2Polynomial and. Rational Functions

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1 Polnomial and Rational Functions A ballista was used in ancient times as a portable rock-throwing machine. Its primar function was to destro the siege weaponr of opposing forces. Skilled artiller men aimed and fired the ballista entirel b ee. What tpe of projectile path do ou think these artiller men preferred a high, arching trajector or a low, relativel level trajector? Wh? If ou move far enough along a curve of the graph of a rational function, there is a straight line that ou will increasingl approach but never cross or touch. This line is called an asmptote. In Chapter, ou will learn how to use asmptotes as aids for sketching graphs of rational functions. Charles & Josette Lenars/CORBIS 5

2 6 CHAPTER Polnomial and Rational Functions Section. Quadratic Functions Analze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of quadratic functions. Use quadratic functions to model and solve real-life problems. The Graph of a Quadratic Function In this and the net section, ou will stud the graphs of polnomial functions. In Chapter, ou were introduced to the following basic functions. f a b f c f Linear function Constant function Squaring function These functions are eamples of polnomial functions. Definition of Polnomial Function Let n be a nonnegative integer and let a n, a n,..., a, a, a 0 be real numbers with a n 0. The function f a n n a n n... a a a 0 is called a polnomial function of with degree n. Polnomial functions are classified b degree. For instance, a constant function has degree 0 and a linear function has degree. In this section, ou will stud second-degree polnomial functions, which are called quadratic functions. For instance, each of the following functions is a quadratic function. f 6 g h 9 k m Note that the squaring function is a simple quadratic function that has degree. Definition of Quadratic Function Let a, b, and c be real numbers with a 0. The function f a b c is called a quadratic function. The graph of a quadratic function is a special tpe of U-shaped curve that is called a parabola. Parabolas occur in man real-life applications especiall those involving reflective properties of satellite dishes and flashlight reflectors. You will stud these properties in Section..

3 SECTION. Quadratic Functions 7 All parabolas are smmetric with respect to a line called the ais of smmetr, or simpl the ais of the parabola. The point where the ais intersects the parabola is the verte of the parabola, as shown in Figure.. If a > 0, the graph of f a b c is a parabola that opens upward. If a < 0, the graph of f a b c is a parabola that opens downward. Opens upward Ais f() = a + b + c, a < 0 Verte is high point Ais Verte is low point f() = a + b + c, a > 0 Opens downward a > 0 Figure. a < 0 NOTE A precise definition of the terms minimum and maimum will be given in Section 5.. The simplest tpe of quadratic function is f a. Its graph is a parabola whose verte is (0, 0). If a > 0, the verte is the point with the minimum -value on the graph, and if a < 0, the verte is the point with the maimum -value on the graph, as shown in Figure.. f() = a, a > 0 Minimum at (0, 0) Maimum at (0, 0) f() = a, a < 0 Minimum occurs at verte. Figure. Maimum occurs at verte. EXPLORATION. Graph a for a,, 0.5, 0.5,, and. How does changing the value of a affect the graph?. Graph h for h,,, and. How does changing the value of h affect the graph?. Graph k for k,,, and. How does changing the value of k affect the graph?

4 8 CHAPTER Polnomial and Rational Functions Recall from Section. that the graphs of f ± c, f ± c, f, and f are rigid transformations of the graph of f because the do not change the basic shape of the graph. The graph of af is a nonrigid transformation, provided a ±. EXAMPLE Sketching Graphs of Quadratic Functions Sketch the graph of each function and compare it with the graph of. a. f b. g c. h d. k NOTE In parts (c) and (d) of Eample, note that the coefficient a determines how widel the parabola given b f a opens. If a is small, the parabola opens more widel than if a is large. Solution a. To obtain the graph of f, reflect the graph of in the -ais. Then shift the graph upward one unit, as shown in Figure.(a). b. To obtain the graph of g, shift the graph of two units to the left and three units downward, as shown in Figure.(b). c. Compared with, each output of h shrinks b a factor of creating the broader parabola shown in Figure.(c). d. Compared with, each output of k stretches b a factor of, creating the narrower parabola shown in Figure.(d). (0, ) = f() = + g() = ( + ) =, (, ) (a) (b) = k() = h() = = (c) Figure. (d)

5 SECTION. Quadratic Functions 9 The Standard Form of a Quadratic Function The standard form of a quadratic function is f a h k. This form is especiall convenient because it identifies the verte of the parabola. Standard Form of a Quadratic Function The quadratic function given b f a h k, a 0 is in standard form. The graph of f is a parabola whose ais is the vertical line h and whose verte is the point h, k. If a > 0, the parabola opens upward, and if a < 0, the parabola opens downward. The standard form of a quadratic function is useful for sketching a parabola because it identifies four basic transformations of the graph of.. The factor a produces a vertical stretch or shrink.. If a < 0, the graph is reflected in the -ais.. The factor h represents a horizontal shift of h units.. The term k represents a vertical shift of k units. To graph a parabola, it is helpful to begin b writing the quadratic function in standard form using the process of completing the square, as shown in Eample. EXAMPLE Using Standard Form to Graph a Parabola f() = ( + ) Sketch the graph of f 8 7 and identif the verte and the ais of the parabola. Solution Begin b writing the quadratic function in standard form. Notice that the first step in completing the square is to factor out an coefficient of that is not. f 8 7 Write original function. 7 Factor out of -terms. 7 Add and subtract within parentheses. (, ) Figure. = = 7 Regroup terms. 8 7 Simplif. Write in standard form. From this form, ou can see that the graph of f is a parabola that opens upward and has its verte at h, k,. This corresponds to a left shift of two units and a downward shift of one unit relative to the graph of, as shown in Figure.. In the figure, ou can see that the ais of the parabola is the vertical line through the verte,.

6 0 CHAPTER Polnomial and Rational Functions To find the -intercepts of the graph of f a b c, ou can solve the equation a b c 0. If a b c does not factor, ou can use the Quadratic Formula to find the -intercepts. However, remember that a parabola ma not have an -intercepts. EXAMPLE Finding the Verte and -Intercepts of a Parabola Sketch the graph of f 6 8 and identif the verte and -intercepts. Solution As in Eample, begin b writing the quadratic function in standard form. f Write original function. Factor out of -terms. Add and subtract 9 within parentheses. Figure.5 f() = ( ) + (, ) (, 0) (, 0) 5 = Regroup terms. Write in standard form. From this form, ou can see that the verte is,. To find the -intercepts of the graph, solve the equation Write original equation. Factor out. Factor. Set st factor equal to 0. Set nd factor equal to 0. The -intercepts are, 0 and, 0. So, the graph of f is a parabola that opens downward, as shown in Figure.5. EXAMPLE Writing the Equation of a Parabola (, ) = f() Write the standard form of the equation of the parabola whose verte is, and that passes through the point 0, 0, as shown in Figure.6. Solution Because the verte of the parabola is at h, k,, the equation has the form f a. Substitute for h and k in standard form. Because the parabola passes through the point 0, 0, it follows that f 0 0. (0, 0) 0 a 0 0 a a Substitute 0 for. Simplif. Subtract from each side. Figure.6 Substitution into the standard form ields f. Substitute for a in standard form. So, the equation of this parabola is f.

7 SECTION. Quadratic Functions Application Man applications involve finding the maimum or minimum value of a quadratic function. Some quadratic functions are not easil written in standard form. For such functions, it is useful to have an alternative method for finding the verte. For a quadratic function in the form f a b c, the verte occurs when b a. Verte of a Parabola The verte of the graph of f is b a, f a b a b c. EXAMPLE 5 The Maimum Height of a Baseball A baseball is hit at a point feet above the ground at a velocit of 00 feet per second and at an angle of 5 with respect to the ground. The path of the baseball is given b the function f 0.00 where f is the height of the baseball (in feet) and is the horizontal distance from home plate (in feet). a. What is the maimum height reached b the baseball? b. How far does the baseball travel horizontall? Solution For this quadratic function, ou have f a b c So, a 0.00 and b. Because the function has a maimum at b a, ou can conclude that the baseball reaches its maimum height when it is feet from home plate, where is Height (in feet) f() = (56.5, 8.5) Distance (in feet) Figure.7 b a Substitute for a and b feet. a. To find the maimum height, ou must determine the value of the function when f feet b. The path of the baseball is shown in Figure.7. You can estimate from the graph in Figure.7 that the ball hits the ground at a distance of about 0 feet from home plate. The actual distance is the -intercept of the graph of f, which ou can find b solving the equation and taking the positive solution, 5.5 feet.

8 CHAPTER Polnomial and Rational Functions Eercises for Section. In Eercises 8, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] (a) (c) (e) (g) (, ) (, 0) (, 0) (, ) (b) (d) (f) (h) (0, ) (0, ). f. f. f. f 5. f ( ) 6. f 7. f 8. f Eploration In Eercises 9, graph each equation. Compare the graph of each function with the graph of. 9. (a) f (b) g (c) h (d) k 0. (a) f (b) g (c) h (d) k 6 6 (, 0) 6 8 (, ) 6. (a) f (b) g (c) h (d) k. (a) f (b) g (c) (d) h k In Eercises 8, sketch the graph of the quadratic function b hand. Identif the verte and an -intercepts.. f 6. h 5 5. f 6 6. f 7. f f 6 9. h g. f 5. f. f 5. f 5. h 6. f f f 6 In Eercises 9 6, use a graphing utilit to graph the quadratic function. Identif the verte and an -intercepts. Then check our results analticall b completing the square. 9. f 0. f 0. g 8. f 0. f 6. f 5. g 6. f In Eercises 7, find the standard form of the equation of the parabola (0, ) 8 (, 0) (, 0) 6 (0, ) (, 0) 6 8

9 SECTION. Quadratic Functions (, ) (, 0). (, ). 6 (, 0) In Eercises 5, find the quadratic function that has the given verte and whose graph passes through the given point. Verte (, 0) Point , 5,,, 5, 0, 9,, 0, 7, 5 8.,, , 5, 5, 0 6, 6, 0, 7, (, 0) 0, Writing About Concepts (, 0) (0, ) (, ) 6 6 (, ) In Eercises 5 6, (a) determine the -intercepts, if an, of the graph visuall, (b) eplain how the -intercepts relate to the solutions of the quadratic equation when 0, and (c) find the -intercepts analticall to confirm our results Writing About Concepts (continued) Write the quadratic equation f a b c in standard form to verif that the verte occurs at b a, f b a (a) Is it possible for the graph of a quadratic equation to have onl one -intercept? Eplain. (b) Is it possible for the graph of a quadratic equation to have no -intercepts? Eplain

10 CHAPTER Polnomial and Rational Functions In Eercises 65 7, use a graphing utilit to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when f 66. f f f f f 5 7. f f In Eercises 7 78, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given -intercepts. (There are man correct answers.) 7., 0,, , 0, 5, , 0, 0, 0 76., 0, 8, 0 77., 0,, , 0,, 0 In Eercises 79 8, find two positive real numbers whose product is a maimum. 79. The sum is The sum is The sum of the first and twice the second is. 8. The sum of the first and three times the second is. Geometr In Eercises 8 and 8, consider a rectangle of length and perimeter P. (a) Write the area A as a function of and determine the domain of the function. (b) Graph the area function. (c) Find the length and width of the rectangle of maimum area. 8. P 00 feet 8. P 6 meters 85. Numerical, Graphical, and Analtic Analsis A rancher has 00 feet of fencing to enclose two adjacent rectangular corrals (see figure). (a) Complete si rows of a table such as the one below, showing possible values for,, and the area of the corral. Area (b) Use a graphing utilit to generate additional rows of the table. Use the table to estimate the dimensions that will enclose the maimum area. (c) Write the area A as a function of. (d) Use a graphing utilit to graph the area function. Use the graph to approimate the dimensions that will produce the maimum enclosed area. (e) Write the area function in standard form to find analticall the dimensions that will produce the maimum area. 86. Geometr An indoor phsical fitness room consists of a rectangular region with a semicircle on each end (see figure). The perimeter of the room is to be a 00-meter single-lane running track. (a) Determine the radius of the semicircular ends of the room. Determine the distance, in terms of, around the inside edge of the two semicircular parts of the track. (b) Use the result of part (a) to write an equation, in terms of and, for the distance traveled in one lap around the track. Solve for. (c) Use the result of part (b) to write the area A of the rectangular region as a function of. What dimensions will produce a maimum area of the rectangle? 87. Maimum Revenue Find the number of units sold that ields a maimum revenue for a compan that produces health food supplements. The total revenue R (in dollars) is given b R where is the number of units sold.

11 SECTION. Quadratic Functions Maimum Revenue Find the number of units sold that ields a maimum revenue for a sporting goods manufacturer. The total revenue R is given b R where is the number of units sold. 89. Minimum Cost The dail production cost C (in dollars) for a manufacturer of lighting fitures is given b C where is the number of units produced (see figure). How man fitures should be produced each da to ield a minimum cost? 9. Maimum Profit The profit P (in hundreds of dollars) that a custom ccle shop makes depends on the amount (in hundreds of dollars) the compan spends on advertising according to the model P What amount for advertising ields a maimum profit? 9. Height of a Ball The height (in feet) of a ball thrown b a child is given b where is the horizontal distance (in feet) from the point at which the ball is thrown (see figure). C 800 Cost (in dollars) 90. Minimum Cost The dail production cost C (in dollars) for a tetile manufacturer is given b C 00, , where is the number of units produced. How man units should be produced each da to ield a minimum cost? 9. Maimum Profit The profit P (in dollars) for a plastics molding compan is given b P ,000, where is the number of units sold (see figure). What sales level will ield a maimum profit? Profit (in dollars) ,000,000 0,000,000 5,000,000 0,000,000 5,000,000 P Number of units 00,000 00,000 00,000 00, ,000 Number of units (a) How high is the ball when it leaves the child s hand? (b) What is the maimum height of the ball? (c) How far from the child does the ball strike the ground? 9. Path of a Diver The path of a diver is given b 9 9 where is the height (in feet) and is the horizontal distance from the end of the diving board (in feet). (a) How high is the diver when he leaves the diving board? (b) What is the maimum height of the diver? (c) How far from the end of the diving board does the diver hit the surface of the water? 95. Forestr The number of board-feet V in a 6-foot log is approimated b the model V , 5 0 where is the diameter (in inches) of the log at the small end. (One board-foot is a measure of volume equivalent to a board that is inches wide, inches long, and inch thick.) (a) Use a graphing utilit to graph the function. (b) Estimate the number of board-feet in a 6-foot log with a diameter of 6 inches. (c) Estimate the diameter of a 6-foot log that produced 500 board-feet.

12 6 CHAPTER Polnomial and Rational Functions 96. Wind Drag The number of horsepower required to overcome wind drag on an automobile is approimated b 0.00s 0.005s 0.09, where s is the speed of the car (in miles per hour). (a) Use a graphing utilit to graph the function. (b) Graphicall estimate the maimum speed of the car if the power required to overcome wind drag is not to eceed 0 horsepower. Verif our estimate analticall. 97. Graphical Analsis From 960 to 00, the per capita consumption C of cigarettes b Americans (age 8 and older) can be modeled b C.t 0.5t 87, 0 s 00 0 t where t is the ear, with t 0 corresponding to 960. (Source: Tobacco Outlook Report) (a) Use a graphing utilit to graph the model. (b) Use the graph of the model to approimate the maimum average annual consumption. Beginning in 966, all cigarette packages were required b law to carr a health warning. Do ou think the warning had an effect? Eplain. (c) In 000, the U.S. population (age 8 and over) was 09,8,09. Of those, about 8,08,590 were smokers. What was the average annual cigarette consumption per smoker in 000? What was the average dail cigarette consumption per smoker? 98. Maimum Fuel Econom A stud was done to compare the speed (in miles per hour) with the mileage (in miles per gallon) of an automobile. The results are shown in the table. (Source: Federal Highwa Administration) Speed, Mileage, Speed, Mileage, (a) Use the regression feature of a graphing utilit to find a quadratic model for the data. (b) Use a graphing utilit to plot the data and graph the model in the same viewing window. (c) Estimate the speed for which the miles per gallon is greatest. Verif analticall. 99. Data Analsis The numbers (in thousands) of hairdressers and cosmetologists in the United States for the ears 996 through 00 are shown in the table. (Source: U.S. Bureau of Labor Statistics) Year Number, (a) Use a graphing utilit to create a scatter plot of the data. Let t represent the ear, with t 6 corresponding to 996. (b) Use the regression feature of a graphing utilit to find a quadratic model for the data. (c) Use a graphing utilit to graph the model in the same viewing window as the scatter plot. How well does the model fit the data? (d) Use the trace feature of the graphing utilit to approimate the ear in which the number of hairdressers and cosmetologists was the least. (e) Use the model to predict the number of hairdressers and cosmetologists in 00. True or False? In Eercises 00 0, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 00. The function given b f has no -intercepts. 0. The graphs of two functions given b f 0 7 and g 0 have the same ais of smmetr. 0. A quadratic function of must have eactl one -intercept. 0. All quadratic functions that have the form a, where a 0, have the same -intercepts.

13 SECTION. Polnomial Functions of Higher Degree 7 Section. NOTE A precise definition of the term continuous is given in Section.. Polnomial Functions of Higher Degree Use transformations to sketch graphs of polnomial functions. Determine the end behavior of graphs of polnomial functions using the Leading Coefficient Test. Use zeros of polnomial functions as sketching aids. Graphs of Polnomial Functions In this section, ou will stud basic features of the graphs of polnomial functions. The first feature is that 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.8. (a) Polnomial functions have continuous graphs. Figure.8 (b) Functions with graphs that are not continuous are not polnomial functions. The second feature is that the graph of a polnomial function has onl smooth, rounded turns, as shown in Figure.9(a). A polnomial function cannot have a sharp turn. For instance, the function given b f, which has a sharp turn at the point 0, 0, as shown in Figure.9(b), is not a polnomial function. 6 5 f() = (0, 0) (a) Polnomial functions have graphs with smooth, rounded turns. Figure.9 (b) Functions whose graphs have sharp turns are not polnomial functions. The graphs of polnomial functions of degree greater than are more difficult to analze than the graphs of polnomials of degree 0,, or. However, using the features presented in this section, together with point plotting, intercepts, and smmetr, ou should be able to make reasonabl accurate sketches b hand. In Chapter 5, ou will learn more techniques for analzing the graphs of polnomial functions.

14 8 CHAPTER Polnomial and Rational Functions The polnomial functions that have the simplest graphs are monomials of the form f n where n is an integer greater than zero. From Figure.0, ou can see that when n is even the graph is similar to the graph of f and when n is odd the graph is similar to the graph of f. Moreover, the greater the value of n, the flatter the graph near the origin. Polnomial functions of the form f n are often referred to as power functions. = (, ) = = 5 (, ) = (, ) (, ) (a) If n is even, the graph of n touches the ais at the -intercept. Figure.0 (b) If n is odd, the graph of n crosses the ais at the -intercept. EXAMPLE Sketching Transformations of Power Functions Sketch the graph of each function. a. f 5 b. h Solution a. Because the degree of f 5 is odd, its graph is similar to the graph of. As shown in Figure.(a), the graph of f 5 is a reflection in the -ais of the graph of 5. b. The graph of h, as shown in Figure.(b), is a left shift of one unit of the graph of. h() = ( + ) = (, ) = 5 f() = 5 (, ) (0, ) (, ) (, 0) (a) Figure. (b)

15 SECTION. Polnomial Functions of Higher Degree 9 EXPLORATION For each polnomial function, identif the degree of the function and whether the degree of the function is even or odd. Identif the leading coefficient and whether it is greater than 0 or less than 0. Use a graphing utilit to graph each function. Describe the relationship between the degree and the sign of the leading coefficient of the polnomial function and the right- and left-hand behaviors of the graph of the function. a. f b. f 5 5 c. f 5 5 d. f 5 e. f f. f g. f The Leading Coefficient Test In Eample, note that both 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 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 a 0 eventuall rises or falls in the following manner.. When n is odd: f() as f() as f() as f() as If the leading coefficient is If the leading coefficient is positive a n > 0, the graph negative a n < 0, the graph falls to the left and rises to rises to the left and falls to the right. the right.. When n is even: f() as f() as f() as f() as If the leading coefficient is If the leading coefficient is positive a n > 0, the graph negative a n < 0, the graph rises to the left and right. falls to the left and right. The dashed portions of the graphs indicate that the test determines onl the right-hand and left-hand behaviors of the graph. The notation f as indicates that the graph rises without bound to the right. The notations f as, f as, and f as have similar meanings. You will stud precise definitions of these concepts in Section 5.5.

16 50 CHAPTER Polnomial and Rational Functions EXAMPLE Appling the Leading Coefficient Test Describe the right-hand and left-hand behaviors of the graph of f. Solution 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.. f() = + Figure. 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 using other tests. EXAMPLE Appling the Leading Coefficient Test Describe the right-hand and left-hand behaviors of the graph of each function. a. f 5 b. f 5 Solution a. Because the degree is even and the leading coefficient is positive, the graph rises to the left and right, as shown in Figure.(a). b. 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.(b). f() = 5 + f() = 5 (a) Figure. (b)

17 SECTION. Polnomial Functions of Higher Degree 5 Real Zeros of Polnomial Functions It can be shown that for a polnomial function f of degree n, the following statements are true. (Remember that the zeros of a function of are the -values for which the function is zero.). The graph of f has, at most, n turning points. (Turning points are points at which the graph changes from increasing to decreasing, or vice versa.). The function f has, at most, n real zeros. (You will stud this result in detail in Section.5 on the Fundamental Theorem of Algebra.) Finding the zeros of polnomial functions is one of the most important problems in algebra. There is a strong interpla between graphical and analtic approaches to this problem. Sometimes ou can use information about the graph of a function to help find its zeros, and in other cases ou can use information about the zeros of a function to help sketch its graph. Real Zeros of Polnomial Functions If f is a polnomial function and a is a real number, the following statements are equivalent.. 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. NOTE In the equivalent statements above, notice that finding real zeros of polnomial functions is closel related to factoring and finding -intercepts. EXAMPLE Finding the Real Zeros of a Polnomial Function Turning point (, 0) Figure. Turning point (0, 0) f() = + Turning point (, 0) Find all real zeros of f (). Use the graph in Figure. to determine the number of turning points of the graph of the function. Solution In this case, the polnomial factors as shown. f Write original function. Remove common monomial factor. Factor completel. So, the real zeros are 0,, and, and the corresponding -intercepts are 0, 0,, 0, and, 0, as shown in Figure.. Note in the figure that the graph has three turning points. This is consistent with the fact that a fourthdegree polnomial can have at most three turning points. NOTE In Eample, note that because k is even, the factor ields the repeated zero 0. The graph touches the -ais at 0, as shown in Figure.. Repeated Zeros A factor a k, k >, ields a repeated zero a of multiplicit k.. 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.

18 5 CHAPTER Polnomial and Rational Functions EXAMPLE 5 Sketching the Graph of a Polnomial Function Sketch the graph of f. Solution. 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(a)].. Find the Real Zeros of the Polnomial. B factoring f ou can see that the zeros of f are 0 and (both of odd multiplicit). So, the -intercepts occur at and, 0. 0, 0 Add these points to our graph, as shown in Figure.5(a).. Plot a Few Additional Points. Use the zeros of the polnomial to find the test intervals. In each test interval, choose a representative -value and evaluate the polnomial function, as shown in the table. TECHNOLOGY Eample 5 uses an analtic approach to describe the graph of the function. A graphing utilit is a complement to this approach. Remember that an important aspect of using a graphing utilit is to find a viewing window that shows all significant features of the graph. For instance, which of the graphs in Figure.6 shows all of the significant features of the function in Eample 5?. Representative Point on Test Interval -Value Value of f Sign Graph, 0 f 7 Positive, 7 0, f Negative,.5 f Positive.5,.6875,. Draw the Graph. Draw a continuous curve through the points, as shown in Figure.5(b). Because both zeros are of odd multiplicit, ou know that the graph should cross the - ais at 0 and (a) Up to left Up to right (0, 0) ( ), f() = (b) Figure (a) Figure.5 NOTE If ou are unsure of the shape of a portion of the graph of a polnomial function, plot some additional points, such as the point 0.5, 0.5, as shown in Figure.5(b). (b)

19 SECTION. Polnomial Functions of Higher Degree 5 Before appling the Leading Coefficient Test to a polnomial function, it is a good idea to check that the polnomial function is written in standard form. EXAMPLE 6 Sketching the Graph of a Polnomial Function Sketch the graph of f 6 9. Solution. 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(a)].. Find the Real Zeros of the Polnomial. B factoring f ou can see that the zeros of f are 0 (odd multiplicit) and (even multiplicit). So, the -intercepts occur at and 0, 0, 0. Add these points to our graph, as shown in Figure.7(a).. Plot a Few Additional Points. Use the zeros of the polnomial to find the test intervals. In each test interval, choose a representative -value and evaluate the polnomial function, as shown in the table. NOTE Observe in Eample 6 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 of and to the right of the zero. This suggests that if the 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 the zero is of even multiplicit, then the sign of f does not change from one side of the zero to the other side. Representative Point on Test Interval -Value Value of f Sign Graph, f 0.5 Positive 0.5, 0, 0.5 f 0.5 Negative 0.5, f Negative,,. Draw the Graph. Draw a continuous curve through the points, as shown in Figure.7(b). As indicated b the multiplicities of the zeros, the graph crosses the -ais at 0, 0 but does not cross the -ais at, f() = Up to left Down to right (0, 0) (, 0 ) (a) Figure.7 (b)

20 5 CHAPTER Polnomial and Rational Functions Eercises for Section. In Eercises 8, match the polnomial function with its graph. [The graphs are labeled (a) through (h).] (a) (c) (e) (g) (b) (d) (f) (h). f. f. f 5. f 5. f 6. f 7. f 8. f In Eercises 9, sketch the graph of n and each transformation (a) f (b) f (c) f (d) f (a) f 5 (b) f 5 (c) f 5 (d) f. (a) f (b) f (c) f (d) f. 6 (a) f 8 6 (b) f 6 (c) f 6 (d) f 6 In Eercises, determine the right-hand and lefthand behaviors of the graph of the polnomial function.. f 5. f 5. g h f. 5 f f 6 5 f 5 h t t 5t f s 7 8 s 5s 7s Graphical Analsis In Eercises 6, use a graphing utilit to graph the functions f and g in the same viewing window. Zoom out sufficientl far to show that the right-hand and left-hand behaviors of f and g appear to be identical.. f 9, g. f, g 5. f 6, g 6. f 6, g In Eercises 7, find all the real zeros of the polnomial function. 7. f 5 8. f 9 9. h t t 6t 9 0. f 0 5. f. f 5. f. g 5 5. f t t t t 6. f 0 7. g t t 8. f g t t 5 6t 9t 0. f 0. f f 5 00

21 SECTION. Polnomial Functions of Higher Degree 55 Graphical Analsis In Eercises 6, (a) use a graphing utilit to graph the function, (b) use the graph to approimate an -intercepts of the graph, and (c) set 0 and solve the resulting equation. Compare the result with an -intercepts of the graph In Eercises 7 56, find a polnomial function that has the given zeros. (There are man correct answers.) 7. 0, , 9., 6 50., ,, 5. 0,, 5 5.,,, 0 5.,, 0,, 55., 56., 5, 5 In Eercises 57 66, find a polnomial of degree n that has the given zeros. (There are man correct answers.) Zeros , 0, 60.,, ,, 6. 0,, 6.,, ± 6.,, ± ,, 66.,, 5, 6 In Eercises 67 80, 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. 67. f g 69. f t t t g f 7. f 7. f f 5 Degree n n n n n n n n n 5 n f f f 78. h 79. g t t t 80. g 0 In Eercises 8 8, use a graphing utilit to graph the function. Use the zero or root feature to approimate the zeros of the function. Determine the multiplicit of each zero. 8. f 8. f 8. g h 5 5 Writing About Concepts 85. Describe a polnomial function that could represent the graph. Indicate the degree of the function and the sign of its leading coefficient. (a) (b) (c) (d) 86. Consider the function given b f. Eplain how the graph of g differs (if it does) from the graph of f. Determine whether g is odd, even, or neither. (a) g f (b) g f (c) g f (d) g f (e) g f (f) g f (g) g f (h) g f f

22 56 CHAPTER Polnomial and Rational Functions 87. Numerical and Graphical Analsis An open bo is to be made from a square piece of material, 6 inches on a side, b cutting equal squares of length from the corners and turning up the sides (see figure). (a) Complete four rows of a table such as the one below. (b) Use a graphing utilit to generate additional rows of the table. Use the table to estimate the dimensions that will produce a maimum volume. (c) Verif that the volume of the bo is given b the function Determine the domain of the function. (d) Use a graphing utilit to graph V and use the graph to estimate the value of for which V is maimum. Compare our result with that of part (b). 88. Geometr An open bo with locking tabs is to be made from a square piece of material inches on a side. This is to be done b cutting equal squares from the corners and folding along the dashed lines shown in the figure. 6 V 6. in. (a) Verif that the volume of the bo is given b the function V 8 6. in. Bo Bo Bo Height Width Volume (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. 89. Revenue The total revenue R (in millions of dollars) for a compan is related to its advertising epense b the function given b R where is the amount spent on advertising (in tens of thousands of dollars). Use the graph of this 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) 00, , R Advertising epense (in tens of thousands of dollars) 90. Sports The number of males N (in millions) participating in high school athletics from 990 to 00 can be approimated b the function given b N t 0.06t 0.0t. where t is the ear, with t 0 corresponding to 990. Use a graphing utilit to graph the function and estimate the ear when participation is growing most rapidl. This point is called the point of diminishing returns because the increase in the number of participants will be less with each additional ear. Hint: Use a viewing window in which 0 and.5. (Source: National Federation of State High School Associations) True or False? In Eercises 9 9, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 9. A fifth-degree polnomial can have five turning points in its graph. 9. It is possible for a sith-degree polnomial to have onl one zero. 9. The graph of a third-degree polnomial function must fall to the left and rise to the right. 9. If a function has a repeated zero of even multiplicit, then the graph of the function onl touches the -ais at that point, but does not cross the -ais at that point.

23 SECTION. Polnomial and Snthetic Division 57 Section. f() = (, 0 ) ( ) Figure.8, 0 (, 0) Polnomial and Snthetic Division Divide polnomials using long division. Use snthetic division to divide polnomials b binomials of the form Use the Remainder Theorem and the Factor Theorem. Use polnomial division to answer questions about real-life problems. Long Division of Polnomials In this section, ou will stud two procedures for dividing polnomials. These procedures are especiall valuable in factoring and finding the zeros of polnomial functions. To begin, suppose ou are given the graph of Notice that a zero of f occurs at, as shown in Figure.8. Because is a zero of f, ou know that is a factor of f. This means that there eists a second-degree polnomial q such that To find q, ou can use long division, as shown in Eample. EXAMPLE Long Division of Polnomials Divide b, and then use the result to factor the polnomial completel. Solution f f q. 6 7 ) From this division, ou can conclude that Think Think Think and b factoring the quadratic 6 7, ou have Multipl: 6. Subtract. Multipl: 7. Subtract. Multipl:. Subtract. k. NOTE that this factorization agrees with the graph shown in Figure.8 in that the three -intercepts occur at, and,.

24 58 CHAPTER Polnomial and Rational Functions In Eample, is a factor of the polnomial 6 9 6, and the long division process produces a remainder of zero. Often, long division will produce a nonzero remainder. For instance, if ou divide 5 b, ou obtain the following. Divisor Quotient Dividend Remainder In fractional form, ou can write this result as shown. Dividend 5 Divisor This implies that ) 5 Quotient 5 Remainder Divisor 5 ( Multipl each side b. which illustrates the following theorem, called the Division Algorithm. The Division Algorithm If f and d are polnomials such that d 0, and the degree of d is less than or equal to the degree of f, there eist unique polnomials q and r such that f d q r Dividend Quotient Divisor Remainder where r 0 or the degree of r is less than the degree of d. If the remainder r is zero, d divides evenl into f. The Division Algorithm can also be written as f r q d d. In the Division Algorithm, the rational epression f d is improper because the degree of f is greater than or equal to the degree of d. On the other hand, the rational epression r d is proper because the degree of r is less than the degree of d. Here are some eamples. 5 Improper rational epression Proper rational epression

25 SECTION. Polnomial and Snthetic Division 59 EXAMPLE Long Division of Polnomials Divide b. Solution Because there is no -term or -term in the dividend, ou need to line up the subtraction b using zero coefficients (or leaving spaces) for the missing terms. So, divides evenl into and ou can write Check ) 0 0 0, 0. You can check the result of a division problem b multipling. EXAMPLE Long Division of Polnomials Divide 5 b. Solution Note that the first subtraction eliminated two terms from the dividend. When this happens, the quotient skips a term. You can write the result as Check )

26 60 CHAPTER Polnomial and Rational Functions NOTE Snthetic division works onl for divisors of the form k. You cannot use snthetic division to divide a polnomial b a quadratic such as. Snthetic Division There is a nice shortcut for long division of polnomials when dividing b divisors of the form k. This shortcut is called snthetic division. The pattern for snthetic division of a cubic polnomial is summarized below. (The pattern for higher-degree polnomials is similar.) Snthetic Division (for a Cubic Polnomial) To divide a b c d b k, use the following pattern. k a b c d ka a r Remainder Coefficients of quotient Coefficients of dividend Vertical pattern:add terms. Diagonal pattern: Multipl b k. EXAMPLE Using Snthetic Division Use snthetic division to divide 0 b. Solution You should set up the arra as shown below. Note that a zero is included for each missing term in the dividend. 0 0 Then, use the snthetic division pattern b adding terms in columns and multipling the results b. Divisor: Dividend: Remainder: So, ou have Check 0 Quotient:. 0 0

27 SECTION. Polnomial and Snthetic Division 6 The Remainder and Factor Theorems The remainder obtained in the snthetic division process has an important interpretation, as described in the Remainder Theorem. THEOREM. The Remainder Theorem If a polnomial f is divided b k, then the remainder is r f k. Proof From the Division Algorithm, ou have f k q r and because either r 0 or the degree of r is less than the degree of k, ou know that r must be a constant. That is, r r. Now, b evaluating f at k, ou have f k k k q k r 0 q k r r. The Remainder Theorem tells ou that snthetic division can be used to evaluate a polnomial function. That is, to evaluate a polnomial function f when k, divide f b k. The remainder will be f k, as shown in Eample 5. EXAMPLE 5 Using the Remainder Theorem Use the Remainder Theorem to evaluate the following function at. Solution f Using snthetic division, ou obtain the following. 8 6 Because the remainder is r 9, ou can conclude that f This means that, 9 is a point on the graph of f. You can check this b substituting in the original function. Another important theorem is the Factor Theorem, which is stated below. This theorem states that ou can test to see whether a polnomial has k as a factor b evaluating the polnomial at k. If the result is 0, k is a factor. THEOREM. The Factor Theorem A polnomial f has a factor k if and onl if f k 0.

28 6 CHAPTER Polnomial and Rational Functions Proof Using the Division Algorithm with the factor k, ou have f k q r. B the Remainder Theorem, r r f k, and ou have f k q f k where q is a polnomial of lesser degree than f. If f k 0, then f k q and ou see that k is a factor of f. Conversel, if k is a factor of f, division of f b k ields a remainder of 0. So, b the Remainder Theorem, ou have f k 0. EXAMPLE 6 Factoring a Polnomial: Repeated Division Show that and are factors of Then find the remaining factors of f. f f() = (, 0 ) 0 (, 0) (, 0) (, 0) Figure.9 Solution Use snthetic division with the factor. Use the result of this division to perform snthetic division again with the factor. 7 6 Because the resulting quadratic epression factors as 5 the complete factorization of f is f. 0 remainder, so f 0 and is a factor. 0 remainder, so f 0 and is a factor. Note that this factorization implies that f has four real zeros:,,, and. This is confirmed b the graph of f, which is shown in Figure.9. Uses of the Remainder in Snthetic Division The remainder r, obtained in the snthetic division of f b k, provides the following information.. The remainder r gives the value of f at k. That is, r f k.. If r 0, k is a factor of f.. If r 0, k, 0 is an -intercept of the graph of f.

29 SECTION. Polnomial and Snthetic Division 6 Application EXAMPLE 7 Take-Home Pa The 00 monthl take-home pa for an emploee who is single and claimed one deduction is given b the function , where represents the take-home pa (in dollars) and represents the gross monthl salar (in dollars). Find a function that gives the take-home pa as a percent of the gross monthl salar. Solution Because the gross monthl salar is given b and the take-home pa is given b, the percent P of gross monthl salar that the person takes home is P The graphs of and P are shown in Figure.0(a) and (b), respectivel. Note in Figure.0(b) that as a person s gross monthl salar increases, the percent that he or she takes home decreases. P Take-home pa (in dollars) = Take-home pa (as percent of gross) P = Gross monthl salar (in dollars) Gross monthl salar (in dollars) (a) Figure.0 (b) Throughout this tet, the importance of developing several problem-solving strategies is emphasized. In the eercises for this section, tr using more than one strateg to solve several of the eercises. For instance, if ou find that k divides evenl into f (with no remainder), tr sketching the graph of f. You should find that k, 0 is an -intercept of the graph. Your problem-solving skills will be enhanced, too, b using a graphing utilit to verif algebraic calculations, and conversel, to verif graphing utilit results b analtic methods.

30 6 CHAPTER Polnomial and Rational Functions Eercises for Section. Analtic Analsis In Eercises, use long division to verif that..... Graphical and Analtic Analsis In Eercises 5 and 6, use a graphing utilit to graph the two equations in the same viewing window. Use the graphs to verif that the epressions are equivalent. Use long division to verif that analticall , 5,,, 5 5, 5, In Eercises 7 0, use long division to divide In Eercises 8, use snthetic division to divide In Eercises 9 6, write the function in the form f k q r for the given value of k, and demonstrate that f k r. Function 9. f 0. f 5 8. f f 0. f. f 5 5. f 6 6. f Value of k k k k k 5 k k 5 k k In Eercises 7 50, use snthetic division to find each function value. Verif using another method. 7. f 0 (a) f (b) f (c) f (d) f 8 8. g 6 (a) g (b) g (c) g (d) g

31 SECTION. Polnomial and Snthetic Division h 5 0 (a) h (b) h (c) h (d) h f (a) f (b) f (c) f 5 (d) f 0 In Eercises 5 58, use snthetic division to show that the given value of is a solution of the third-degree polnomial equation, and use the result to factor the polnomial completel. List all real zeros of the function. Polnomial Equation Value of In Eercises 59 66, (a) verif the given factors of the function f, (b) find the remaining factors of f, (c) use our results to write the complete factorization of f, (d) list all real zeros of f, and (e) confirm our results b using a graphing utilit to graph the function. Function Factors f 5 f 9 6 f ,, 5, 6. f 8 7 0, f 6 9 f f 0 5 f 8, 5, 5, 5, Graphical Analsis In Eercises 67 70, (a) use the zero or root feature of a graphing utilit to approimate the zeros of the function accurate to three decimal places, (b) determine one or more of the eact zeros and use snthetic division to verif our result, and (c) factor the polnomial completel. 67. f g h t t t 7t 70. f s s s 0s In Eercises 7 76, simplif the rational epression Writing About Concepts In Eercises 77 and 78, perform the division b assuming that n is a positive integer n 9 n 7 n 7 n n n 5 n 6 n 79. Briefl eplain what it means for a divisor to divide evenl into a dividend. 80. Briefl eplain how to check polnomial division, and justif our reasoning. Give an eample. In Eercises 8 and 8, find the constant c such that the denominator will divide evenl into the numerator. 8. c 8. 5 c 5 In Eercises 8 and 8, answer the questions about the division. k 8. What is the remainder when k? Eplain our reasoning. 8. If it is necessar to find f, is it easier to evaluate the function directl or to use snthetic division? Eplain our reasoning.

32 66 CHAPTER Polnomial and Rational Functions 85. Data Analsis The average monthl basic rates R (in dollars) for cable television in the United States for the ears 990 through 00 are shown in the table, where t represents the time (in ears), with t 0 corresponding to 990. (Source: Kagan World Media) Year, t 0 5 Rate, R Year, t Rate, R (a) Use a graphing utilit to create a scatter plot of the data. (b) Use the regression feature of the graphing utilit to find a cubic model for the data. Then graph the model in the same viewing window as the scatter plot. Compare the model with the data. (c) Use the model to create a table of estimated values of R. Compare the estimated values with the actual data. (d) Use snthetic division to evaluate the model for the ear 005. Even though the model is relativel accurate for estimating the given data, would ou use this model to predict future cable rates? Eplain our reasoning. 86. Data Analsis The numbers M (in thousands) of United States militar personnel on active dut for the ears 990 through 00 are shown in the table, where t represents the time (in ears), with t 0 corresponding to 990. (Source: U.S. Department of Defense) (c) Use the model to create a table of estimated values of M. Compare the estimated values with the actual data. (d) Use snthetic division to evaluate the model for the ear 005. Even though the model is relativel accurate for estimating the given data, would ou use this model to predict the number of militar personnel in the future? Eplain our reasoning. True or False? In Eercises 87 90, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 87. If 7 is a factor of some polnomial function f, then 7 is a zero of f. 88. is a factor of the polnomial If k is a zero of a function f, then f k To divide b using snthetic division, the setup would appear as shown. Year, t 0 Personnel, M Year, t Personnel, M Year, t Personnel, M (a) Use a graphing utilit to create a scatter plot of the data. (b) Use the regression feature of the graphing utilit to find a cubic model for the data. Then graph the model in the same viewing window as the scatter plot. Compare the model with the data.

33 SECTION. Comple Numbers 67 Section. Comple Numbers Use the imaginar unit i to write comple numbers. Add, subtract, and multipl comple numbers. Use comple conjugates to write the quotient of two comple numbers in standard form. Find comple solutions of quadratic equations. The Imaginar Unit i You have learned that some quadratic equations have no real solutions. For instance, the quadratic equation 0 Equation with no real solution has no real solution because there is no real number that can be squared to produce. To overcome this deficienc, mathematicians created an epanded sstem of numbers using the imaginar unit i, defined as i Imaginar unit where i. B adding real numbers to real multiples of this imaginar unit, the set of comple numbers is obtained. Each comple number can be written in the standard form a bi. The real number a is called the real part of the comple number a bi, and the number bi (where b is a real number) is called the imaginar part of the comple number. Definition of a Comple Number If a and b are real numbers, the number a bi is a comple number, and it is said to be written in standard form. If b 0, the number a bi a is a real number. If b 0, the number a bi is called an imaginar number. A number of the form bi, where b 0, is called a pure imaginar number. The set of real numbers is a subset of the set of comple numbers, as shown in Figure.. This is true because ever real number a can be written as a comple number using b 0. That is, for ever real number a, ou can write a a 0i. Real numbers Imaginar numbers Comple numbers Figure. Equalit of Comple Numbers Two comple numbers a bi and c di, written in standard form, are equal to each other a bi c di if and onl if a c and b d. Equalit of two comple numbers

34 68 CHAPTER Polnomial and Rational Functions Operations with Comple Numbers To add (or subtract) two comple numbers, ou add (or subtract) the real and imaginar parts of the numbers separatel. Addition and Subtraction of Comple Numbers If a bi and c di are two comple numbers written in standard form, their sum and difference are defined as follows. Sum: a bi c di a c b d i Difference: a bi c di a c b d i The additive identit in the comple number sstem is zero (the same as in the real number sstem). Furthermore, the additive inverse of the comple number a bi is (a bi) a bi. Additive Inverse So, ou have a bi a bi 0 0i 0. EXAMPLE Adding and Subtracting Comple Numbers Perform the operations on the comple numbers. a. i i b. i i c. i 5 i d. i i 7 i Solution a. Remove parentheses. i i Group like terms. i 5 i Write in standard form. b. i i i i Remove parentheses. c. d. i i i i i i Group like terms. Write in standard form. i 5 i i 5 i 5 i i 0 i i i i 7 i i i 7 i 7 i i i 0 0i 0 Note in Eample (b) that the sum of two comple numbers can be a real number.

35 SECTION. Comple Numbers 69 EXPLORATION Complete the table. i i i i i i i 5 i 6 i 7 i 8 i 9 i 0 i i What pattern do ou see? Write a brief description of how ou would find i raised to an positive integer power. Man of the properties of real numbers are valid for comple numbers as well. Here are some eamples. Associative Properties of Addition and Multiplication Commutative Properties of Addition and Multiplication Distributive Propert of Multiplication Over Addition Notice below how these properties are used when two comple numbers are multiplied. a bi c di a c di bi c di ac ad i bc i bd i ac ad i bc i bd ac bd ad i bc i ac bd ad bc i Distributive Propert Distributive Propert i Commutative Propert Associative Propert Rather than tring to memorize this multiplication rule, ou should simpl remember how the Distributive Propert is used to multipl two comple numbers. The procedure is similar to multipling two polnomials and combining like terms. EXAMPLE Multipling Comple Numbers Multipl the comple numbers. a. i b. i i c. d. i i e. i i i Solution a. i i 8 i Distributive Propert Simplif. b. i i i Multipl. i Simplif. c. i i 8 6i i i 8 6i i Product of binomials i 8 6i i i Group like terms. Write in standard form. d. i i 9 6i 6i i 9 6i 6i Product of binomials i 9 Simplif. Write in standard form. e. i 9 6i 6i i 9 6i 6i Product of binomials i 9 i 5 i Simplif. Write in standard form.

36 70 CHAPTER Polnomial and Rational Functions Comple Conjugates Notice in Eample (d) that the product of two comple numbers can be a real number. This occurs with pairs of comple numbers of the form a bi and a bi, called comple conjugates. a bi a bi a abi abi b i a b a b EXAMPLE Multipling Conjugates Multipl each comple number b its comple conjugate. a. i b. i Solution a. The comple conjugate of i is i. i i i b. The comple conjugate of i is i. i i i 6 9i To write the quotient of a bi and c di in standard form, where c and d are not both zero, multipl the numerator and denominator b the comple conjugate of the denominator to obtain a bi a bi c di c di c di c di ac bd bc ad i c d ac bd bc ad i c d c d. Standard form EXAMPLE Writing a Comple Number in Standard Form Write the comple number i i in standard form. Solution i i i i i i 8 i i 6i 6 i 8 i i 6 6 6i i Multipl numerator and denominator b comple conjugate of denominator. Epand. i Simplif and write in standard form.

37 SECTION. Comple Numbers 7 Comple Solutions of Quadratic Equations When using the Quadratic Formula to solve a quadratic equation, ou often obtain a result such as, which ou know is not a real number. B factoring out i, ou can write this number in standard form. i The number i is called the principal square root of. STUDY TIP The definition of principal square root uses the rule for a > 0 and b < 0. This rule is not valid if both a and b are negative. For eample, whereas ab a b i 5i 5i 5i To avoid problems with multipling square roots of negative numbers, be sure to convert to standard form before multipling. Principal Square Root of a Negative Number If a is a positive number, the principal square root of the negative number a is defined as a ai. EXAMPLE 5 Writing Comple Numbers in Standard Form Write each comple number in standard form and simplif. a. b. 8 7 c. Solution a. i i 6 i 6 6 b i 7 i i i i c. i i i i i EXAMPLE 6 Comple Solutions of a Quadratic Equation Solve each quadratic equation. a. 0 b. 5 0 Solution a. 0 ±i Write original equation. Subtract from each side. Etract square roots. b. 5 0 ± 5 ± 56 6 Write original equation. Quadratic Formula Simplif. ± i 6 ± i Write 56 in standard form. Write in standard form.

38 7 CHAPTER Polnomial and Rational Functions Eercises for Section. In Eercises, find real numbers a and b such that the equation is true.. a bi 0 6i. a bi i. a b i 5 8i. a 6 bi 6 5i In Eercises 5 6, write the comple number in standard form i i. i i i 0i In Eercises 7 6, perform the addition or subtraction and write the result in standard form i 6 i 8. i 5 6i 9. 8 i i 0. i 6 i i. i 7i i 5.6.i 5.8.i In Eercises 7 0, perform the operation and write the result in standard form i i. 6 i i. 6i 5 i. 8i 9 i i 0i i 8. i 9. i i 0. i i In Eercises 8, write the comple conjugate of the comple number. Then multipl the number b its comple conjugate.. 6 i. 7 i. 5i. i In Eercises 9 58, write the quotient in standard form i i i i 5. i 6 7i 5. i i i 8 6i 56. i i 57. i 5i 58. 5i i In Eercises 59 6, perform the operation and write the result in standard form. 59. i i i i i 8i 6. i i 5 i i i i In Eercises 6 7, use the Quadratic Formula to solve the quadratic equation t t In Eercises 7 76, write the power of i as i, i,, or. 7. i 0 7. i i i 67 In Eercises 77 8, simplif the comple number and write it in standard form. 77. i i 78. 6i i 79. i 80. 5i i 8. i True or False? In Eercises 85 88, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 85. There is no comple number that is equal to its comple conjugate.

39 SECTION. Comple Numbers i is a solution of i i 50 i 7 i 09 i Writing About Concepts 89. Show that the product of a comple number and its comple conjugate is a real number. 90. Describe the error. 9. Eplain how to perform operations with comple numbers, that is, add, subtract, multipl, and divide. 9. Show that the comple conjugate of the sum of two comple numbers a b i and a b i is the sum of their comple conjugates. 9. Impedance The opposition to current in an electrical circuit is called its impedance. The impedance z in a parallel circuit with two pathwas satisfies the equation z z z where z is the impedance (in ohms) of pathwa and z is the impedance of pathwa. Use the table to determine the impedance of each parallel circuit. The impedance of each pathwa is found b adding the impedance of each component in the pathwa. (a) (b) Smbol Impedance Resistor Inductor Capacitor aω a 5 Ω Ω 6 Ω bω bi Ω Ω 0 Ω cω ci a bi Section Project: The Mandelbrot Set Graphing utilities can be used to draw pictures of fractals in the comple plane. The most famous fractal is called the Mandelbrot Set, after the Polish-born mathematician Benoit Mandelbrot. To construct the Mandelbrot Set, consider the following sequence of numbers. c, c c, c c c, c c c c,... The behavior of this sequence depends on the value of the comple number c. For some values of c this sequence is bounded, which means that the absolute value of each number a bi a b in the sequence is less than some fied number N. For other values of c the sequence is unbounded, which means that the absolute values of the terms of the sequence become infinitel large. If the sequence is bounded, the comple number c is in the Mandelbrot Set. If the sequence is unbounded, the comple number c is not in the Mandelbrot Set. (a) The pseudo code below can be translated into a program for a graphing utilit. (Programs for several models of graphing calculators can be found at our website college.hmco.com. ) The program determines whether the comple number c is in the Mandelbrot Set. To run the program for c 0.i, enter for A and 0. for B. Press ENTER to see the first term of the sequence. Press ENTER again to see the second term of the sequence. Continue pressing ENTER. If the terms become large, the sequence is unbounded. For the number c 0.i, the terms are 0.i, i, i, i,..., and so the sequence is bounded. So, c 0.i is in the Mandelbrot Set. Program. Enter the real part A.. Enter the imaginar part B.. Store A in C.. Store B in D. 5. Store 0 in N (number of term). 6. Label. 7. Increment N. 8. Displa N. 9. Displa A. 0. Displa B.. Store A in F.. Store B in G.. Store F G C in A.. Store FG D in B. 5. Goto Label. 9 Ω 0 Ω (b) Use a graphing calculator program or a computer program to determine whether the comple numbers c, c 0.5i, and c 0. 0.i are in the Mandelbrot Set.

40 7 CHAPTER Polnomial and Rational Functions Section.5 The Fundamental Theorem of Algebra Understand and use the Fundamental Theorem of Algebra. Find all the zeros of a polnomial function. Write a polnomial function with real coefficients, given its zeros. The Fundamental Theorem of Algebra You know that an nth-degree polnomial can have at most n real zeros. In the comple number sstem, this statement can be improved. That is, in the comple number sstem, ever nth-degree polnomial function has precisel n zeros. This important result is derived from the Fundamental Theorem of Algebra, first proved b the German mathematician Carl Friedrich Gauss ( ). THEOREM. The Fundamental Theorem of Algebra If f is a polnomial of degree n, where n > 0, then f has at least one zero in the comple number sstem. Using the Fundamental Theorem of Algebra and the equivalence of zeros and factors, ou obtain the Linear Factorization Theorem. (A proof is given in Appendi A.) NOTE The Fundamental Theorem of Algebra and the Linear Factorization Theorem tell ou onl that the zeros or factors of a polnomial eist, not how to find them. Such theorems are called eistence theorems. To find the zeros of a polnomial function, ou still must rel on other techniques. THEOREM. Linear Factorization Theorem If f is a polnomial of degree n, where n > 0, then f has precisel n linear factors f a n c c... c n where c, c,..., c n are comple numbers. EXAMPLE Zeros of Polnomial Functions Find the zeros of (a) f, (b) f 6 9, (c) and (d) f. f, Solution a. The first-degree polnomial f has eactl one zero:. b. Counting multiplicit, the second-degree polnomial function f 6 9 has eactl two zeros: and. (This is called a repeated zero. ) c. The third-degree polnomial function f i i has eactl three zeros: 0, i, and i. d. The fourth-degree polnomial function f i i has eactl four zeros:,, i, and i.

41 SECTION.5 The Fundamental Theorem of Algebra 75 The Rational Zero Test The Rational Zero Test relates the possible rational zeros of a polnomial (having integer coefficients) to the leading coefficient and to the constant term of the polnomial. Recall that a rational number is an real number that can be written as the ratio of two integers. The Fogg Art Museum JEAN LE ROND D ALEMBERT (77 78) d Alembert worked independentl of Carl Gauss in tring to prove the Fundamental Theorem of Algebra. His efforts were such that, in France, the Fundamental Theorem of Algebra is frequentl known as the Theorem of d Alembert. The Rational Zero Test If the polnomial f a n n a n n... a a a 0 has integer coefficients, ever rational zero of f has the form Rational zero p q where p and q have no common factors other than, and p a factor of the constant term a 0 q a factor of the leading coefficient a n. To use the Rational Zero Test, ou should first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. Possible rational zeros factors of constant term factors of leading coefficient Having formed this list of possible rational zeros, use a trial-and-error method to determine which, if an, are actual zeros of the polnomial. NOTE When the leading coefficient is, the possible rational zeros are simpl the factors of the constant term. f() = + + EXAMPLE Rational Zero Test with Leading Coefficient of Find the rational zeros of f. Solution Because the leading coefficient is, the possible rational zeros are ±, the factors of the constant term. Possible rational zeros: ± Figure. B testing these possible zeros, ou can see that neither works. f is not a zero. f is not a zero. So, ou can conclude that the given polnomial has no rational zeros. Note from the graph of f in Figure. that f does have one real zero between and 0. However, b the Rational Zero Test, ou know that this real zero is not a rational number. The net few eamples show how snthetic division can be used to test for rational zeros.

42 76 CHAPTER Polnomial and Rational Functions EXAMPLE Rational Zero Test with Leading Coefficient of Find the rational zeros of f 6. f() = (, 0) (, 0) Figure. Solution Because the leading coefficient is, the possible rational zeros are the factors of the constant term. Possible rational zeros: ±, ±, ±, ±6 A test of these possible zeros shows that and are the onl two that work. To test that and are zeros of f, ou can appl snthetic division, as shown. So, ou have 0 0 f Because the factor produces no real zeros, ou can conclude that and are the onl real zeros of f, which is verified in Figure.. If the leading coefficient of a polnomial is not, the list of possible rational zeros can increase dramaticall. In such cases, the search can be shortened in several was: () a programmable graphing utilit can be used to speed up the calculations; () a graph, drawn either b hand or with a graphing utilit, can give a good estimate of the locations of the zeros; and () snthetic division can be used to test the possible rational zeros and to assist in factoring the polnomial. EXAMPLE Using the Rational Zero Test Find the rational zeros of f 8. Solution The leading coefficient is and the constant term is. Possible rational zeros: B snthetic division, ou can determine that is a zero. 5 So, f factors as Factors of ±, ± Factors of ±, ± ±, ±, ±, ± f 5 and ou can conclude that the rational zeros of f are,, and.

43 SECTION.5 The Fundamental Theorem of Algebra 77 Conjugate Pairs In Eample (c) and (d) on page 7, note that the pairs of comple zeros are conjugates. That is, the are of the form a bi and a bi. NOTE Be sure ou see that this result is true onl if the polnomial function has real coefficients. For instance, the result applies to the function f but not to the function g i. THEOREM.5 Comple Zeros Occur in Conjugate Pairs Let f be a polnomial function that has real coefficients. If a bi, where b 0, is a zero of the function, the conjugate a bi is also a zero of the function. EXAMPLE 5 Finding a Polnomial with Given Zeros Find a fourth-degree polnomial function with real coefficients that has,, and i as zeros. Solution Because i is a zero and the polnomial is stated to have real coefficients, ou know that the conjugate i must also be a zero. So, from the Linear Factorization Theorem, f can be written as f a i i. For simplicit, let a to obtain f Factoring a Polnomial The Linear Factorization Theorem shows that ou can write an nth-degree polnomial as the product of n linear factors. f a n c c c... c n However, this result includes the possibilit that some of the values of c i are comple. The following theorem sas that even if ou do not want comple factors, ou can still write f as the product of linear and/or quadratic factors. THEOREM.6 Factors of a Polnomial Ever polnomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. Proof To begin, use the Linear Factorization Theorem to conclude that f can be completel factored in the form f d c c c... c n. If each c k is real, there is nothing more to prove. If an c k is comple c k a bi, b 0, then, because the coefficients of f are real, ou know that the conjugate c j a bi is also a zero. B multipling the corresponding factors, ou obtain c a a b k c j a bi a bi where each coefficient of the quadratic epression is real.

44 78 CHAPTER Polnomial and Rational Functions A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Be sure ou see that this is not the same as being irreducible over the rationals. For eample, the quadratic i i is irreducible over the reals (and therefore over the rationals). On the other hand, the quadratic is irreducible over the rationals, but reducible over the reals. EXAMPLE 6 Finding the Zeros of a Polnomial Function Find all the zeros of f 6 60 given that i is a zero of f. Solution Because comple zeros occur in conjugate pairs, ou know that i is also a zero of f. This means that both i and i are factors of f. Multipling these two factors produces i i i i Using long division, ou can divide 0 into f to obtain the following. 0 ) 6 60 So, ou have f i and ou can conclude that the zeros of f are i, i,, and. 0 In Eample 6, if ou had not been told that i is a zero of f, ou could still find all zeros of the function b using snthetic division to find the real zeros and. Then ou could factor the polnomial as 0. Finall, b using the Quadratic Formula, ou could determine that the zeros are,, i, and i.

45 SECTION.5 The Fundamental Theorem of Algebra 79 Eample 7 shows how to find all the zeros of a polnomial function, including comple zeros. EXAMPLE 7 Finding the Zeros of a Polnomial Function Write f 5 8 as the product of linear factors, and list all of its zeros. Solution The possible rational zeros are ±, ±, ±, and ±8. Snthetic division produces the following is a zero. (, 0) Figure. f() = (, 0) STUDY TIP In Eample 7, the fifthdegree polnomial function has three real zeros. In such cases, ou can use the zoom and trace features or the zero or root feature of a graphing utilit to approimate the real zeros. You can then use these real zeros to determine the comple zeros analticall. So, ou have 0 f 5 8. B factoring as 0 i i is a repeated zero. is a zero. ou obtain f i i which gives the following five zeros of f.,,, i, and i Note from the graph of f shown in Figure. that the real zeros are the onl ones that appear as -intercepts. TECHNOLOGY You can use the table feature of a graphing utilit to help ou determine which of the possible rational zeros are zeros of the polnomial in Eample 7. The table should be set to ask mode. Then enter each of the possible rational zeros in the table. When ou do this, ou will see that there are two rational zeros, and, as shown in the table below.

46 80 CHAPTER Polnomial and Rational Functions Before concluding this section, here are two additional hints that can help ou find the real zeros of a polnomial.. If the terms of f have a common monomial factor, it should be factored out before appling the tests in this section. For instance, b writing f 5 5 ou can see that 0 is a zero of f and that the remaining zeros can be obtained b analzing the cubic factor.. If ou are able to find all but two zeros of f, ou can alwas use the Quadratic Formula on the remaining quadratic factor. For instance, if ou succeeded in writing f 5 ou can appl the Quadratic Formula to to conclude that the two remaining zeros are 5 and 5. EXAMPLE 8 Using a Polnomial Model Figure.5 You are designing candle-making kits. Each kit will contain 5 cubic inches of candle wa and a mold for making a pramid-shaped candle. You want the height of the candle to be inches less than the length of each side of the candle s square base, as shown in Figure.5. What should the dimensions of our candle mold be? Solution The volume of a pramid is given b V Bh, where B is the area of the base and h is the height. The area of the base is and the height is. So, the volume of the pramid is V Bh. Substituting 5 for the volume ields 5 Substitute 5 for V The possible rational zeros are ±, ±, ±5, ±5, ±5, and ±75. Multipl each side b. Write in general form. Using snthetic division, ou can determine that 5 is a solution and ou have The two solutions of the quadratic factor are imaginar and can be discarded. You can conclude that the base of the candle mold should be 5 inches b 5 inches and the height of the mold should be 5 inches.

47 SECTION.5 The Fundamental Theorem of Algebra 8 Eercises for Section.5 In Eercises 6, find all the zeros of the function.. f 6. f. g ). f f 6 i i 6. h t t t t i t i In Eercises 7 0, use the Rational Zero Test to list all possible rational zeros of f. Verif that the zeros of f shown on the graph are contained in the list. 7. f In Eercises 0, find all the real zeros of the function... f 6 6 g. f 7 6. C h t t t t 0 p h 9 0 f 9 9 f f In Eercises, find all real solutions of the polnomial equation.. z z z f f f In Eercises 5 8, (a) list the possible rational zeros of f, (b) sketch the graph of f so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of f. 5. f 6. f f f 5 In Eercises 9, (a) list the possible rational zeros of f, (b) use a graphing utilit to graph f so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of f. 9. f 8 0. f 7. f 5 7. f 7 8 Graphical Analsis In Eercises 6, (a) use the zero or root feature of a graphing utilit to approimate the zeros of the function accurate to three decimal places and (b) determine one of the eact zeros and use snthetic division to verif our result, and then factor the polnomial completel.. f. P t t 7t 5. h g

48 8 CHAPTER Polnomial and Rational Functions i In Eercises 7, find a polnomial function with real coefficients that has the given zeros. (There are man correct answers.) 7., 5i, 5i 8., i, i 9. 6, 5 i, 5 i 0., i,.,, i. 5, 5, i In Eercises 6, write the polnomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completel factored form.. f 6 7. f 8 ( Hint: One factor is 6. ) 5. f 5 6 ( Hint: One factor is. ) 6. f 0 ( Hint: One factor is. ) In Eercises 7 5, use the given zero to find all the zeros of the function. Function 7. f f f g g 0 5. h f 5 5. f 0 Zero 5i i i 5 i i i i i In Eercises 55 7, find all the zeros of the function and write the polnomial as a product of linear factors. 55. f f h 58. g f f f z z z 6. h() 6. g f h h f g g h f f 9 00 In Eercises 7 78, find all the zeros of the function. When there is an etended list of possible rational zeros, use a graphing utilit to graph the function in order to discard an rational zeros that are obviousl not zeros of the function. 7. f f s s 5s s f f f g In Eercises 79 8, find all the real zeros of the function. 79. f 80. f z z z 7z 9 8. f g 5 0 In Eercises 8 86, find all the rational zeros of the polnomial function. 8. P f f 86. f z z 6 z z 6 6z z z In Eercises 87 90, match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: 0; Irrational zeros: (b) Rational zeros: ; Irrational zeros: 0 (c) Rational zeros: ; Irrational zeros: (d) Rational zeros: ; Irrational zeros: f 88. f 89. f 90. f

49 SECTION.5 The Fundamental Theorem of Algebra 8 Writing About Concepts 9. A third-degree polnomial function f has real zeros,, and, and its leading coefficient is negative. Write an equation for f. Sketch the graph of f. How man polnomial functions are possible for f? 9. Sketch the graph of a fifth-degree polnomial function, whose leading coefficient is positive, that has one root at of multiplicit. 9. Use the information in the table. Interval,,,, Value of f Positive Negative Negative Positive (a) What are the real zeros of the polnomial function f? (b) What can be said about the behavior of the graph of f at? (c) What is the least possible degree of f? Eplain our reasoning. Can the degree of f ever be odd? Eplain our reasoning. (d) Is the leading coefficient of f positive or negative? Eplain our reasoning. (e) Write an equation for f. (f) Sketch a graph of the function in part (e). 9. Use the information in the table. Interval,, 0 0,, Value of f Negative Positive Positive Positive (a) What are the real zeros of the polnomial function f? (b) What can be said about the behavior of the graph of f at 0 and? (c) What is the least possible degree of f? Eplain our reasoning. Can the degree of f ever be even? Eplain our reasoning. (d) Is the leading coefficient of f positive or negative? Eplain our reasoning. (e) Write an equation for f. (f) Sketch a graph of the function in part (e). 95. Geometr A rectangular package to be sent b a deliver service (see figure) can have a maimum combined length and girth (perimeter of a cross section) of 0 inches. (a) Show that the volume of the package is given b V 0. (b) Use a graphing utilit to graph the function and approimate the dimensions of the package that ield a maimum volume. (c) Find values of such that V,500. Which of these values is a phsical impossibilit in the construction of the package? Eplain our reasoning. 96. Geometr An open bo is to be made from a rectangular piece of material, 5 centimeters b 9 centimeters, b cutting equal squares from the corners and turning up the sides. (a) Let represent the length of the sides of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open bo. (b) Use the diagram to write the volume V of the bo as a function of. Determine the domain of the function. (c) Sketch the graph of the function and approimate the dimensions of the bo that ield a maimum volume. (d) Find values of such that V 56. Which of these values is a phsical impossibilit in the construction of the bo? Eplain our reasoning. 97. Cost The ordering and transportation cost C (in thousands of dollars) for the components used in manufacturing a product is given b C , where is the order size (in hundreds). In Section 5., ou will learn that the cost is a minimum when , Use a calculator to approimate the optimal order size to the nearest hundred units.

50 8 CHAPTER Polnomial and Rational Functions 98. Profit A compan that produces portable cassette plaers estimates that the profit P (in dollars) for selling a particular model is given b P ,000, where is the advertising epense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that ield a profit of $,500, Profit A compan that manufactures biccles estimates that the profit P (in dollars) for selling a particular model is given b P ,000, where is the advertising epense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that ield a profit of $800, Sports The average salaries s (in millions of dollars) for major league baseball plaers for the ears 995 through 00 are shown in the table, where t represents the ear, with t 5 corresponding to 995. (Source: Major League Baseball Plaers Association) Year, t Average Salar, s Year, t 9 0 Average Salar, s True or False? In Eercises 0 and 0, decide whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 0. It is possible for a third-degree polnomial function with integer coefficients to have no real zeros. 0. If iis a zero of the function given b f i i then i must also be a zero of f. 0. (a) Find a quadratic function f (with integer coefficients) that has ± bi as zeros. Assume that b is a positive integer. (b) Find a quadratic function f (with integer coefficients) that has a ± bi as zeros. Assume that b is a positive integer. 0. The graph of one of the following functions is shown below. Identif the function shown in the graph. Eplain wh each of the others is not the correct function. Use a graphing utilit to verif our result. (a) f ).5 (b) g ).5 (c) h ).5 (d) k ).5 0 (a) Use the regression feature of a graphing utilit to find a cubic model for the data. (b) Use the graphing utilit to create a scatter plot of the data. Then graph the model and the scatter plot in the same viewing window. How do the compare? (c) According to the model found in part (a), in what ear did the average salar reach $. million? (d) According to the model found in part (a), in what ear did the average salar reach $.0 million? (e) According to the right-hand behavior of the model, will the average salaries of major league baseball plaers continue to increase? Eplain our reasoning

51 SECTION.6 Rational Functions 85 Section.6 Rational Functions Find the domains of rational functions. Find the horizontal and vertical asmptotes of graphs of rational functions. Analze and sketch graphs of rational functions. Sketch graphs of rational functions that have slant asmptotes. Use rational functions to model and solve real-life problems. Introduction A rational function can be written in the form f N() D() where N and D are polnomials and D is not the zero polnomial. In this section, it is assumed that N and D have no common factors. In general, the domain of a rational function of includes all real numbers ecept -values that make the denominator zero. Much of the discussion of rational functions will focus on their graphical behavior near these -values ecluded from the domain. EXAMPLE Finding the Domain of a Rational Function Find the domain of f and discuss the behavior of f near an ecluded -values. Solution Because the denominator is zero when 0, the domain of f is all real numbers ecept 0. To determine the behavior of f near this ecluded value, evaluate f to the left and right of 0, as indicated in the tables below. approaches 0 from the left f f() = f Figure.6 approaches 0 from the right. Note that as approaches 0 from the left, f decreases without bound. In contrast, as approaches 0 from the right, f increases without bound. The graph of f is shown in Figure.6.

52 86 CHAPTER Polnomial and Rational Functions Vertical asmptote: = 0 Figure.7 f() = Horizontal asmptote: = 0 Horizontal and Vertical Asmptotes In Eample, the behavior of f near 0 is denoted as follows. f as 0 f as f decreases without bound f increases without bound as approaches 0 from the left. as approaches 0 from the right. The line 0 is a vertical asmptote of the graph of f, as shown in Figure.7. From this figure, ou can see that the graph of f also has a horizontal asmptote the line 0. This means that the values of f approach zero as increases or decreases without bound. f 0 as f 0 as 0 f approaches 0 as decreases without bound. f approaches 0 as increases without bound. Vertical and Horizontal Asmptotes. The line a is a vertical asmptote of the graph of f if f or f as a, either from the right or from the left.. The line b is a horizontal asmptote of the graph of f if f b as or. NOTE A more precise discussion of a vertical asmptote is given in Section.5. A more precise discussion of horizontal asmptote is given in Section 5.5. Eventuall (as or ), the distance between the horizontal asmptote and the points on the graph must approach zero. Figure.8 shows the horizontal and vertical asmptotes of the graphs of three rational functions. f() = + + Vertical asmptote: = Horizontal asmptote: = f() = + Horizontal asmptote: = 0 f() = ( ) Vertical asmptote: = Horizontal asmptote: = 0 (a) Figure.8 (b) (c) The graphs of f in Figure.7 and f in Figure.8(a) are hperbolas. You will stud hperbolas in Section..

53 SECTION.6 Rational Functions 87 Asmptotes of a Rational Function Let f be the rational function given b f N D a n n a n n... a a 0 b m m b m m... b b 0 where N and D have no common factors.. The graph of f has vertical asmptotes at the zeros of D.. The graph of f has one or no horizontal asmptote determined b comparing the degrees of N and D. a. If n < m, the graph of f has the line 0 (the -ais) as a horizontal asmptote. b. If n m, the graph of f has the line a n b m as a horizontal asmptote. c. If n > m, the graph of f has no horizontal asmptote. EXAMPLE Finding Horizontal and Vertical Asmptotes (a) Horizontal asmptote: = Vertical asmptote: = (b) Figure.9 f() = + + Horizontal asmptote: = 0 f() = Vertical asmptote: = Find all horizontal and vertical asmptotes of the graph of each rational function. a. f b. f Solution a. For this rational function, the degree of the numerator is less than the degree of the denominator, so the graph has the line 0 as a horizontal asmptote. To find an vertical asmptotes, set the denominator equal to zero and solve the resulting equation for. 0 Set denominator equal to zero. 0 Factor. Because this equation has no real solutions, ou can conclude that the graph has no vertical asmptote. The graph of the function is shown in Figure.9(a). b. For this rational function, the degree of the numerator is equal to the degree of the denominator. The leading coefficient of the numerator is and the leading coefficient of the denominator is, so the graph has the line as a horizontal asmptote. To find an vertical asmptotes, set the denominator equal to zero and solve the resulting equation for. 0 Set denominator equal to zero. 0 Factor. 0 Set st factor equal to 0. 0 Set nd factor equal to 0. This equation has two real solutions, and. So, the graph has the lines and as vertical asmptotes. The graph of the function is shown in Figure.9(b).

54 88 CHAPTER Polnomial and Rational Functions Analzing Graphs of Rational Functions STUDY TIP Testing for smmetr can be useful, especiall for simple rational functions. For eample, the graph of f is smmetric with respect to the origin, and the graph of g is smmetric with respect to the -ais. Guidelines for Analzing Graphs of Rational Functions Let f N D, where N and D are polnomials with no common factors.. Find and plot the -intercept (if an) b evaluating f 0.. Find the zeros of the numerator (if an) b solving the equation N 0. Then plot the corresponding -intercepts.. Find the zeros of the denominator (if an) b solving the equation D 0. Then sketch the corresponding vertical asmptotes.. Find and sketch the horizontal asmptote (if an) b using the rule for finding the horizontal asmptote of a rational function. 5. Test for smmetr. 6. Plot at least one point between and one point beond each -intercept and vertical asmptote. 7. Use smooth curves to complete the graph between and beond the vertical asmptotes. TECHNOLOGY PITFALL Some graphing utilities have difficult graphing rational functions that have vertical asmptotes. Often, the utilit will connect parts of the graph that are not supposed to be connected. For instance, Figure.0(a) shows the graph of f. Notice that the graph should consist of two separated portions one to the left of and the other to the right of. To eliminate this problem, ou can tr changing the mode of the graphing utilit to dot mode. The problem with this mode is that the graph is then represented as a collection of dots [as shown in Figure.0(b)] rather than as a smooth curve (a) Figure.0 (b) The concept of test intervals from Section. can be etended to graphing of rational functions. To do this, use the fact that a rational function can change signs onl at its zeros and its undefined values (the -values for which its denominator is zero). Between two consecutive zeros of the numerator and the denominator, a rational function must be entirel positive or entirel negative. This means that when the zeros of the numerator and denominator of a rational function are put in order, the divide the real number line into test intervals in which the function has no sign changes. A representative -value is chosen to determine if the value of the rational function is positive (the graph lies above the -ais) or negative (the graph lies below the -ais).

55 SECTION.6 Rational Functions 89 EXAMPLE Sketching the Graph of a Rational Function Sketch the graph of g and state its domain. Solution -Intercept: 0,, because g 0 -Intercept: None, because 0 Vertical asmptote:, zero of denominator Horizontal asmptote: = 0 g() = 6 Horizontal asmptote: 0, because degree of N < degree of D Additional points: Representative Point on Test Interval -Value Value of g Sign Graph, g 0.5 Negative, 0.5 Figure. Vertical asmptote: =, g Positive, B plotting the intercepts, asmptotes, and a few additional points, ou can obtain the graph shown in Figure.. The domain of g is all real numbers ecept. NOTE The graph of g in Eample is a vertical stretch and a right shift of the graph of f because g f. EXAMPLE Sketching the Graph of a Rational Function Sketch the graph of f and state its domain. Solution -Intercept: None, because 0 is not in the domain -Intercept :, 0, because 0 Vertical asmptote: 0, zero of denominator Horizontal asmptote:, because degree of N degree of D Vertical asmptote: = 0 Figure. Horizontal asmptote: = f() = Additional points: Representative Point on Test Interval -Value Value of f Sign Graph, 0 0, f f Positive Negative,,, f.75 Positive,.75 B plotting the intercepts, asmptotes, and a few additional points, ou can obtain the graph shown in Figure.. The domain of f is all real numbers ecept 0.

56 90 CHAPTER Polnomial and Rational Functions EXAMPLE 5 Sketching the Graph of a Rational Function Sketch the graph of f. Solution Factoring the denominator, ou have f. Horizontal asmptote: = 0 Vertical asmptote: = Vertical asmptote: = f() = -Intercept: 0, 0, because f 0 0 -Intercept: 0, 0 Vertical asmptotes:,, zeros of denominator Horizontal asmptote: 0, because degree of N < degree of D Additional points: Representative Point on Test Interval -Value Value of f Sign Graph, f 0. Negative, 0., f Positive 0.5, 0. 0, f 0.5 Negative, 0.5, f 0.75 Positive, 0.75 Figure. The graph is shown in Figure.. EXAMPLE 6 Sketching the Graph of a Rational Function Sketch the graph of f 9. Solution Factoring the numerator and denominator, ou have f. -Intercept: 0, 9, because f 0 9 -Intercepts:, 0 and, 0 Vertical asmptotes:,, zeros of denominator Horizontal asmptote:, because degree of N degree of D Vertical asmptote: = 8 Vertical asmptote: = Smmetr: With respect to -ais, because Additional points: f f Horizontal asmptote: = 6 Representative Point on Test Interval -Value Value of f Sign Graph, 6 f Positive 6, f() = 9,,,,.5 f.5. Negative.5,. 0.5 f 0.5. Positive 0.5,..5 f.5. Negative.5,. 6 f Positive 6, 0.8 Figure. The graph is shown in Figure..

57 SECTION.6 Rational Functions 9 Vertical asmptote: = Figure.5 f() = + NOTE A more detailed eplanation of the term slant asmptote is given in Section 5.6. Slant asmptote: = Slant Asmptotes Consider a rational function whose denominator is of degree or greater. If the degree of the numerator is eactl one more than the degree of the denominator, the graph of the function has a slant (or oblique) asmptote. For eample, the graph of f has a slant asmptote, as shown in Figure.5. To find the equation of a slant asmptote, use long division. For instance, b dividing into, ou obtain f. Slant asmptote As increases or decreases without bound, the remainder term approaches 0, so the graph of f approaches the line, as shown in Figure.5. EXAMPLE 7 A Rational Function with a Slant Asmptote Sketch the graph of f. Solution Factoring the numerator as allows ou to recognize the -intercepts. Using long division f allows ou to recognize that the line is a slant asmptote of the graph. -Intercept: 0,, because f 0 5 Slant asmptote: = -Intercepts:, 0 and, 0 Vertical asmptote:, zero of denominator Slant asmptote: Additional points: 5 Representative Point on Test Interval -Value Value of f Sign Graph, f. Negative,. Vertical asmptote: = f() =,,, 0.5 f Positive 0.5,.5.5 f.5.5 Negative.5,.5 f Positive, Figure.6 The graph is shown in Figure.6.

58 9 CHAPTER Polnomial and Rational Functions Applications There are man eamples of asmptotic behavior in real life. For instance, Eample 8 shows how a vertical asmptote can be used to analze the cost of removing pollutants from smokestack emissions. EXAMPLE 8 Cost-Benefit Model A utilit compan burns coal to generate electricit. The cost of removing a certain percent of the pollutants from smokestack emissions is tpicall not a linear function. That is, if it costs C dollars to remove 5% of the pollutants, it would cost more than C dollars to remove 50% of the pollutants. As the percent of removed pollutants approaches 00%, the cost tends to increase without bound, becoming prohibitive. The cost C (in dollars) of removing p% of the smokestack pollutants is given b C 80,000p 00 p, Sketch the graph of this function. You are a member of a state legislature considering a law that would require utilit companies to remove 90% of the pollutants from their smokestack emissions. The current law requires 85% removal. How much additional cost would the utilit compan incur as a result of the new law? Solution The graph of this function is shown in Figure.7. Note that the graph has a vertical asmptote at p 00. Because the current law requires 85% removal, the current cost to the utilit compan is C 80,000(85) Evaluate C at p 85. If the new law increases the percent removal to 90%, the cost to the utilit compan will be C 80,000(90) p < 00. $5,. $70,000. Evaluate C at p 90. So, the new law would require the utilit compan to spend an additional 70,000 5, $66,667. Subtract 85% removal cost from 90% removal cost. 000 C Cost (in thousands of dollars) C = 80,000p 00 p 85% 90% Figure Percent of pollutants removed p

59 SECTION.6 Rational Functions 9 EXAMPLE 9 Ultraviolet Radiation Eposure time (in hours) T Figure.8 T = 0.7s +.8 s Sunsor Scale reading s For a person with sensitive skin, the amount of time T (in hours) the person can be eposed to the sun with minimal burning can be modeled b T 0.7s.8, 0 < s 0 s where s is the Sunsor Scale reading. The Sunsor Scale is based on the level of intensit of UVB ras. (Source: Sunsor, Inc.) a. Find the amount of time a person with sensitive skin can be eposed to the sun with minimal burning for s 0, s 5, and s 00. b. If the function was valid for all s > 0, what would be the horizontal asmptote of this function, and what would it represent? Solution a. When s 0, T.75 hours When s 5, T. hours When s 00, T 0.6 hour. 00 b. As shown in Figure.8, the horizontal asmptote is the line T 0.7. This line represents the shortest possible eposure time with minimal burning. Eercises for Section.6 In Eercises 6, (a) complete each table, (b) determine the vertical and horizontal asmptotes of the graph of the function, and (c) find the domain of the function f f f f. f. f 5. f. f 5. f 6. f In Eercises 7, find the domain of the function and identif an horizontal and vertical asmptotes. 7. f 8. f 9. 5 f 0. f.. f f.. f 5 f 9

60 9 CHAPTER Polnomial and Rational Functions In Eercises 5 0, match the rational function with its graph. [The graphs are labeled (a) through (f).] (a) (b) (c) (d) (e) (f ) 5. f f f 0. In Eercises, find the zeros (if an) of the rational function.. g.. f. 6 6 f 5 f g 8 In Eercises 5, (a) identif all intercepts, (b) find an vertical and horizontal asmptotes, (c) check for smmetr, (d) plot additional solution points as needed, and (e) sketch the graph of the rational function. 5. f h C 0. P f h 5 f g. g. f.. f t t f 9 t 5. h 6. g g s s 8. f s 9. ( ) g 0. h. f. f. 6. f f 5 5 In Eercises 5 50, sketch the graph of the function. State the domain of the function and identif an vertical and horizontal asmptotes. 5. h t 6. g t 7. f t t 8. f t f 0 f 5 In Eercises 5 58, (a) identif all intercepts, (b) find an vertical and slant asmptotes, (c) check for smmetr, (d) plot additional solution points as needed, and (e) sketch the graph of the rational function. 5. f 5. f 5. g 5. h g f f 5 5 f Writing About Concepts In Eercises 59 6, write a rational function f that has the specified characteristics. 59. Vertical asmptotes:,

61 SECTION.6 Rational Functions 95 Writing About Concepts (continued) 60. Vertical asmptote: None Horizontal asmptote: 0 6. Vertical asmptote: None Horizontal asmptote: 6. Vertical asmptotes: 0, 5 Horizontal asmptote: 6. Give an eample of a rational function whose domain is the set of all real numbers. Give an eample of a rational function whose domain is the set of all real numbers ecept. 6. Describe what is meant b an asmptote of a graph. In Eercises 65 68, use a graphing utilit to graph the rational function. Give the domain of the function and identif an asmptotes. Then zoom out sufficientl far so that the graph appears as a line. Identif the line f f g 68. h Graphical Reasoning In Eercises 69 7, (a) use the graph to determine an -intercepts of the rational function and (b) set 0 and solve the resulting equation to confirm our result in part (a) Pollution The cost C (in millions of dollars) of removing p% of the industrial and municipal pollutants discharged into a river is given b C 55p 00 p, (a) Find the cost of removing 0% of the pollutants. (b) Find the cost of removing 0% of the pollutants. (c) Find the cost of removing 75% of the pollutants. (d) According to this model, would it be possible to remove 00% of the pollutants? Eplain. 7. Reccling In a pilot project, a rural township is given reccling bins for separating and storing recclable products. The cost C (in dollars) for suppling bins to p% of the population is given b C 5,000p 00 p, (a) Find the cost of suppling bins to 5% of the population. (b) Find the cost of suppling bins to 50% of the population. (c) Find the cost of suppling bins to 90% of the population. (d) According to this model, would it be possible to suppl bins to 00% of the residents? Eplain. 75. Population Growth The game commission introduces 00 deer into newl acquired state game lands. The population N of the herd is given b N 0 5 t 0.0t, where t is the time in ears. (a) Find the population when t is 5, 0, and 5. (b) What is the limiting size of the herd as time increases? 76. Concentration of a Miture A 000-liter tank contains 50 liters of a 5% brine solution. You add liters of a 75% brine solution to the tank. (a) Show that the concentration C, the proportion of brine to total solution, in the final miture is given b C p < 00. t 0 0 p < 00. (b) Determine the domain of the function based on the phsical constraints of the problem. (c) Graph the concentration function. As the tank is filled, what happens to the rate at which the concentration of brine is increasing? What percent does the concentration of brine appear to approach?

62 96 CHAPTER Polnomial and Rational Functions 77. Page Design A rectangular page is designed to contain 6 square inches of print. The margins at the top and bottom of the page are each inch deep. The margins on each side are inches wide. What should the dimensions of the page be so that the least amount of paper is used? 78. Geometr A rectangular region of length and width has an area of 500 square meters. (a) Write the width as a function of. (b) Determine the domain of the function based on the phsical constraints of the problem. (c) Sketch a graph of the function and determine the width of the rectangle when 0 meters. 79. Medicine The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is given b C t t t > 0. t 50, (a) Determine the horizontal asmptote of the graph of the function and interpret its meaning in the contet of the problem. (b) Use a graphing utilit to graph the function and approimate the time when the bloodstream concentration is greatest. 80. Average Speed A driver averaged 50 miles per hour on the round trip between home and a cit 00 miles awa. The average speeds for going and returning were and miles per hour, respectivel. (a) Show that 5 5. (b) Determine the vertical and horizontal asmptotes of the graph of the function. Analtic, Numerical, and Graphical Analsis In Eercises 8 8, (a) determine the domains of f and g, (b) find an vertical asmptotes of the graph of f, (c) compare the functions b completing the table, (d) use a graphing utilit to graph f and g in the same viewing window, and (e) eplain wh the graphing utilit ma not show the difference in the domains of f and g. 8. f g, f g f, f g f, f g f f g , g g g True or False? In Eercises 85 88, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 85. A polnomial can have infinitel man vertical asmptotes. 86. The graph of a rational function never intersects one of its vertical asmptotes. 87. It is possible that the graph of a rational function has no vertical asmptotes. 88. The graph of the rational function f has a vertical asmptote at. In Eercises 89 and 90, use a graphing utilit to graph the function. Eplain wh there is no vertical asmptote when a superficial eamination of the function ma indicate that there should be one h 90. g

63 SECTION.6 Rational Functions Business Partnerships The numbers P (in thousands) of business partnerships in the United States for the ears 997 through 00 are shown in the table. (Source: U.S. Internal Revenue Service) Year Partnerships, P Year Partnerships, P 08 6 A model for the data is given b P t t where t is the time in ears, with t 7 corresponding to 997. (a) Use a graphing utilit to plot the data and graph the model in the same viewing window. (b) Use the model to estimate the number of partnerships in 008. (c) Would this model be useful for estimating the number of partnerships after 008? Eplain our reasoning. (d) Use the regression feature of a graphing utilit to find a linear model for the data. (e) Which model do ou think is a better fit for the data? Eplain our reasoning. 9. Militar The numbers M (in thousands) of United States militar reserve personnel for the ears 995 through 00 are shown in the table. (Source: U.S. Department of Defense) Year Personnel, M Year Personnel, M A model for the data is given b M 5.5.7t t where t is the time (in ears), with t 5 corresponding to 995. (a) Use a graphing utilit to plot the data and graph the model in the same viewing window. (b) Use the model to estimate the number of reserve personnel in 006. (c) Use the regression feature of a graphing utilit to find a cubic model for the data. (d) Which model do ou think is a better fit for the data? Eplain our reasoning. Section Project: Rational Functions The numbers N (in thousands) of insured commercial banks in the United States for the ears 99 through 00 are shown in the table. (Source: U.S. Federal Deposit Insurance Corporation) Year Banks, N Year Banks, N For each of the following, let t represent 99. (a) Use the regression feature of a graphing utilit to find a linear model for the data. Use a graphing utilit to plot the data points and graph the linear model in the same viewing window. (b) In order to find a rational model to fit the data, use the following steps. Add a third row to the table with entries N. Again use a graphing utilit to find a linear model to fit the new set of data. Use t for the independent variable and N for the dependent variable. The resulting linear model has the form at b. N Solve this equation for N. This is our rational model. (c) Use a graphing utilit to plot the original data t, N and graph our rational model in the same viewing window. (d) Use the table feature of a graphing utilit to show the actual data and the predicted number of banks based on each model for each of the ears in the given table. Which model do ou prefer? Eplain wh ou chose the model ou did.

64 98 CHAPTER Polnomial and Rational Functions Review Eercises for Chapter Focus on Stud Capsule in Appendi C. Graphical Reasoning In parts (a) (d) of Eercises and, use a graphing utilit to graph the equation in the same viewing window with. Describe how each graph differs from the graph of. (a) (b). (c) (d). (a) (b) (c) (d) In Eercises 6, find the quadratic function that has the indicated verte and whose graph passes through the given point.. Verte:, ; Point:,. Verte:, ; Point: 0, 5. Verte:, ; Point:, 6. Verte:, ; Point:, 6 In Eercises 7 6, write the quadratic function in standard form and sketch its graph. 7. g 8. f 6 9. f h. f t t t. f 8. h. h f 5 f 6 7. Geometr The perimeter of a rectangle is 00 meters. (a) Draw a rectangle that gives a visual representation of the problem. Label the length and width in terms of and, respectivel. (b) Write as a function of. Use the result to write the area A as a function of. (c) Of all possible rectangles with perimeters of 00 meters, find the dimensions of the one with the maimum area. 8. Maimum Profit A real estate office handles 50 apartment units. When the rent is $50 per month, all units are occupied. However, for each $0 increase in rent, one unit becomes vacant. Each occupied unit requires an average of $8 per month for service and repairs. What rent should be charged to obtain the maimum profit? 9. Minimum Cost The dail production cost C (in dollars) for a manufacturer is given b C 0, , where is the number of units produced. How man units should be produced each da to ield a minimum cost? 0. Sociolog The average age of the groom at a first marriage for a given age of the bride can be approimated b the model , 0 5, where is the age of the groom and is the age of the bride. For what age of the bride is the average age of the groom 6? (Source: U.S. Census Bureau) In Eercises 6, sketch the graphs of n and the transformation ,,,, 5, 5, f f f f f 5 f 5 In Eercises 7 0, determine the right-hand and left-hand behaviors of the graph of the polnomial function. 7. f f g h In Eercises 6, find the zeros of the function and sketch its graph.. f. f. f t t t. f 8 5. f 0 6. g In Eercises 7, use long division to divide In Eercises 6, use snthetic division to divide

65 REVIEW EXERCISES In Eercises 7 and 8, use snthetic division to determine whether the given values of are zeros of the function. 7. f 0 9 (a) (b) (c) 0 (d) 8. f (a) (b) (c) (d) In Eercises 9 and 50, use snthetic division to find each function value. 9. f 0 0 (a) f (b) f 50. g t t 5 5t 8t 0 (a) g (b) g In Eercises 5 5, (a) verif the given factor(s) of the function f, (b) find the remaining factors of f, (c) use our results to write the complete factorization of f, (d) list all real zeros of f, and (e) confirm our results b using a graphing utilit to graph the function. Function 5. f f f 7 5. f Factor(s) 6 5 Data Analsis In Eercises 55 58, use the following information. The total annual attendance A (in millions) at women s Division I basketball games for the ears 990 through 00 is shown in the table. The variable t represents the ear, with t 0 corresponding to 990. (Source: NCAA) Year, t Attendance, A Year, t Attendance, A Use the regression feature of a graphing utilit to find a cubic model for the data. 56. Use a graphing utilit to plot the data and graph the model in the same viewing window. Compare the model with the data. 57. Use the model to create a table of estimated values of A. Compare the estimated values with the actual data. 58. Use snthetic division to evaluate the model for the ear 008. Do ou think the model is accurate in predicting the future attendance? Eplain our reasoning. In Eercises 59 6, write the comple number in standard form i i 6. i i In Eercises 6 7, perform the operations and write the result in standard form. 6. i i i i 6i 5 i 66. 5i 8i 67. i 6 i i i i 69. i 6 i i i i i 5 i i In Eercises 7 76, find all solutions of the equation In Eercises 77 8, determine the number of zeros of the function, then find the zeros. 77. f f 79. f f f 8 5 i i f 6 i i In Eercises 8 and 8, use the Rational Zero Test to list all possible rational zeros of f. 8. f f 8 5 In Eercises 85 88, find all the real zeros of the function. 85. f f f f

66 00 CHAPTER Polnomial and Rational Functions In Eercises 89 and 90, find a polnomial with real coefficients that has the given zeros. 89.,, i 90. In Eercises 9 and 9, write the polnomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completel factored form. 9. f Hint: One factor is. 9. f 5 Hint: One factor is. 9. Write quadratic equations that have (a) two distinct real solutions, (b) two comple solutions, and (c) no real solution. 9. What is the degree of a function that has eactl two real zeros and two comple zeros? In Eercises 95 98, find the domain of the rational function. In Eercises 99 0, identif an horizontal or vertical asmptotes. 99. f g 0. g In Eercises 0, identif intercepts, check for smmetr, identif an vertical or horizontal asmptotes, and sketch the graph of the rational function.,, i 95. f 96. f 97. f 98. f f 0. f h 06. f 07. p 08. h g. 5 f f 6 In Eercises 6, find the equation of the slant asmptote and sketch the graph of the rational function.. f. f 5. f Average Cost A business has a cost of C for producing units. The average cost per unit is given b C C , when > 0. Determine the average cost per unit as increases without bound. (Find the horizontal asmptote.) 8. Seizure of Illegal Drugs The cost C (in millions of dollars) for the federal government to seize p% of an illegal drug as it enters the countr is given b C 58p 00 p, (a) Find the cost of seizing 5% of the drug. (b) Find the cost of seizing 50% of the drug. (c) Find the cost of seizing 75% of the drug. (d) According to this model, would it be possible to seize 00% of the drug? 9. Numerical and Graphical Analsis A right triangle is formed in the first quadrant b the - and -aes and a line through the point,. (a) Draw a diagram that illustrates the problem. Label the known and unknown quantities. (b) Verif that the area A of the triangle is given b A, (c) Create a table of values showing the areas for various values of. Start the table at.5 and increment in steps of 0.5. Continue until ou can approimate the dimensions of the triangle of minimum area. (d) Use a graphing utilit to graph the area function. Use the graph to approimate the dimensions of the triangle of minimum area. (e) Determine the slant asmptote of the area function. Eplain its meaning. 0. Rising Water The rise of distilled water in tubes of diameter inches is approimated b the model , f 0 0 p < 00. > 0 >. where is measured in inches. Approimate the diameter of the tube that will cause the water to rise 0. inch.

67 P.S. Problem Solving 0 P.S. Problem Solving. The profit P (in millions of dollars) for a compan is modeled b a quadratic function of the form P at bt c where t represents the ear. If ou were president of the compan, which of the models below would ou prefer? Eplain our reasoning. (a) (b) (c) (d) a is positive and a is positive and a is negative and a is negative and. (a) Assume that the function given b f a b c, a 0 has two real zeros. Show that the -coordinate of the verte of the graph is the average of the zeros of f. (b) Use a graphing utilit to demonstrate the result of part (a) for f.. Given the function f a h k, state the values of a, h, and k that give a reflection in the -ais with either a shrink or a stretch of the graph of the function f.. Eplore the transformations of the form g a h 5 k. (a) Use a graphing utilit to graph the functions 5 and 5 5. t b a. t b a. t b a. t b a. Determine whether the graphs are increasing or decreasing. Eplain our reasoning. (b) Will the graph of g alwas be increasing or decreasing? If so, is this behavior determined b a, h, or k? Eplain our reasoning. (c) Use a graphing utilit to graph the function H 5. Use the graph and the result of part (b) to determine whether H can be written in the form H a h 5 k. Eplain our reasoning. 5. Use the form f k q r to create a cubic function that (a) passes through the point, 5 and rises to the right, and (b) passes through the point, and falls to the right. (There are man correct answers.) 6. The growth of a red oak tree is approimated b the function G 0.00t 0.7t 0.58t 0.89 where G is the height of the tree (in feet) and t t is its age (in ears). (a) Use a graphing utilit to graph the function. Hint: Use a viewing window in which 0 5 and (b) 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 size will be less with each additional ear. (c) Using calculus, the point of diminishing returns can also be found b finding the verte of the parabola given b 0.009t 0.7t Find the verte of this parabola. (d) Compare our results from parts (b) and (c). 7. Consider the function f. (a) Use a graphing utilit to graph the function. Does the graph have a vertical asmptote at? (b) Rewrite the function in simplified form. (c) Use the zoom and trace features to determine the value of the graph near. 8. A wire 00 centimeters in length is cut into two pieces. One piece is bent to form a square and the other to form a circle. Let equal the length of the wire used to form the square. (a) Write the function that represents the combined area of the two figures. (b) Determine the domain of the function.