3.2 Polynomial Functions of Higher Degree

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1 71_00.qp 1/7/06 1: PM Page 6 Section. Polnomial Functions of Higher Degree 6. Polnomial Functions of Higher Degree What ou should learn Graphs of Polnomial Functions You should be able to sketch accurate graphs of polnomial functions of degrees 0, 1, and. The graphs of polnomial functions of degree greater than are more difficult to sketch b hand. However, in this section ou will learn how to recognize some of the basic features of the graphs of polnomial functions. Using these features along with point plotting, intercepts, and smmetr, ou should be able to make reasonabl accurate sketches b hand. The graph of a polnomial function is continuous. Essentiall, this means that the graph of a polnomial function has no breaks, holes, or gaps, as shown in Figure.1. Informall, ou can sa that a function is continuous if its graph can be drawn with a pencil without lifting the pencil from the paper. (a) Polnomial functions have continuous graphs. 䊏 䊏 䊏 䊏 Use transformations to sketch graphs of polnomial functions. Use the Leading Coefficient Test to determine the end behavior of graphs of polnomial functions. Find and use zeros of polnomial functions as sketching aids. Use the Intermediate Value Theorem to help locate zeros of polnomial functions. Wh ou should learn it You can use polnomial functions to model various aspects of nature, such as the growth of a red oak tree, as shown in Eercise 9 on page 7. (b) Functions with graphs that are not continuous are not polnomial functions. Figure.1 Another feature of the graph of a polnomial function is that it has onl smooth, rounded turns, as shown in Figure.15(a). It cannot have a sharp turn such as the one shown in Figure.15(b). Sharp turn (a) Polnomial functions have graphs with smooth, rounded turns. Figure.15 (b) Functions with graphs that have sharp turns are not polnomial functions. Leonard Lee Rue III/Earth Scenes

2 71_00.qp 1/7/06 1: PM Page 6 6 Chapter Polnomial and Rational Functions Librar of Parent Functions: Polnomial Function The graphs of polnomial functions of degree 1 are lines, and those of functions of degree are parabolas. The graphs of all polnomial functions are smooth and continuous. A polnomial function of degree n has the form f a n n a n 1 n 1... a a 1 a 0 where n is a positive integer and a n 0. The polnomial functions that have the simplest graphs are monomials of the form f n, where n is an integer greater than zero. If n is even, the graph is similar to the graph of f and touches the ais at the -intercept. If n is odd, the graph is similar to the graph of f and crosses the ais at the -intercept. The greater the value of n, the flatter the graph near the origin. The basic characteristics of the cubic function f are summarized below. A review of polnomial functions can be found in the Stud Capsules. Graph of f Domain:, Range:, Intercept: 0, 0 (0, 0) Increasing on, 1 Odd function f() = Origin smmetr Eploration Use a graphing utilit to graph n for n,, and 8. (Use the viewing window and 1 6. ) Compare the graphs. In the interval 1, 1, which graph is on the bottom? Outside the interval 1, 1, which graph is on the bottom? Use a graphing utilit to graph n for n, 5, and 7. (Use the viewing window and. ) Compare the graphs. In the intervals, 1 and 0, 1, which graph is on the bottom? In the intervals 1, 0 and 1,, which graph is on the bottom? Eample 1 Transformations of Monomial Functions Sketch the graphs of (a) f 5, (b) g 1, and (c) h 1. a. Because the degree of f 5 is odd, the graph is similar to the graph of. Moreover, the negative coefficient reflects the graph in the -ais, as shown in Figure.16. b. The graph of g 1 is an upward shift of one unit of the graph of, as shown in Figure.17. c. The graph of h 1 is a left shift of one unit of the graph of, as shown in Figure.18. Prerequisite Skills If ou have difficult with this eample, review shifting and reflecting of graphs in Section 1.5. f() = 5 ( 1, 1) 1 1 (0, 0) (1, 1) 5 g() = + 1 (0, 1) 1 1 h() = ( + 1) 5 (0, 1) (, 1) 1 ( 1, 0) 1 Figure.16 Figure.17 Now tr Eercise 9. Figure.18

3 71_00.qp 1/7/06 1: PM Page 65 Section. Polnomial Functions of Higher Degree 65 The Leading Coefficient Test In Eample 1, note that all three graphs eventuall rise or fall without bound as moves to the right. Whether the graph of a polnomial eventuall rises or falls can be determined b the polnomial function s degree (even or odd) and b its leading coefficient, as indicated in the Leading Coefficient Test. Leading Coefficient Test As moves without bound to the left or to the right, the graph of the polnomial function f a n n... a 1 a 0, a n 0, eventuall rises or falls in the following manner. 1. When n is odd: If the leading coefficient is If the leading coefficient is positive a n > 0, the graph falls negative a n < 0, the graph rises to the left and rises to the right. to the left and falls to the right.. When n is even: f() as f() as f() as f() as f() as f() as f() as f() as Eploration For each function, identif the degree of the function and whether the degree of the function is even or odd. Identif the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utilit to graph each function. Describe the relationship between the degree and sign of the leading coefficient of the function and the right- and lefthand behavior of the graph of the function. a. 1 b c. 5 5 d. 5 e. f. 1 g. h. 6 5 STUDY TIP The notation f as indicates that the graph falls to the left. The notation f as indicates that the graph rises to the right. If the leading coefficient is If the leading coefficient is positive a n > 0, the graph negative a n < 0, the graph falls rises to the left and right. to the left and right. Note that the dashed portions of the graphs indicate that the test determines onl the right-hand and left-hand behavior of the graph. A review of the shapes of the graphs of polnomial functions of degrees 0, 1, and ma be used to illustrate the Leading Coefficient Test. As ou continue to stud polnomial functions and their graphs, ou will notice that the degree of a polnomial plas an important role in determining other characteristics of the polnomial function and its graph.

4 71_00.qp 1/7/06 1: PM Page Chapter Polnomial and Rational Functions Eample Appling the Leading Coefficient Test Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of each polnomial function. a. f b. f 5 c. f 5 a. Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right, as shown in Figure.19. b. Because the degree is even and the leading coefficient is positive, the graph rises to the left and right, as shown in Figure.0. c. Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right, as shown in Figure.1. A good test of students understanding is to present a graph of a function without giving its equation, and ask the students what the can tell ou about the function s degree and leading coefficient b looking at the graph. You might want to displa a few such graphs on an overhead projector during class for practice. f() = + f() = f() = Figure.19 Figure.0 Figure.1 Now tr Eercise 15. In Eample, note that the Leading Coefficient Test onl tells ou whether the graph eventuall rises or falls to the right or left. Other characteristics of the graph, such as intercepts and minimum and maimum points, must be determined b other tests. Zeros of Polnomial Functions It can be shown that for a polnomial function f of degree n, the following statements are true. 1. The function f has at most n real zeros. (You will stud this result in detail in Section. on the Fundamental Theorem of Algebra.). The graph of f has at most n 1 relative etrema (relative minima or maima). Recall that a zero of a function f is a number for which f 0. Finding the zeros of polnomial functions is one of the most important problems in algebra. You have alread seen that there is a strong interpla between graphical and algebraic approaches to this problem. Sometimes ou can use information about the graph of a function to help find its zeros. In other cases, ou can use information about the zeros of a function to find a good viewing window. Eploration For each of the graphs in Eample, count the number of zeros of the polnomial function and the number of relative etrema, and compare these numbers with the degree of the polnomial. What do ou observe? Additional Eamples Describe the right-hand and left-hand behavior of the graph of each function. a. b. c. f f 5 f 5 a. The graph falls to the left and right. b. The graph rises to the left and falls to the right. c. The graph falls to the left and rises to the right.

5 71_00.qp 1/7/06 1: PM Page 67 Section. Polnomial Functions of Higher Degree 67 Real Zeros of Polnomial Functions If f is a polnomial function and a is a real number, the following statements are equivalent. 1. a is a zero of the function f.. a is a solution of the polnomial equation f 0.. a is a factor of the polnomial f.. a, 0 is an -intercept of the graph of f. Finding zeros of polnomial functions is closel related to factoring and finding -intercepts, as demonstrated in Eamples,, and 5. TECHNOLOGY SUPPORT For instructions on how to use the zero or root feature, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. Eample Finding Zeros of a Polnomial Function Find all real zeros of f. Algebraic f Write original function. Substitute 0 for f. Remove common monomial factor. Factor completel. So, the real zeros are 0,, and 1, and the corresponding -intercepts are 0, 0,, 0, and 1, 0. Check Now tr Eercise. 0 is a zero. is a zero. 1is a zero. Graphical Use a graphing utilit to graph. In Figure., the graph appears to have the -intercepts 0, 0,, 0, and 1, 0. Use the zero or root feature, or the zoom and trace features, of the graphing utilit to verif these intercepts. Note that this third-degree polnomial has two relative etrema, at 0.55, 0.6 and 1.,.11. ( 0.55, 0.6) (0, 0) (, 0) 6 6 ( 1, 0) (1.,.11) = Figure. Eample Analzing a Polnomial Function Find all real zeros and relative etrema of f Substitute 0 for f. Remove common monomial factor. Factor completel. So, the real zeros are 0, 1, and 1, and the corresponding -intercepts are 0, 0, 1, 0, and 1, 0, as shown in Figure.. Using the minimum and maimum features of a graphing utilit, ou can approimate the three relative etrema to be 0.71, 0.5, 0, 0, and 0.71, 0.5. Now tr Eercise 5. ( 0.71, 0.5) (0.71, 0.5) ( 1, 0) (1, 0) (0, 0) f() = + Figure.

6 71_00.qp 1/7/06 1: PM Page Chapter Polnomial and Rational Functions Repeated Zeros For a polnomial function, a factor of a k, k > 1, ields a repeated zero a of multiplicit k. 1. If k is odd, the graph crosses the -ais at a.. If k is even, the graph touches the -ais (but does not cross the -ais) at a. Eample 5 Finding Zeros of a Polnomial Function Find all real zeros of f 5 1. Use a graphing utilit to obtain the graph shown in Figure.. From the graph, ou can see that there are three zeros. Using the zero or root feature, ou can determine that the zeros are approimatel 1.86, 0.5, and.11. It should be noted that this fifth-degree polnomial factors as f The three zeros obtained above are the zeros of the cubic factor 1 (the quadratic factor 1 has two comple zeros and so no real zeros). Now tr Eercise 7. STUDY TIP In Eample, note that because k is even, the factor ields the repeated zero 0. The graph touches (but does not cross) the -ais at 0, as shown in Figure f() = 5 1 Figure. Eample 6 Finding a Polnomial Function with Given Zeros Find polnomial functions with the following zeros. (There are man correct solutions.) a. 1 b., 11, 11,, a. Note that the zero 1 corresponds to either 1 or 1. To avoid fractions, choose the second factor and write f b. For each of the given zeros, form a corresponding factor and write f Now tr Eercise 55. Prerequisite Skills If ou have difficult with Eample 6(b), review special products in Section P.. Eploration Use a graphing utilit to graph 1 1. Predict the shape of the curve 1, and verif our answer with a graphing utilit.

7 71_00.qp 1//07 1:1 PM Page 69 Section. Polnomial Functions of Higher Degree 69 Note in Eample 6 that there are man polnomial functions with the indicated zeros. In fact, multipling the functions b an real number does not change 1 the zeros of the function. For instance, multipl the function from part (b) b to obtain f Then find the zeros of the function. You will obtain the zeros, 11, and 11, as given in Eample 6. Eample 7 Sketching the Graph of a Polnomial Function Sketch the graph of f b hand. 1. Appl the Leading Coefficient Test. Because the leading coefficient is positive and the degree is even, ou know that the graph eventuall rises to the left and to the right (see Figure.5).. Find the Real Zeros of the Polnomial. B factoring f ou can see that the real zeros of f are 0 (of odd multiplicit ) and (of odd multiplicit 1). So, the -intercepts occur at and, 0. 0, 0 Add these points to our graph, as shown in Figure.5.. Plot a Few Additional Points. To sketch the graph b hand, find a few additional points, as shown in the table. Be sure to choose points between the zeros and to the left and right of the zeros. Then plot the points (see Figure.6) f Draw the Graph. Draw a continuous curve through the points, as shown in Figure.6. Because both zeros are of odd multiplicit, ou know that the graph should cross the -ais at 0 and. If ou are unsure of the shape of a portion of the graph, plot some additional points. TECHNOLOGY TIP It is eas to make mistakes when entering functions into a graphing utilit. So, it is important to have an understanding of the basic shapes of graphs and to be able to graph simple polnomials b hand. For eample, suppose ou had entered the function in Eample 7 as 5. B looking at the graph, what mathematical principles would alert ou to the fact that ou had made a mistake? Eploration Partner Activit Multipl three, four, or five distinct linear factors to obtain the equation of a polnomial function of degree,, or 5. Echange equations with our partner and sketch, b hand, the graph of the equation that our partner wrote. When ou are finished, use a graphing utilit to check each other s work. Figure.5 Figure.6 Now tr Eercise 71. Activities 1. Find all of the real zeros of f Answer: 1, 0, 6. Determine the right-hand and left-hand behavior of f Answer: The graph rises to the left and right.. Find a polnomial function of degree that has zeros of 0,, and 1. Answer: f 5

8 71_00.qp 1/7/06 1: PM Page Chapter Polnomial and Rational Functions Eample 8 Sketching the Graph of a Polnomial Function Sketch the graph of f Appl the Leading Coefficient Test. Because the leading coefficient is negative and the degree is odd, ou know that the graph eventuall rises to the left and falls to the right (see Figure.7).. Find the Real Zeros of the Polnomial. B factoring f ou can see that the real zeros of f are 0 (of odd multiplicit 1) and (of even multiplicit ). So, the -intercepts occur at and, 0. 0, 0 Add these points to our graph, as shown in Figure.7.. Plot a Few Additional Points. To sketch the graph b hand, find a few additional points, as shown in the table. Then plot the points (see Figure.8.). Draw the Graph. Draw a continuous curve through the points, as shown in Figure.8. As indicated b the multiplicities of the zeros, the graph crosses the -ais at 0, 0 and touches (but does not cross) the -ais at, f STUDY TIP Observe in Eample 8 that the sign of f is positive to the left of and negative to the right of the zero 0. Similarl, the sign of f is negative to the left and to the right of the zero. This suggests that if a zero of a polnomial function is of odd multiplicit, then the sign of f changes from one side of the zero to the other side. If a zero is of even multiplicit, then the sign of f does not change from one side of the zero to the other side. The following table helps to illustrate this result f 0 1 Sign 1 f Sign 9 f( ) = + 6 This sign analsis ma be helpful in graphing polnomial functions. 6 5 Up to left (0, 0) Down to right (, 0) Figure.7 Figure.8 Now tr Eercise 7. TECHNOLOGY TIP Remember that when using a graphing utilit to verif our graphs, ou ma need to adjust our viewing window in order to see all the features of the graph.

9 71_00.qp 1/7/06 1: PM Page 71 Section. Polnomial Functions of Higher Degree 71 The Intermediate Value Theorem The Intermediate Value Theorem concerns the eistence of real zeros of polnomial functions. The theorem states that if a, f a and b, f b are two points on the graph of a polnomial function such that f a f b, then for an number d between f a and f b there must be a number c between a and b such that f c d. (See Figure.9.) fb ( ) fc () = d f( a) Intermediate Value Theorem Let a and b be real numbers such that a < b. If f is a polnomial function such that f a f b, then in the interval a, b, f takes on ever value between f a and f b. Figure.9 a cb This theorem helps ou locate the real zeros of a polnomial function in the following wa. If ou can find a value a at which a polnomial function is positive, and another value b at which it is negative, ou can conclude that the function has at least one real zero between these two values. For eample, the function f 1 is negative when and positive when 1. Therefore, it follows from the Intermediate Value Theorem that f must have a real zero somewhere between and 1. Eample 9 Approimating the Zeros of a Function Find three intervals of length 1 in which the polnomial f 1 5 is guaranteed to have a zero. Graphical Use a graphing utilit to graph 1 5 as shown in Figure = Numerical Use the table feature of a graphing utilit to create a table of function values. Scroll through the table looking for consecutive function values that differ in sign. For instance, from the table in Figure.1 ou can see that f 1 and f 0 differ in sign. So, ou can conclude from the Intermediate Value Theorem that the function has a zero between 1 and 0. Similarl, f 0 and f 1 differ in sign, so the function has a zero between 0 and 1. Likewise, f and f differ in sign, so the function has a zero between and. So, ou can conclude that the function has zeros in the intervals 1, 0, 0, 1, and,. Figure.0 From the figure, ou can see that the graph crosses the -ais three times between 1 and 0, between 0 and 1, and between and. So, ou can conclude that the function has zeros in the intervals 1, 0, 0, 1, and,. Now tr Eercise 79. Figure.1

10 71_00.qp 1/7/06 1: PM Page 7 7 Chapter Polnomial and Rational Functions. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks. 1. The graphs of all polnomial functions are, which means that the graphs have no breaks, holes, or gaps.. The is used to determine the left-hand and right-hand behavior of the graph of a polnomial function.. A polnomial function of degree n has at most real zeros and at most turning points, called.. If a is a zero of a polnomial function f, then the following statements are true. (a) a is a of the polnomial equation f 0. (b) is a factor of the polnomial f. (c) a, 0 is an of the graph of f. 5. If a zero of a polnomial function is of even multiplicit, then the graph of f the -ais, and if the zero is of odd multiplicit, then the graph of f the -ais. 6. The Theorem states that if f is a polnomial function such that f a f b, then in the interval a, b, f takes on ever value between f a and f b. In Eercises 1 8, match the polnomial function with its graph. [The graphs are labeled (a) through (h).] (a) (c) (e) (g) f. f 9 (b) (d) (f) (h) f 5. f 1 5. f 1 6. f 1 7. f 8. f In Eercises 9 and 10, sketch the graph of n and each specified transformation. 9. (a) f (b) f (c) f 1 (d) f 10. (a) f 5 (b) f 5 (c) f (d) f 1 1 Graphical Analsis In Eercises 11 1, use a graphing utilit to graph the functions f and g in the same viewing window. Zoom out far enough so that the right-hand and left-hand behaviors of f and g appear identical. Show both graphs. 11. f 9 1, g 1. f 1, g 1 1. f 16, g 1. f 6, g

11 71_00.qp 1/7/06 1: PM Page 7 Section. Polnomial Functions of Higher Degree 7 In Eercises 15, use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polnomial function. Use a graphing utilit to verif our result. 15. f h g f f f h t t 5t f s 7 8 s 5s 7s 1 In Eercises, find all the real zeros of the polnomial function. Determine the multiplicit of each zero. Use a graphing utilit to verif our result.. f 5. f 9 5. h t t 6t 9 6. f f 8. f 1 9. f t t t t 0. f 0 1. f 1 5. f 5 8 Graphical Analsis In Eercises, (a) find the zeros algebraicall, (b) use a graphing utilit to graph the function, and (c) use the graph to approimate an zeros and compare them with those from part (a).. f 1. g g t 1 t f g t t 5 6t 9t 9. f 0 0. f f In Eercises 5 8, use a graphing utilit to graph the function and approimate (accurate to three decimal places) an real zeros and relative etrema. 5. f f f f In Eercises 9 58, find a polnomial function that has the given zeros. (There are man correct answers.) 9. 0, 50. 7, 51. 0,, 5. 0,, 5 5.,,, 0 5., 1, 0, 1, 55. 1, , 6 57., 5, 5 58., 7, 7 In Eercises 59 6, find a polnomial function with the given zeros, multiplicities, and degree. (There are man correct answers.) 59. Zero:, multiplicit: 60. Zero:, multiplicit: 1 Zero: 1, multiplicit: 1 Zero:, multiplicit: Degree: Degree: 61. Zero:, multiplicit: 6. Zero: 5, multiplicit: Zero:, multiplicit: Zero: 0, multiplicit: Degree: Degree: 5 6. Zero: 1, multiplicit: 6. Zero: 1, multiplicit: Zero:, multiplicit: 1 Zero:, multiplicit: Degree: Degree: Rises to the left, Falls to the left, Falls to the right Falls to the right In Eercises 65 68, sketch the graph of a polnomial function that satisfies the given conditions. If not possible, eplain our reasoning. (There are man correct answers.) 65. Third-degree polnomial with two real zeros and a negative leading coefficient 66. Fourth-degree polnomial with three real zeros and a positive leading coefficient 67. Fifth-degree polnomial with three real zeros and a positive leading coefficient 68. Fourth-degree polnomial with two real zeros and a negative leading coefficient In Eercises 69 78, sketch the graph of the function b (a) appling the Leading Coefficient Test, (b) finding the zeros of the polnomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. 69. f g 71. f 7. f 7. f f f h 5 8 g t 1 t t 78. g

12 71_00.qp 1/7/06 1: PM Page 7 7 Chapter Polnomial and Rational Functions In Eercises 79 8, (a) use the Intermediate Value Theorem and a graphing utilit to find graphicall an intervals of length 1 in which the polnomial function is guaranteed to have a zero, and (b) use the zero or root feature of the graphing utilit to approimate the real zeros of the function. Verif our answers in part (a) b using the table feature of the graphing utilit. in. in. 79. f 80. f g 8. h 10 In Eercises 8 90, use a graphing utilit to graph the function. Identif an smmetr with respect to the -ais, - ais, or origin. Determine the number of -intercepts of the graph. 8. f 6 8. h 85. g t 1 t t 86. g f 88. f 89. g h Numerical and Graphical Analsis An open bo is to be made from a square piece of material 6 centimeters on a side b cutting equal squares with sides of length from the corners and turning up the sides (see figure). 6 (a) Verif that the volume of the bo is given b the function V 6. (b) Determine the domain of the function V. (c) Use the table feature of a graphing utilit to create a table that shows various bo heights and the corresponding volumes V. Use the table to estimate a range of dimensions within which the maimum volume is produced. (d) Use a graphing utilit to graph V and use the range of dimensions from part (c) to find the -value for which V is maimum. 9. Geometr An open bo with locking tabs is to be made from a square piece of material inches on a side. This is done b cutting equal squares from the corners and folding along the dashed lines, as shown in the figure. Figure for 9 (a) Verif that the volume of the bo is given b the function V (b) Determine the domain of the function V. (c) Sketch the graph of the function and estimate the value of for which V is maimum. 9. Revenue The total revenue R (in millions of dollars) for a compan is related to its advertising epense b the function R , 0 00 where is the amount spent on advertising (in tens of thousands of dollars). Use the graph of the function shown in the figure to estimate the point on the graph at which the function is increasing most rapidl. This point is called the point of diminishing returns because an epense above this amount will ield less return per dollar invested in advertising. Revenue (in millions of dollars) R Advertising epense (in tens of thousands of dollars) 9. Environment The growth of a red oak tree is approimated b the function G 0.00t 0.17t 0.58t 0.89 where G is the height of the tree (in feet) and t t is its age (in ears). Use a graphing utilit to graph the function and estimate the age of the tree when it is growing most rapidl. This point is called the point of diminishing returns because the increase in growth will be less with each additional ear. (Hint: Use a viewing window in which 0 5 and )

13 71_00.qp 1/7/06 1: PM Page 75 Section. Polnomial Functions of Higher Degree 75 Data Analsis In Eercises 95 98, use the table, which shows the median prices (in thousands of dollars) of new privatel owned U.S. homes in the Northeast 1 and in the South for the ears 1995 through 00. The data can be approimated b the following models t 6.99t 1 5.9t t 0.58t 1 8.5t In the models, t represents the ear, with t 5 corresponding to (Sources: National Association of Realtors) 95. Use a graphing utilit to plot the data and graph the model for 1 in the same viewing window. How closel does the model represent the data? 96. Use a graphing utilit to plot the data and graph the model for in the same viewing window. How closel does the model represent the data? 97. Use the models to predict the median prices of new privatel owned homes in both regions in 010. Do our answers seem reasonable? Eplain. 98. Use the graphs of the models in Eercises 95 and 96 to write a short paragraph about the relationship between the median prices of homes in the two regions. Snthesis Year, t True or False? In Eercises 99 10, determine whether the statement is true or false. Justif our answer. 99. It is possible for a sith-degree polnomial to have onl one zero The graph of the function f rises to the left and falls to the right The graph of the function f 1 crosses the -ais at The graph of the function f 1 touches, but does not cross, the -ais. 10. The graph of the function f crosses the -ais at. 10. The graph of the function f 1 rises to the left and falls to the right. Librar of Parent Functions In Eercises , determine which polnomial function(s) ma be represented b the graph shown. There ma be more than one correct answer (a) f 1 (b) f 1 (c) f 1 (d) f 1 (e) f (a) f (b) f (c) f (d) f (e) f 107. (a) f 1 (b) f 1 (c) f 1 (d) f 1 (e) f 1 Skills Review In Eercises , let f 1 and g 8. Find the indicated value f g 109. g f fg f g g f 0 In Eercises , solve the inequalit and sketch the solution on the real number line. Use a graphing utilit to verif our solution graphicall < f g