%
Section. Polnomial Functions and Modeling 7 7. U.S. Farms. As the number of farms has decreased in the United States, the average size of the remaining farms has grown larger, as shown below. YEAR AVERAGE ACREAGE PER FARM 900 7 90 9 90 9 90 7 90 7 90 6 960 97 970 7 980 6 990 60 000 Source: U.S. Departments of Agriculture and of Commerce a) Use a graphing calculator to fit linear, quadratic, cubic, and quartic functions to the data. Let represent the number of ears since 900. b) With each function found in part (a), estimate the average acreage in 00 and in 00 and determine which function gives the most realistic estimates. Cubic;, 0; answers ma var 7. Funeral Costs. Since the Federal Trade Commission began regulating funeral directors in 98, the average cost of a funeral for an adult has greatl increased, as shown in the following table. YEAR AVERAGE COST OF FUNERAL b) With each function found in part (a), estimate the cost of a funeral in 00 and in 00 and determine which function gives the most realistic estimates. Linear; $6, $700; answers ma var 7. Mortgage Debt. Mortgage debt is mounting in the United States, as shown in the table below. YEAR MORTGAGE DEBT (IN BILLIONS) 99 $ 99 09 99 8 99 77 996 86 997 0 998 7 999 660 000 6887 00 796 a) Use a graphing calculator to fit linear and quadratic functions to the data. Let represent the number of ears since 99. b) Use the functions found in part (a) to estimate the debt in 00. Compare the estimates and determine which model gives the most realistic estimate. Quadratic; $,8 billion; answers ma var 76. Dog Years. A dog s life span is tpicall much shorter than that of a human. Age equivalents for dogs and humans are given in the table on the net page. 980 $809 98 77 99 7 99 6 996 78 998 00 00 80 Source: National Funeral Directors Association a) Use a graphing calculator to fit linear, quadratic, cubic, and quartic functions to the data. Let represent the number of ears since 980. Answers to Eercises 7(a), 7(a), and 7(a) can be found on p. IA-6.
7 Chapter Polnomial and Rational Functions AGE OF DOG, X (IN YEARS) HUMAN AGE, h (IN YEARS) 0. 0. 0 6 0 8 8 0 6 7 8 9 06 Source: Countr,December 99, p. 60 a) Use a graphing calculator to fit linear and cubic functions to the data. Graph the functions with the data to determine which function has the better fit. b) Use the function found in part (a) to estimate the equivalent human age for dogs that are, 0, and ears old. 6 r; 7 r; 76 r Collaborative Discussion and Writing 77. How is the range of a polnomial function related to the degree of the polnomial? Skill Maintenance Find the distance between the pair of points. 79., and 0, [.] 80., and, [.] 6 8. Find the center and the radius of the circle 9. [.] Center:, ;radius: 7 8. The diameter of a circle connects the points 6, and, on the circle. Find the coordinates of the center of the circle and the length of the radius. [.] Center:, ; radius: Solve. 8. [.6], or, 8. [.6], or, 8. 6 7 [.6] or, or 86. [.6],,, or, Snthesis 87. In earl 00, $000 was deposited at a certain interest rate compounded annuall. One ear later, $00 was deposited in another account at the same rate. At the end of that ear, there was a total of $7.80 in both accounts. What is the annual interest rate? 7% 78. Polnomial functions are continuous. Discuss what continuous means in terms of the domains of the functions and the characteristics of their graphs. Answer to Eercise 76(a) can be found on p. IA-6.
Section. Graphing Polnomial Functions 7. Graphing Polnomial Functions Graph polnomial functions. Use the intermediate value theorem to determine whether a function has a real zero between two given real numbers. Graphing Polnomial Functions In addition to using the leading-term test and finding the zeros of the function, it is helpful to consider the following facts when graphing a polnomial function. If P is a polnomial function of degree n, the graph of the function has: at most n real zeros, and thus at most n-intercepts; at most n turning points. (Turning points on a graph, also called relative maima and minima, occur when the function changes from decreasing to increasing or from increasing to decreasing.) EXAMPLE Graph the polnomial function h. Solution. First, we use the leading-term test to determine the end behavior of the graph. The leading term is. The degree,, is even, and the coefficient,, is negative. Thus the end behavior of the graph as l and as l can be sketched as follows.. The zeros of the function are the first coordinates of the -intercepts of the graph. To find the zeros, we solve h 0 b factoring and using the principle of zero products. 0 0 0 or 0 Factoring Using the principle of zero products 0 or. The zeros of the function are 0 and. Note that the multiplicit of 0 is and the multiplicit of is. The -intercepts are 0, 0 and,0.
76 Chapter Polnomial and Rational Functions. The zeros divide the -ais into three intervals:,0, 0,, and., 0 0 The sign of h is the same for all values of in each of the three intervals. That is, h is positive for all -values in an interval or h is negative for all -values in an interval. To determine which, we choose a test value for from each interval and find h. X 8 Y TEST FUNCTION LOCATION OF INTERVAL VALUE, VALUE, h SIGN OF h POINTS ON GRAPH,0 Below -ais X 0,, Above -ais 8 Below -ais 0 0 This test-point procedure also gives us three points to plot. In this case, we have,,,, and, 8.. To determine the -intercept, we find h 0 : h h 0 0 0 0. The -intercept is 0, 0.. A few additional points are helpful when completing the graph.. h 0. 0. 0. 0. 0... (0, 0) (, ) (, 0) h() (, ) 6 7 8 (, 8)
Section. Graphing Polnomial Functions 77 0 6. The degree of h is. The graph of h can have at most -intercepts and at most turning points. In fact, it has -intercepts and turning point. The zeros, 0 and, each have odd multiplicities, for 0 and for. Since the multiplicities are odd, the graph crosses the -ais at 0 and. The end behavior of the graph is what we described in step (). As l and l, h l. The graph appears to be correct. The following is a procedure for graphing polnomial functions. To graph a polnomial function:. Use the leading-term test to determine the end behavior.. Find the zeros of the function b solving f 0. An real zeros are the first coordinates of the -intercepts.. Use the -intercepts (zeros) to divide the -ais into intervals and choose a test point in each interval to determine the sign of all function values in that interval.. Find f 0. This gives the -intercept of the function.. If necessar, find additional function values to determine the general shape of the graph and then draw the graph. 6. As a partial check, use the facts that the graph has at most n -intercepts and at most n turning points. Multiplicit of zeros can also be considered in order to check where the graph crosses or is tangent to the -ais. We can also check the graph with a graphing calculator. EXAMPLE Graph the polnomial function f 8. Solution. The leading term is. The degree,, is odd, and the coefficient,, is positive. Thus the end behavior of the graph will appear as follows.. To f ind the zeros, we solve f 0. Here we can use factoring b grouping. 8 0 0 0 0 Factoring b grouping Factoring a difference of squares The zeros are,, and. Each is of multiplicit. The -intercepts are,0,, 0, and, 0.
78 Chapter Polnomial and Rational Functions. The zeros divide the -ais into four intervals:,,,,, and,., 0 We choose a test value for from each interval and find f. 8 X X Y 9 TEST FUNCTION LOCATION OF INTERVAL VALUE, VALUE, f SIGN OF f POINTS ON GRAPH,,, Below -ais Above -ais 9 Below -ais, Above -ais 0 The test values and corresponding function values also give us four points on the graph:,,,,, 9, and,.. To determine the -intercept, we find f 0 : f 8 f 0 0 0 8 0. The -intercept is 0,.. We find a few additional points and complete the graph.. f 9.. 0. 7.. 7 (.,.) (, 0) (, ) (, 0) (, 0) (0, ) f() 8 6 7 (., 7) 8 (0., 7.) (., 9) 9 (, 9)
Section. Graphing Polnomial Functions 79 8 6 6. The degree of f is. The graph of f can have at most -intercepts and at most turning points. It has -intercepts and turning points. Each zero has a multiplicit of ; thus the graph crosses the -ais at,, and. The graph has the end behavior described in step (). As l, h l, and as l, h l. The graph appears to be correct. Some polnomials are difficult to factor. In the net eample, the polnomial is given in factored form. In Sections. and., we will learn methods that facilitate determining factors of such polnomials. EXAMPLE Graph the polnomial function g 7 6. Solution. The leading term is. The degree,, is even, and the coefficient,, is positive. The sketch below shows the end behavior.. To find the zeros, we solve g 0: 0. The zeros are,, and ; is of multiplicit ; the others are of multiplicit. The -intercepts are, 0,, 0, and, 0.. The zeros divide the -ais into four intervals:,,,,,, and,. 0 We choose a test value for from each interval and find g. 7 6 X.. X Y.86 6 6.6 TEST FUNCTION LOCATION OF INTERVAL VALUE, VALUE, g SIGN OF g POINTS ON GRAPH,..9 Above -ais, 6 Below -ais, Below -ais,. 6.6 Above -ais 0
80 Chapter Polnomial and Rational Functions 7 6 0 8 The test values and corresponding function values also give us four points on the graph:.,.9,, 6,,, and., 6.6.. To determine the -intercept, we find g 0 : g 7 6 g 0 0 7 0 0 0 6 6. The -intercept is 0, 6.. We find a few additional points and draw the graph. g 0.. 0..8..6.... (.,.9) g() 7 6 0 8 (., 6.6) 6 (, 0) (, 0) (, 0) (, ) 6 (, 6) 8 0 (0.,.8) ( 0., 6.) 6 (0, 6) 8 Stud Tip Visualization is one ke to understanding and retaining new concepts. This tet has an eceptional art package with precise color-coding to streamline the learning process. Take time to stud each art piece and observe the concept that is illustrated. 6. The degree of g is. The graph of g can have at most -intercepts and at most turning points. It has -intercepts and turning points. One of the zeros,, has a multiplicit of, so the graph is tangent to the -ais at. The other zeros, and, each have a multiplicit of so the graph crosses the -ais at and. The graph has the end behavior described in step (). As l and as l, g l. The graph appears to be correct. The Intermediate Value Theorem Polnomial functions P are continuous, hence their graphs are unbroken. The domain of a polnomial function, unless restricted b the statement of the function, is,. Suppose two function values P a and P b have opposite signs. Since P is continuous, its graph must be a curve from a, P a to b, P b without a break. Then it follows that the curve must cross the -ais at some point c between a and b that is, the function has a zero at c between a and b. P(a) P(b) (a, P(a)); P(a) 0 a c b (c, P(c)); P(c) 0 (b, P(b)); P(b) 0
Section. Graphing Polnomial Functions 8 The Intermediate Value Theorem For an polnomial function P with real coefficients, suppose that for a b, P a and P b are of opposite signs. Then the function has a real zero between a and b. 0 8 6 6 8 f() 6 0 g() b EXAMPLE Using the intermediate value theorem, determine, if possible, whether the function has a real zero between a and b. a) f 6; a, b b) f 6; a, b c) g ; a, d) g ; a, b Solution We find f a and f b or g a and g b and determine whether the differ in sign. The graphs of f and g at left provide a visual check of the conclusions. a) f 6, f 6 8 Note that f is negative and f is positive. B the intermediate value theorem, since f and f have opposite signs, then f has a zero between and. The graph confirms this. b) f 6 6, f 6 8 Both f and f are positive. Thus the intermediate value theorem does not allow us to determine whether there is a real zero between and. Note that the graph of f shows that there are two zeros between and. c) g 7, g 8 g Since g and have opposite signs, g has a zero between and. The graph confirms this. d) g, g 8 8 Both g and g are negative. This does not necessaril mean that there is not a zero between and. The graph of g does show that there are no zeros between and, but the function values and 8 do not allow us to use the intermediate value theorem to determine this.
8 Chapter Polnomial and Rational Functions A 8 6 6 6 8 0 Visualizing the Graph Match the function with its graph. F B 7 6 6 7. f H. f 6 D. f J G 8 7 6 6 7 C 00 7 0 0 7 00. f B. f A 6. f 6 9 C H D 6 6 7. f I 8. f E 9. f 7 6 G I 0. f 7 F E Answers on page A-0 J 6
Section. Graphing Polnomial Functions 8. Eercise Set For each function in Eercises 6, state: a) the maimum number of real zeros that the function can have, b) the maimum number of -intercepts that the graph of the function can have, and c) the maimum number of turning points that the graph of the function can have.. f 6 (a) ; (b) ; (c). f 6 (a) 6; (b) 6; (c). f (a) ; (b) ; (c). f 0 (a) 0; (b) 0; (c) 9. f (a) ; (b) ; (c) 6. f (a) ; (b) ; (c) In Eercises 7, use the leading-term test and our knowledge of -intercepts to match the function with one of graphs (a) (f ), which follow. a) 00 b) 0 6 6 c) 00 d) 0 00 00 e) 6 f) 6 6 8 7. f (d) 8. f 0. 6 7 (a) 9. f (f) 0. f 6 (c) 0 0 0 0 0 0 0 0 0 0. f 0 (b). f 0. 7 0. 6 0. 6 (e) Graph each polnomial function. Follow the steps outlined in the procedure on page 77.. f. g. h 6. g 7. h 8. g 9. f 0. h. g. f. f. h. f 6. g 9 7. g 8. h 9. h 0. g 8. f 6 8 7. h Using the intermediate value theorem, determine, if possible, whether the function f has a real zero between a and b.. f 9 ; a, b. f 9 ; a, b. f ; a, b Answers to Eercises can be found on p. IA-6.
8 Chapter Polnomial and Rational Functions 6. f ; a, b 7. f 6; a, b 8. f 7 ; a, b 9. f ; a, b 0. f ; a, b Collaborative Discussion and Writing. Eplain how to find the zeros of a polnomial function from its graph.. Is it possible for the graph of a polnomial function to have no -intercepts? no -intercepts? Eplain our answer. Skill Maintenance Match the equation with one of the graphs (a) ( f), which follow. a) b) c). [.] d. [.] f. 6 [.] e 6. 6 7. [.] b 8. [.] c Solve. 9. [.] 0. 0 [.], 0,. 6 0 [.] e) d) f) 9 0. [.] 96 0, [.] a Answers to Eercises 6 0 can be found on p. IA-6.