Polynomial and Rational Functions

Transcription

1 Polnomial and Rational Functions 3 A LOOK BACK In Chapter, we began our discussion of functions. We defined domain and range and independent and dependent variables; we found the value of a function and graphed functions. We continued our stud of functions b listing the properties that a function might have, like being even or odd, and we created a librar of functions, naming ke functions and listing their properties, including their graphs. Ragweed plus bad air will equal fall miser A hot, dr, smogg summer is adding up to an insufferable fall for allerg sufferers. Eperts sa people with allergies to ragweed, who normall sneeze and wheeze through September, will be dabbing their ees and nose well into October, even though pollen counts are actuall no worse than usual. The difference this ear is that a summer of heav smog has made the pollen in the air feel much thicker, making it more irritating. Smog plas an etremel important role in allerg suffering, sas Frances Coates, owner of Ottawa-based Aerobiolog Research, which measures pollen levels. It holds particles in the air a lot longer. Low rainfall delaed the ragweed season until the end of August. It usuall hits earl in the month. SOURCE: Adapted b Louise Surette from Toronto Star, September 8, 00. See Chapter Project. A LOOK AHEAD In this chapter, we look at two general classes of functions: polnomial functions and rational functions, and eamine their properties. Polnomial functions are arguabl the simplest epressions in algebra. For this reason, the are often used to approimate other, more complicated, functions. Rational functions are simpl ratios of polnomial functions. We also introduce methods that can be used to determine the zeros of polnomial functions that are not factored completel. We end the chapter with a discussion of comple zeros. OUTLINE 3. Quadratic Functions and Models 3. Polnomial Functions and Models 3.3 Properties of Rational Functions 3.4 The Graph of a Rational Function; Inverse and Joint Variation 3.5 Polnomial and Rational Inequalities 3.6 The Real Zeros of a Polnomial Function 3.7 Comple Zeros; Fundamental Theorem of Algebra Chapter Review Chapter Test Chapter Projects Cumulative Review 49

2 50 CHAPTER 3 Polnomial and Rational Functions 3. Quadratic Functions and Models PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section., pp. 5 7) Completing the Square (Appendi, Section A.5, pp ) Quadratic Equations (Appendi, Section A.5, pp ) Graphing Techniques: Transformations (Section.6, pp. 8 6) Now work the Are You Prepared? problems on page 63. OBJECTIVES Graph a Quadratic Function Using Transformations Identif the Verte and Ais of Smmetr of a Quadratic Function 3 Graph a Quadratic Function Using Its Verte, Ais, and Intercepts 4 Use the Maimum or Minimum Value of a Quadratic Function to Solve Applied Problems 5 Use a Graphing Utilit to Find the Quadratic Function of Best Fit to Data A quadratic function is a function that is defined b a second-degree polnomial in one variable. A quadratic function is a function of the form f = a + b + c () where a, b, and c are real numbers and a Z 0. The domain of a quadratic function is the set of all real numbers. Man applications require a knowledge of quadratic functions. For eample, suppose that Teas Instruments collects the data shown in Table, which relate the number of calculators sold at the price p (in dollars) per calculator. Since the price of a product determines the quantit that will be purchased, we treat price as the independent variable. The relationship between the number of calculators sold and the price p per calculator ma be approimated b the linear equation =,000-50p Table Price per Calculator, p (Dollars) Number of Calculators,,00 0,5 9,65 8,73 8,087 7,05 6,439 Then the revenue R derived from selling calculators at the price p per calculator is equal to the unit selling price p of the product times the number of units actuall sold. That is,

3 SECTION 3. Quadratic Functions and Models 5 Figure 800, So, the revenue R is a quadratic function of the price p. Figure illustrates the graph of this revenue function, whose domain is 0 p 40, since both and p must be nonnegative. Later in this section we shall determine the price p that maimizes revenue. A second situation in which a quadratic function appears involves the motion of a projectile. Based on Newton s second law of motion (force equals mass times acceleration, F = ma), it can be shown that, ignoring air resistance, the path of a projectile propelled upward at an inclination to the horizontal is the graph of a quadratic function. See Figure for an illustration. Later in this section we shall analze the path of a projectile. Figure Path of a cannonball R = p Rp =,000-50pp = -50p +,000p Graph a Quadratic Function Using Transformations We know how to graph the quadratic function f =. Figure 3 shows the graph of three functions of the form f = for a =, a = a, a 7 0, and a = 3., Notice that the larger the value of a, the narrower the graph is, and the smaller the value of a, the wider the graph is. Figure 3 Figure 4 8 Y 3 3 Y Y Figure 5 Graphs of a quadratic function, f() = a + b + c, a Z 0 Ais of smmetr Verte is lowest point (a) Opens up a 0 Verte is highest point Ais of smmetr (b) Opens down a 0 Y * We shall stud parabolas using a geometric definition later in this book. 8 Y 3 3 Y Figure 4 shows the graphs of f = a for a 6 0. Notice that these graphs are reflections about the -ais of the graphs in Figure 3. Based on the results of these two figures, we can draw some general conclusions about the graph of f = a. First, as ƒaƒ increases, the graph becomes taller (a vertical stretch), and as ƒaƒ gets closer to zero, the graph gets shorter (a vertical compression). Second, if a is positive, then the graph opens up, and if a is negative, then the graph opens down. The graphs in Figures 3 and 4 are tpical of the graphs of all quadratic functions, which we call parabolas. * Refer to Figure 5, where two parabolas are pictured. The one on the left opens up and has a lowest point; the one on the right opens down and has a highest point. The lowest or highest point of a parabola is called the verte.

4 5 CHAPTER 3 Polnomial and Rational Functions The vertical line passing through the verte in each parabola in Figure 5 is called the ais of smmetr (usuall abbreviated to ais) of the parabola. Because the parabola is smmetric about its ais, the ais of smmetr of a parabola can be used to find additional points on the parabola when graphing b hand. The parabolas shown in Figure 5 are the graphs of a quadratic function f = a + b + c, a Z 0. Notice that the coordinate aes are not included in the figure. Depending on the values of a, b, and c, the aes could be placed anwhere. The important fact is that the shape of the graph of a quadratic function will look like one of the parabolas in Figure 5. In the following eample, we use techniques from Section.6 to graph a quadratic function f = a + b + c, a Z 0. In so doing, we shall complete the square and write the function f in the form f = a - h + k. EXAMPLE Graphing a Quadratic Function Using Transformations Graph the function f = Find the verte and ais of smmetr. Solution We begin b completing the square on the right side. f = = = = Factor out the from + 8. Complete the square of ( + 4). Notice that the factor of requires that 8 be added and subtracted. () The graph of f can be obtained in three stages, as shown in Figure 6. Now compare this graph to the graph in Figure 5(a). The graph of f = is a parabola that opens up and has its verte (lowest point) at -, -3. Its ais of smmetr is the line = -. Figure 6 (, ) 3 (, ) (, ) 3 (, ) ( 3, ) 3 (, ) Ais of Smmetr3 (0, 0) 3 (0, 0) 3 3 (, 0) 3 ( 3, ) 3 (, ) 3 (a) Multipl b ; Vertical stretch 3 (b) Replace b ; Shift left units 3 (c) ( ) Subtract 3; Shift down 3 units 3 Verte (, 3) (d) ( ) 3 CHECK: Use a graphing utilit to graph f = and use the MINIMUM command to locate its verte. NOW WORK PROBLEM 7. The method used in Eample can be used to graph an quadratic function f = a + b + c, a Z 0, as follows:

5 SECTION 3. Quadratic Functions and Models 53 f = a + b + c = aa + b a b + c = aa + b a + b b b + c - a 4a 4a = aa + b a b = aa + b a b + + c - b 4a 4ac - b 4a Factor out a from a + b. Complete the square b adding and subtracting Factor a b Look closel at this step! 4a. c - b 4a = c # 4a 4a - b 4ac - b = 4a 4a Based on these results, we conclude the following: b 4ac - b If h = - and k =, then a 4a f = a + b + c = a - h + k (3) The graph of f = a - h + k is the parabola = a shifted horizontall h units replace b - h and verticall k units add k. As a result, the verte is at h, k, and the graph opens up if a 7 0 and down if a 6 0. The ais of smmetr is the vertical line = h. For eample, compare equation (3) with equation () of Eample. f = = = a - h + k We conclude that a =, so the graph opens up. Also, we find that h = - and k = -3, so its verte is at -, -3. Identif the Verte and Ais of Smmetr of a Quadratic Function We do not need to complete the square to obtain the verte. In almost ever case, it is easier to obtain the verte of a quadratic function f b remembering that its b -coordinate is h = - The -coordinate can then be found b evaluating f a. at - b b to find k = fa - a a b. We summarize these remarks as follows: Properties of the Graph of a Quadratic Function Verte = a - f = a + b + c, a Z 0 b a, fa - b a bb Ais of smmetr: the line = - b a (4) Parabola opens up if a 7 0; the verte is a minimum point. Parabola opens down if a 6 0; the verte is a maimum point.

6 54 CHAPTER 3 Polnomial and Rational Functions EXAMPLE Locating the Verte without Graphing Without graphing, locate the verte and ais of smmetr of the parabola defined b f = Does it open up or down? Solution For this quadratic function, a = -3, b = 6, and c =. The -coordinate of the verte is b h = - a = = The -coordinate of the verte is b k = fa - b = f = = 4 a The verte is located at the point, 4. The ais of smmetr is the line =. Because a = , the parabola opens down. 3 Graph a Quadratic Function Using Its Verte, Ais, and Intercepts The information we gathered in Eample, together with the location of the intercepts, usuall provides enough information to graph f = a + b + c, a Z 0, b hand. The -intercept is the value of f at = 0; that is, f0 = c. The -intercepts, if there are an, are found b solving the quadratic equation a + b + c = 0 This equation has two, one, or no real solutions, depending on whether the discriminant b - 4ac is positive, 0, or negative. The graph of f has -intercepts, as follows: The -intercepts of a Quadratic Function. If the discriminant b - 4ac 7 0, the graph of f = a + b + c has two distinct -intercepts and so will cross the -ais in two places.. If the discriminant b - 4ac = 0, the graph of f = a + b + c has one -intercept and touches the -ais at its verte. 3. If the discriminant b - 4ac 6 0, the graph of f = a + b + c has no -intercept and so will not cross or touch the -ais. Figure 7 illustrates these possibilities for parabolas that open up. Figure 7 f() = a + b + c, a 7 0 Ais of smmetr = b a Ais of smmetr = b a Ais of smmetr = b a -intercept ( b, f b ( a a )) (a) b 4ac > 0 Two -intercepts -intercept ( b, 0 a ) (b) b 4ac = 0 One -intercept ( b, f( b a a )) (c) b 4ac < 0 No -intercepts

7 SECTION 3. Quadratic Functions and Models 55 EXAMPLE 3 Graphing a Quadratic Function b Hand Using Its Verte, Ais, and Intercepts Use the information from Eample and the locations of the intercepts to graph f = Determine the domain and the range of f. Determine where f is increasing and where it is decreasing. Solution In Eample, we found the verte to be at, 4 and the ais of smmetr to be =. The -intercept is found b letting = 0. The -intercept is f0 =. The -intercepts are found b solving the equation f = 0. This results in the equation = 0 a = -3, b = 6, c = The discriminant b - 4ac = = 36 + = , so the equation has two real solutions and the graph has two -intercepts. Using the quadratic formula, we find that Figure 8 Ais of smmetr (, 4) 4 (0, ) (, ) 4 4 ( 0.5, 0) (.5, 0) and = -b + 3b - 4ac a = -b - 3b - 4ac a = = The -intercepts are approimatel -0.5 and.5. The graph is illustrated in Figure 8. Notice how we used the -intercept and the ais of smmetr, =, to obtain the additional point, on the graph. The domain of f is the set of all real numbers. Based on the graph, the range of f is the interval - q, 44. The function f is increasing on the interval - q, and decreasing on the interval, q. CHECK: Graph f = using a graphing utilit. Use ZERO (or ROOT) to locate the two -intercepts. Use MAXIMUM to locate the verte. = = L -0.5 L.5 NOW WORK PROBLEM 35. If the graph of a quadratic function has onl one -intercept or no -intercepts, it is usuall necessar to plot an additional point to obtain the graph b hand. EXAMPLE 4 Graphing a Quadratic Function b Hand Using Its Verte, Ais, and Intercepts Graph f = b determining whether the graph opens up or down and b finding its verte, ais of smmetr, -intercept, and -intercepts, if an. Determine the domain and the range of f. Determine where f is increasing and where it is decreasing. Solution For f = , we have a =, b = -6, and c = 9. Since a = 7 0, the parabola opens up. The -coordinate of the verte is h = - b a = - -6 = 3

8 56 CHAPTER 3 Polnomial and Rational Functions Figure 9 (0, 9) 6 3 Ais of smmetr 3 (3, 0) 6 (6, 9) The -coordinate of the verte is k = f3 = = 0 So, the verte is at 3, 0. The ais of smmetr is the line = 3. The -intercept is f0 = 9. Since the verte 3, 0 lies on the -ais, the graph touches the -ais at the -intercept. B using the ais of smmetr and the -intercept at 0, 9, we can locate the additional point 6, 9 on the graph. See Figure 9. The domain of f is the set of all real numbers. Based on the graph, the range of f is the interval 30, q. The function f is decreasing on the interval - q, 3 and increasing on the interval 3, q. Graph the function in Eample 4 b completing the square and using transformations. Which method do ou prefer? NOW WORK PROBLEM 43. EXAMPLE 5 Graphing a Quadratic Function b Hand Using Its Verte, Ais, and Intercepts NOTE In Eample 5, since the verte is above the -ais and the parabola opens up, we can conclude the graph of the quadratic function will have no -intercepts. Figure 0 Ais of smmetr ( ) 4, (0, ) ( ) 7, 4 8 Graph f = + + b determining whether the graph opens up or down and b finding its verte, ais of smmetr, -intercept, and -intercepts, if an. Determine the domain and the range of f. Determine where f is increasing and where it is decreasing. Solution For f = + +, we have a =, b =, and c =. Since a = 7 0, the parabola opens up. The -coordinate of the verte is The -coordinate of the verte is k = fa - h = - b a = b = a 6 b + a - 4 b + = 7 8 So, the verte is at a - The ais of smmetr is the line = - The 4, 7 8 b. 4. -intercept is f0 =. The -intercept(s), if an, obe the equation + + = 0. Since the discriminant b - 4ac = - 4 = , this equation has no real solutions, and therefore the graph has no -intercepts. We use the point 0, and the ais of smmetr = - to locate the additional point a - on the graph. See Figure 0. 4, b The domain of f is the set of all real numbers. Based on the graph, the range of f is the interval c 7 The function f is decreasing on the interval a - q, - 8, q b. 4 b and increasing on the interval a - 4, q b. NOW WORK PROBLEM 47. Given the verte h, k and one additional point on the graph of a quadratic function f = a + b + c, a Z 0, we can use f = a - h + k (5) to obtain the quadratic function.

9 SECTION 3. Quadratic Functions and Models 57 Figure 8 4 (0, 3) 4 8 EXAMPLE 6 (, 5) 3 4 Finding the Quadratic Function Given Its Verte and One Other Point Determine the quadratic function whose verte is, -5 and whose -intercept is -3. The graph of the parabola is shown in Figure. Solution The verte is, -5, so h = and k = -5. Substitute these values into equation (5). Equation (5) To determine the value of a, we use the fact that f0 = -3 (the -intercept). f = a = a = 0, = f(0) = -3-3 = a - 5 a = The quadratic function whose graph is shown in Figure is f = a - h + k = = CHECK: Figure shows the graph f = a - h + k f = a h =, k = -5 f = using a graphing utilit. Figure Summar NOW WORK PROBLEM 53. Steps for Graphing a Quadratic Function f a b c, a 0, b Hand. Option STEP : Complete the square in to write the quadratic function in the form STEP : Graph the function in stages using transformations. Option b STEP : Determine the verte a - a, fa - b a bb. b STEP : Determine the ais of smmetr, = - a. STEP 3: Determine the -intercept, f0. f = a - h + k. STEP 4: (a) If b - 4ac 7 0, then the graph of the quadratic function has two -intercepts, which are found b solving the equation a + b + c = 0. (b) If b - 4ac = 0, the verte is the -intercept. (c) If b - 4ac 6 0, there are no -intercepts. STEP 5: Determine an additional point b using the -intercept and the ais of smmetr. STEP 6: Plot the points and draw the graph.

10 58 CHAPTER 3 Polnomial and Rational Functions 4 Use the Maimum or Minimum Value of a Quadratic Function to Solve Applied Problems When a mathematical model leads to a quadratic function, the properties of this quadratic function can provide important information about the model. For eample, for a quadratic revenue function, we can find the maimum revenue; for a quadratic cost function, we can find the minimum cost. To see wh, recall that the graph of a quadratic function b is a parabola with verte at a - This verte is the highest point on the a, fa - b a bb. graph if a 6 0 and the lowest point on the graph if a 7 0. If the verte is the highest point a 6 0, b then fa - is the maimum value of f. If the verte is the a b b lowest point a 7 0, then fa - is the minimum value of f. a b This propert of the graph of a quadratic function enables us to answer questions involving optimization (finding maimum or minimum values) in models involving quadratic functions. f = a + b + c, a Z 0 EXAMPLE 7 Finding the Maimum or Minimum Value of a Quadratic Function Determine whether the quadratic function has a maimum or minimum value. Then find the maimum or minimum value. Solution We compare f = to f = a + b + c. We conclude that a =, b = -4, and c = -5. Since a 7 0, the graph of f opens up, so the verte is a minimum point. The minimum value occurs at The minimum value is fa - = - f = b a = - -4 = 4 = æ a =, b = -4 b a b = f = = = -9 Table CHECK: We can support the algebraic solution using the TABLE feature on a graphing utilit. Create Table b letting Y = From the table we see that the smallest value of occurs when =, leading us to investigate the number further. The smmetr about = confirms that is the -coordinate of the verte of the parabola.we conclude that the minimum value is -9 and occurs at =. NOW WORK PROBLEM 6.

11 SECTION 3. Quadratic Functions and Models 59 EXAMPLE 8 Solution Figure 3 Maimizing Revenue The marketing department at Teas Instruments has found that, when certain calculators are sold at a price of p dollars per unit, the revenue R (in dollars) as a function of the price p is Rp = -50p +,000p What unit price should be established to maimize revenue? If this price is charged, what is the maimum revenue? The revenue R is Rp = -50p +,000p R(p) = ap + bp + c The function R is a quadratic function with a = -50, b =,000, and c = 0. Because a 6 0, the verte is the highest point on the parabola. The revenue R is therefore a maimum when the price p is b p = - a = -, = -, = $70.00 æ a = -50, b =,000 The maimum revenue R is R70 = ,00070 = $735,000 See Figure 3 for an illustration. Revenue (dollars) R 800, , , , , ,000 00,000 00,000 (70, ) p Price p per calculator (dollars) NOW WORK PROBLEM 7. EXAMPLE 9 Maimizing the Area Enclosed b a Fence A farmer has 000 ards of fence to enclose a rectangular field. What are the dimensions of the rectangle that encloses the most area? Figure 4 Solution Figure 4 illustrates the situation. The available fence represents the perimeter of the rectangle. If is the length and w is the width, then w w The area A of the rectangle is + w = 000 (6) A = w

12 60 CHAPTER 3 Polnomial and Rational Functions To epress A in terms of a single variable, we solve equation (6) for w and substitute the result in A = w. Then A involves onl the variable. [You could also solve equation (6) for and epress A in terms of w alone. Tr it!] Then the area A is + w = 000 w = w = = Equation (6) Solve for w. Now, A is a quadratic function of. A = w = = A = a = -, b = 000, c = 0 Figure 5 50,000 Figure 5 shows a graph of A = using a graphing utilit. Since a 6 0, the verte is a maimum point on the graph of A.The maimum value occurs at = - b a = = 500 The maimum value of A is Aa - b a b = A500 = = -50, ,000 = 50,000 The largest rectangle that can be enclosed b 000 ards of fence has an area of 50,000 square ards. Its dimensions are 500 ards b 500 ards. NOW WORK PROBLEM 77. EXAMPLE 0 Analzing the Motion of a Projectile A projectile is fired from a cliff 500 feet above the water at an inclination of 45 to the horizontal, with a muzzle velocit of 400 feet per second. In phsics, it is established that the height h of the projectile above the water is given b h = where is the horizontal distance of the projectile from the base of the cliff. See Figure 6. Figure 6 h () (a) Find the maimum height of the projectile. (b) How far from the base of the cliff will the projectile strike the water?

13 SECTION 3. Quadratic Functions and Models 6 Seeing the Concept Graph h() = , Solution Use MAXIMUM to find the maimum height of the projectile, and use ROOT or ZERO to find the distance from the base of the cliff to where it strikes the water. Compare our results with those obtained in Eample 0. (a) The height of the projectile is given b a quadratic function. We are looking for the maimum value of h. Since a 6 0, the maimum value is obtained at the verte. We compute The maimum height of the projectile is (b) The projectile will strike the water when the height is zero. To find the distance traveled, we need to solve the equation h = = 0 We find the discriminant first. Then, h = = = - b a = - a b = -b ; 3b - 4ac a We discard the negative solution and find that the projectile will strike the water at a distance of about 5458 feet from the base of the cliff. NOW WORK PROBLEM 8. = - ;.4 a - = b = 500 h500 = = = 750 ft b - 4ac = - 4a - b500 = L e EXAMPLE Solution The Golden Gate Bridge The Golden Gate Bridge, a suspension bridge, spans the entrance to San Francisco Ba. Its 746-foot-tall towers are 400 feet apart. The bridge is suspended from two huge cables more than 3 feet in diameter; the 90-foot-wide roadwa is 0 feet above the water. The cables are parabolic in shape * and touch the road surface at the center of the bridge. Find the height of the cable at a distance of 000 feet from the center. See Figure 7. We begin b choosing the placement of the coordinate aes so that the -ais coincides with the road surface and the origin coincides with the center of the bridge. As a result, the twin towers will be vertical (height = 56 feet above the road) and located 00 feet from the center.also, the cable, which has the Figure 7 ( 00, 56) (00, 56) 0' 00' 56' (0, 0) 746'? 000' 00' * A cable suspended from two towers is in the shape of a catenar, but when a horizontal roadwa is suspended from the cable, the cable takes the shape of a parabola.

14 6 CHAPTER 3 Polnomial and Rational Functions shape of a parabola, will etend from the towers, open up, and have its verte at 0, 0. The choice of placement of the aes enables us to identif the equation of the parabola as = a, a 7 0. We can also see that the points -00, 56 and 00, 56 are on the graph. Based on these facts, we can find the value of a in = a. The equation of the parabola is therefore The height of the cable when = 000 is = = = a 56 = a00 a = L 9.3 feet = 00, = 56 The cable is 9.3 feet high at a distance of 000 feet from the center of the bridge. NOW WORK PROBLEM Use a Graphing Utilit to Find the Quadratic Function of Best Fit to Data In Section.4, we found the line of best fit for data that appeared to be linearl related. It was noted that data ma also follow a nonlinear relation. Figures 8(a) and (b) show scatter diagrams of data that follow a quadratic relation. Figure 8 a b c, a 0 (a) a b c, a 0 (b) EXAMPLE Fitting a Quadratic Function to Data A farmer collected the data given in Table 3, which shows crop ields Y for various amounts of fertilizer used,. (a) With a graphing utilit, draw a scatter diagram of the data. Comment on the tpe of relation that ma eist between the two variables. (b) Use a graphing utilit to find the quadratic function of best fit to these data. (c) Use the function found in part (b) to determine the optimal amount of fertilizer to appl.

15 SECTION 3. Quadratic Functions and Models 63 Table 3 Plot Fertilizer, (Pounds/00 ft ) Yield (Bushels) (d) Use the function found in part (b) to predict crop ield when the optimal amount of fertilizer is applied. (e) Draw the quadratic function of best fit on the scatter diagram. Solution (a) Figure 9 shows the scatter diagram, from which it appears that the data follow a quadratic relation, with a 6 0. (b) Upon eecuting the QUADratic REGression program, we obtain the results shown in Figure 0. The output that the utilit provides shows us the equation = a + b + c. The quadratic function of best fit is where represents the amount of fertilizer used and Y represents crop ield. (c) Based on the quadratic function of best fit, the optimal amount of fertilizer to appl is = - b a = Y = (d) We evaluate the function Y for = 3.5. L 3.5 pounds of fertilizer per 00 square feet Y3.5 = L 0.8 bushels If we appl 3.5 pounds of fertilizer per 00 square feet, the crop ield will be 0.8 bushels according to the quadratic function of best fit. (e) Figure shows the graph of the quadratic function found in part (b) drawn on the scatter diagram. Figure 9 5 Figure 0 Figure Assess Your Understanding Are You Prepared? Look again at Figure 0. Notice that the output given b the graphing calculator does not include r, the correlation coefficient. Recall that the correlation coefficient is a measure of the strength of a linear relation that eists between two variables. The graphing calculator does not provide an indication of how well the function fits the data in terms of r since a quadratic function cannot be epressed as a linear function. NOW WORK PROBLEM 0. Answers are given at the end of these eercises. If ou get a wrong answer, read the pages listed in red.. List the intercepts of the equation = To complete the square of - 5, ou add the number (pp. 5 7). (pp ). Solve the equation = To graph = - 4, ou shift the graph of = to the (pp ) a distance of units. (pp. 8 0)

16 64 CHAPTER 3 Polnomial and Rational Functions Concepts and Vocabular 5. The graph of a quadratic function is called a(n). 6. The vertical line passing through the verte of a parabola is called the. 7. The -coordinate of the verte of f = a + b + c, a Z 0, is. 8. True or False: The graph of f = opens up. 9. True or False: The -coordinate of the verte of f = is f. 0. True or False: If the discriminant b - 4ac = 0, the graph of f = a + b + c, a Z 0, will touch the -ais at its verte. Skill Building In Problems 8, match each graph to one the following functions without using a graphing utilit.. f = -. f = f = f = f = + 7. f = - 8. A. B. C. D. 3 (, 0) (, ) E. F. G. H. 3 (0, ) f = + + f = (, ) 3 (0, ) (, 0) 3 (, ) 3 (, ) In Problems 9 34, graph the function f b starting with the graph of = and using transformations (shifting, compressing, stretching, and/or reflection). Verif our results using a graphing utilit. [Hint: If necessar, write f in the form f = a - h + k. ] 9. f = 4 0. f =. f = f = f = f = f = f = f = f = f = f = f = - 3 f = - - f = f = In Problems 35 5, graph each quadratic function b determining whether its graph opens up or down and b finding its verte, ais of smmetr, -intercept, and -intercepts, if an. Determine the domain and the range of the function. Determine where the function is increasing and where it is decreasing. Verif our results using a graphing utilit. 35. f = f = f = f = f = f = f = f = f = f = f = f = f = f = f = f = f = f = 3-8 +

17 SECTION 3. Quadratic Functions and Models 65 In Problems 53 58, determine the quadratic function whose graph is given Verte: (, 3) 8 6 (3, 5) (0, ) 4 In Problems 59 66, determine, without graphing, whether the given quadratic function has a maimum value or a minimum value and then find the value. Use the TABLE feature on a graphing utilit to support our answer. 59. f = f = f = f = f = f = f = f = 4-4 Applications and Etensions Verte: (, ) (0, ) 8 4 (0, 5) 67. The graph of the function f = a + b + c has verte at 0, and passes through the point, 8. Find a, b, and c. 68. The graph of the function f = a + b + c has verte at, 4 and passes through the point -, -8. Find a, b, and c. Verte: (, ) Answer Problems 69 and 70 using the following: A quadratic function of the form f = a + b + c with b - 4ac 7 0 ma also be written in the form f = a - r - r, where r and r are the -intercepts of the graph of the quadratic function. 69. (a) Find a quadratic function whose -intercepts are -3 and with a = ; a = ; a = -; a = 5. (b) How does the value of a affect the intercepts? (c) How does the value of a affect the ais of smmetr? (d) How does the value of a affect the verte? (e) Compare the -coordinate of the verte with the midpoint of the -intercepts. What might ou conclude? 70. (a) Find a quadratic function whose -intercepts are -5 and 3 with a = ; a = ; a = -; a = 5. (b) How does the value of a affect the intercepts? (c) How does the value of a affect the ais of smmetr? (d) How does the value of a affect the verte? (e) Compare the -coordinate of the verte with the midpoint of the -intercepts. What might ou conclude? Verte: (, 3) Verte: ( 3, 5) (0, 4) Verte: (, 6) 6 3 ( 4, ) 7. Maimizing Revenue Suppose that the manufacturer of a gas clothes drer has found that, when the unit price is p dollars, the revenue R (in dollars) is Rp = -4p p What unit price for the drer should be established to maimize revenue? What is the maimum revenue? 7. Maimizing Revenue The John Deere compan has found that the revenue from sales of heav-dut tractors is a function of the unit price p (in dollars) that it charges. If the revenue R is Rp = - p + 900p what unit price p (in dollars) should be charged to maimize revenue? What is the maimum revenue? 73. Demand Equation The price p (in dollars) and the quantit sold of a certain product obe the demand equation p = , (a) Epress the revenue R as a function of. (Remember, R = p. ) (b) What is the revenue if 00 units are sold? (c) What quantit maimizes revenue? What is the maimum revenue? (d) What price should the compan charge to maimize revenue?

18 66 CHAPTER 3 Polnomial and Rational Functions 74. Demand Equation The price p (in dollars) and the quantit sold of a certain product obe the demand equation p = , (a) Epress the revenue R as a function of. (b) What is the revenue if 00 units are sold? (c) What quantit maimizes revenue? What is the maimum revenue? (d) What price should the compan charge to maimize revenue? 75. Demand Equation The price p (in dollars) and the quantit sold of a certain product obe the demand equation = -5p + 00, 0 p 0 (a) Epress the revenue R as a function of. (b) What is the revenue if 5 units are sold? (c) What quantit maimizes revenue? What is the maimum revenue? (d) What price should the compan charge to maimize revenue? 76. Demand Equation The price p (in dollars) and the quantit sold of a certain product obe the demand equation = -0p + 500, 0 p 5 (a) Epress the revenue R as a function of. (b) What is the revenue if 0 units are sold? (c) What quantit maimizes revenue? What is the maimum revenue? (d) What price should the compan charge to maimize revenue? 77. Enclosing a Rectangular Field David has available 400 ards of fencing and wishes to enclose a rectangular area. (a) Epress the area A of the rectangle as a function of the width w of the rectangle. (b) For what value of w is the area largest? (c) What is the maimum area? 78. Enclosing a Rectangular Field Beth has 3000 feet of fencing available to enclose a rectangular field. (a) Epress the area A of the rectangle as a function of where is the length of the rectangle. (b) For what value of is the area largest? (c) What is the maimum area? 79. Enclosing the Most Area with a Fence A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed? (See the figure.) 80. Enclosing the Most Area with a Fence A farmer with 000 meters of fencing wants to enclose a rectangular plot that borders on a straight highwa. If the farmer does not fence the side along the highwa, what is the largest area that can be enclosed? 8. Analzing the Motion of a Projectile A projectile is fired from a cliff 00 feet above the water at an inclination of 45 to the horizontal, with a muzzle velocit of 50 feet per second. The height h of the projectile above the water is given b h = where is the horizontal distance of the projectile from the base of the cliff. (a) How far from the base of the cliff is the height of the projectile a maimum? (b) Find the maimum height of the projectile. (c) How far from the base of the cliff will the projectile strike the water? (d) Using a graphing utilit, graph the function h, (e) When the height of the projectile is 00 feet above the water, how far is it from the cliff? 8. Analzing the Motion of a Projectile A projectile is fired at an inclination of 45 to the horizontal, with a muzzle velocit of 00 feet per second.the height h of the projectile is given b h = where is the horizontal distance of the projectile from the firing point. (a) How far from the firing point is the height of the projectile a maimum? (b) Find the maimum height of the projectile. (c) How far from the firing point will the projectile strike the ground? (d) Using a graphing utilit, graph the function h, (e) When the height of the projectile is 50 feet above the ground, how far has it traveled horizontall? 83. Suspension Bridge A suspension bridge with weight uniforml distributed along its length has twin towers that etend 75 meters above the road surface and are 400 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 00 meters from the center. (Assume that the road is level.) Architecture A parabolic arch has a span of 0 feet and a maimum height of 5 feet. Choose suitable rectangular coordinate aes and find the equation of the parabola. Then calculate the height of the arch at points 0 feet, 0 feet, and 40 feet from the center.

19 SECTION 3. Quadratic Functions and Models Constructing Rain Gutters A rain gutter is to be made of aluminum sheets that are inches wide b turning up the edges 90. What depth will provide maimum cross-sectional area and hence allow the most water to flow? [Hint: Area of an equilateral triangle = a 3 where 4 b, is the length of a side of the triangle.] in. 89. Chemical Reactions A self-cataltic chemical reaction results in the formation of a compound that causes the formation ratio to increase. If the reaction rate V is given b V = ka -, 0 a 86. Norman Windows A Norman window has the shape of a rectangle surmounted b a semicircle of diameter equal to the width of the rectangle (see the figure). If the perimeter of the window is 0 feet, what dimensions will admit the most light (maimize the area)? [Hint: Circumference of a circle = pr; area of a circle = pr, where r is the radius of the circle.] where k is a positive constant, a is the initial amount of the compound, and is the variable amount of the compound, for what value of is the reaction rate a maimum? 90. Calculus: Simpson s Rule The figure shows the graph of = a + b + c. Suppose that the points -h, 0, 0,, and h, are on the graph. It can be shown that the area enclosed b the parabola, the -ais, and the lines = -h and = h is Area = h 3 ah + 6c Show that this area ma also be given b Area = h Constructing a Stadium A track and field plaing area is in the shape of a rectangle with semicircles at each end (see the figure). The inside perimeter of the track is to be 500 meters. What should the dimensions of the rectangle be so that the area of the rectangle is a maimum? ( h, 0 ) (0, ) h (h, ) h 88. Architecture A special window has the shape of a rectangle surmounted b an equilateral triangle (see the figure). If the perimeter of the window is 6 feet, what dimensions will admit the most light? 9. Use the result obtained in Problem 90 to find the area enclosed b f = , the -ais, and the lines = - and =. 9. Use the result obtained in Problem 90 to find the area enclosed b f = + 8, the -ais, and the lines = - and =. 93. Use the result obtained in Problem 90 to find the area enclosed b f = , the -ais, and the lines = -4 and = Use the result obtained in Problem 90 to find the area enclosed b f = , the -ais, and the lines = - and =.

20 68 CHAPTER 3 Polnomial and Rational Functions 95. A rectangle has one verte on the line = 0 -, 7 0, another at the origin, one on the positive -ais, and one on the positive -ais. Find the largest area A that can be enclosed b the rectangle. 96. Let f = a + b + c, where a, b, and c are odd integers. If is an integer, show that f must be an odd integer. [Hint: is either an even integer or an odd integer.] 97. Hunting The function H = models the number of individuals who engage in hunting activities whose annual income is thousand dollars. SOURCE: Based on data obtained from the National Sporting Goods Association. (a) What is the income level for which there are the most hunters? Approimatel how man hunters earn this amount? (b) Using a graphing utilit, graph H = H. Are the number of hunters increasing or decreasing for individuals earning between $0,000 and $40,000? 98. Advanced Degrees The function P = models the percentage of the U.S. population in March 000 whose age is given b that has earned an advanced degree (more than a bachelor s degree). SOURCE: Based on data obtained from the U.S. Census Bureau. (a) What is the age for which the highest percentage of Americans have earned an advanced degree? What is the highest percentage? (b) Using a graphing utilit, graph P = P. Is the percentage of Americans that have earned an advanced degree increasing or decreasing for individuals between the ages of 40 and 50? 99. Male Murder Victims The function M = models the number of male murder victims who are ears of age SOURCE: Based on data obtained from the Federal Bureau of Investigation. (a) Use the model to approimate the number of male murder victims who are = 3 ears of age. (b) At what age is the number of male murder victims 456? (c) Using a graphing utilit, graph M = M. (d) Based on the graph drawn in part (c), describe what happens to the number of male murder victims as age increases. 00. Health Care Ependitures The function H = models the percentage of total income that an individual who is ears of age spends on health care. SOURCE: Based on data obtained from the Bureau of Labor Statistics. (a) Use the model to approimate the percentage of total income that an individual who is = 45 ears of age spends on health care. (b) At what age is the percentage of income spent on health care 0%? (c) Using a graphing utilit, graph H = H. (d) Based on the graph drawn in part (c), describe what happens to the percentage of income spent on health care as individuals age. 0. Life Ccle Hpothesis An individual s income varies with his or her age.the following table shows the median income I of individuals of different age groups within the United States for 00. For each age group, let the class midpoint represent the independent variable,. For the class 65 ears and older, we will assume that the class midpoint is Age 5 4 ears 5 34 ears ears ears ears 65 ears and older SOURCE: U.S. Census Bureau (a) Draw a scatter diagram of the data. Comment on the tpe of relation that ma eist between the two variables. (b) Use a graphing utilit to find the quadratic function of best fit to these data. (c) Use the function found in part (b) to determine the age at which an individual can epect to earn the most income. (d) Use the function found in part (b) to predict the peak income earned. (e) With a graphing utilit, graph the quadratic function of best fit on the scatter diagram. 0. Life Ccle Hpothesis An individual s income varies with his or her age.the following table shows the median income I of individuals of different age groups within the United States for 000. For each age group, the class midpoint represents the independent variable,. For the age group 65 ears and older, we will assume that the class midpoint is Age 5 4 ears 5 34 ears ears ears ears 65 ears and older SOURCE: U.S. Census Bureau Class Midpoint, Class Midpoint, Median Income, I $9,30 $30,50 $38,340 $4,04 $35,637 $9,688 Median Income, I $9,548 $30,633 $37,088 $4,07 $34,44 $9,67

21 SECTION 3. Quadratic Functions and Models 69 (a) Draw a scatter diagram of the data. Comment on the tpe of relation that ma eist between the two variables. (b) Use a graphing utilit to find the quadratic function of best fit to these data. (c) Use the function found in part (b) to determine the age at which an individual can epect to earn the most income. (d) Use the function found in part (b) to predict the peak income earned. (e) With a graphing utilit, graph the quadratic function of best fit on the scatter diagram. (f) Compare the results of parts (c) and (d) in 000 to those of 00 in Problem Height of a Ball A shot-putter throws a ball at an inclination of 45 to the horizontal. The following data represent the height of the ball h at the instant that it has traveled feet horizontall. Distance, Height, h (a) Draw a scatter diagram of the data. Comment on the tpe of relation that ma eist between the two variables. (b) Use a graphing utilit to find the quadratic function of best fit to these data. (c) Use the function found in part (b) to determine how far the ball will travel before it reaches its maimum height. (d) Use the function found in part (b) to find the maimum height of the ball. (e) With a graphing utilit, graph the quadratic function of best fit on the scatter diagram. 04. Miles per Gallon An engineer collects the following data showing the speed s of a Ford Taurus and its average miles per gallon, M. Speed, s Miles per Gallon, M (a) Draw a scatter diagram of the data. Comment on the tpe of relation that ma eist between the two variables. (b) Use a graphing utilit to find the quadratic function of best fit to these data. (c) Use the function found in part (b) to determine the speed that maimizes miles per gallon. (d) Use the function found in part (b) to predict miles per gallon for a speed of 63 miles per hour. (e) With a graphing utilit, graph the quadratic function of best fit on the scatter diagram Discussion and Writing 05. Make up a quadratic function that opens down and has onl one -intercept. Compare ours with others in the class. What are the similarities? What are the differences? 06. On one set of coordinate aes, graph the famil of parabolas f = + + c for c = -3, c = 0, and c =. Describe the characteristics of a member of this famil. 07. On one set of coordinate aes, graph the famil of parabolas f = + b + for b = -4, b = 0, and b = 4. Describe the general characteristics of this famil. 08. State the circumstances that cause the graph of a quadratic function f = a + b + c to have no -intercepts. 09. Wh does the graph of a quadratic function open up if a 7 0 and down if a 6 0? 0. Refer to Eample 8 on page 59. Notice that if the price charged for the calculators is $0 or $40 the revenue is $0. It is eas to eplain wh revenue would be $0 if the price charged is $0, but how can revenue be $0 if the price charged is $40? Are You Prepared? Answers 5. 0, -9, -3, 0, 3, 0. e -4, right; 4 f 4

22 70 CHAPTER 3 Polnomial and Rational Functions 3. Polnomial Functions and Models PREPARING FOR THIS SECTION Before getting started, review the following: Polnomials (Appendi, Section A.3, pp ) Using a Graphing Utilit to Approimate Local Maima and Local Minima (Section.3, pp ) Graphing Techniques: Transformations (Section.6, pp. 8 6) Intercepts (Section., pp. 5 7) Now work the Are You Prepared? problems on page 8. OBJECTIVES Identif Polnomial Functions and Their Degree Graph Polnomial Functions Using Transformations 3 Identif the Zeros of a Polnomial Function and Their Multiplicit 4 Analze the Graph of a Polnomial Function 5 Find the Cubic Function of Best Fit to Data Identif Polnomial Functions and Their Degree Polnomial functions are among the simplest epressions in algebra. The are eas to evaluate: onl addition and repeated multiplication are required. Because of this, the are often used to approimate other, more complicated functions. In this section, we investigate properties of this important class of functions. A polnomial function is a function of the form f = a n n + a n - n - + Á + a + a 0 () where a n, a n -, Á, a, a 0 are real numbers and n is a nonnegative integer. The domain is the set of all real numbers. A polnomial function is a function whose rule is given b a polnomial in one variable. The degree of a polnomial function is the degree of the polnomial in one variable, that is, the largest power of that appears. EXAMPLE Identifing Polnomial Functions Determine which of the following are polnomial functions. For those that are, state the degree; for those that are not, tell wh not. (a) f = (b) g = (c) (d) F = 0 (e) G = 8 (f) h = H = Solution (a) f is a polnomial function of degree 4. (b) g is not a polnomial function because g = =, so the variable is raised to the power, which is not a nonnegative integer. (c) h is not a polnomial function. It is the ratio of two polnomials, and the polnomial in the denominator is of positive degree. (d) F is the zero polnomial function; it is not assigned a degree.

23 SECTION 3. Polnomial Functions and Models 7 (e) G is a nonzero constant function. It is a polnomial function of degree 0 since G = 8 = 8 0. (f) H = = = So H is a polnomial function of degree 5. Do ou see how to find the degree of H without multipling out? NOW WORK PROBLEMS AND 5. Table 4 We have alread discussed in detail polnomial functions of degrees 0,, and. See Table 4 for a summar of the properties of the graphs of these polnomial functions. Degree Form Name Graph No degree f() = 0 Zero function The -ais 0 f() = a 0, a 0 Z 0 Constant function Horizontal line with -intercept a 0 f() = a + a 0, a Z 0 Linear function Nonvertical, nonhorizontal line with slope a and -intercept a 0 f() = a + a + a 0, a Z 0 Quadratic function Parabola: Graph opens up if a 7 0; graph opens down if a 6 0 One objective of this section is to analze the graph of a polnomial function. If ou take a course in calculus, ou will learn that the graph of ever polnomial function is both smooth and continuous. B smooth, we mean that the graph contains no sharp corners or cusps; b continuous, we mean that the graph has no gaps or holes and can be drawn without lifting pencil from paper. See Figures (a) and (b). Figure Corner Cusp Gap Hole (a) Graph of a polnomial function: smooth, continuous (b) Cannot be the graph of a polnomial function Graph Polnomial Functions Using Transformations We begin the analsis of the graph of a polnomial function b discussing power functions, a special kind of polnomial function. In Words A power function is a function that is defined b a single monomial. A power function of degree n is a function of the form f = a n where a is a real number, a Z 0, and n 7 0 is an integer. () Eamples of power functions are f = 3 f = -5 f = 8 3 f = -5 4 degree degree degree 3 degree 4

24 7 CHAPTER 3 Polnomial and Rational Functions The graph of a power function of degree, f = a, is a straight line, with slope a, that passes through the origin. The graph of a power function of degree, f = a, is a parabola, with verte at the origin, that opens up if a 7 0 and down if a 6 0. If we know how to graph a power function of the form f = n, then a compression or stretch and, perhaps, a reflection about the -ais will enable us to obtain the graph of g = a n. Consequentl, we shall concentrate on graphing power functions of the form f = n. We begin with power functions of even degree of the form f = n, n Ú and n even. Eploration Using our graphing utilit and the viewing window -, -4 6, graph the function Y On the same screen, graph Y = g() = 8 = f() = 4.. Now, also on the same screen, graph Y 3 = h() =. What do ou notice about the graphs as the magnitude of the eponent increases? Repeat this procedure for the viewing window -, 0. What do ou notice? Result See Figures 3(a) and (b). Figure 3 6 Y 4 Y 8 Y 3 Y 8 Y 4 4 (a) 0 Y 3 (b) Table 5 The domain of f() = n, n Ú and n even, is the set of all real numbers, and the range is the set of nonnegative real numbers. Such a power function is an even function (do ou see wh?), so its graph is smmetric with respect to the -ais. Its graph alwas contains the origin (0, 0) and the points (-, ) and (, ). For large n, it appears that the graph coincides with the -ais near the origin, but it does not; the graph actuall touches the -ais onl at the origin. See Table 5, where Y and Y 3 = = 8. For close to 0, the values of are positive and close to 0. Also, for large n, it ma appear that for 6 - or for 7 the graph is vertical, but it is not; it is onl increasing ver rapidl. If ou TRACE along one of the graphs, these distinctions will be clear. To summarize: NOTE Don t forget how graphing calculators epress scientific notation. In Table 5, -E -8 means - * 0-8. Properties of Power Functions, f() n, n Is an Even Integer. The graph is smmetric with respect to the -ais, so f is even.. The domain is the set of all real numbers. The range is the set of nonnegative real numbers. 3. The graph alwas contains the points 0, 0,,, and -,. 4. As the eponent n increases in magnitude, the graph becomes more vertical when 6 - or 7 ; but for near the origin, the graph tends to flatten out and lie closer to the -ais.

25 SECTION 3. Polnomial Functions and Models 73 Now we consider power functions of odd degree of the form f = n, n odd. Eploration Using our graphing utilit and the viewing window -, -6 6, graph the function Y On the same screen, graph Y and Y 3 = h() = = g() = 7 = f() = 3.. What do ou notice about the graphs as the magnitude of the eponent increases? Repeat this procedure for the viewing window -, -. What do ou notice? Result The graphs on our screen should look like Figures 4(a) and (b). Figure 4 Y 3 6 Y 7 Y 3 Y 7 Y 3 6 (a) (b) Y 3 The domain and the range of f() = n, n Ú 3 and n odd, are the set of real numbers. Such a power function is an odd function (do ou see wh?), so its graph is smmetric with respect to the origin. Its graph alwas contains the origin (0, 0) and the points (-, -) and (, ). It appears that the graph coincides with the -ais near the origin, but it does not; the graph actuall crosses the -ais onl at the origin. Also, it appears that as increases the graph is vertical, but it is not; it is increasing ver rapidl. TRACE along the graphs to verif these distinctions. To summarize: Properties of Power Functions, f() n, n Is an Odd Integer. The graph is smmetric with respect to the origin, so f is odd.. The domain and the range are the set of all real numbers. 3. The graph alwas contains the points 0, 0,,, and -, As the eponent n increases in magnitude, the graph becomes more vertical when 6 - or 7, but for near the origin, the graph tends to flatten out and lie closer to the -ais. The methods of shifting, compression, stretching, and reflection studied in Section.6, when used with the facts just presented, will enable us to graph polnomial functions that are transformations of power functions. EXAMPLE Graphing Polnomial Functions Using Transformations Graph: f = - 5

26 74 CHAPTER 3 Polnomial and Rational Functions Solution Figure 5 Figure 5 shows the required stages. (0, 0) (0, 0) (, ) (, ) (a) 5 (, ) (, ) Multipl b ; reflect about -ais (b) 5 (, ) (0, ) (, 0) Add ; shift up unit (c) 5 5 CHECK: Verif the result of Eample b graphing Y = f = - 5. EXAMPLE 3 Graphing Polnomial Functions Using Transformations Graph: f = - 4 Solution Figure 6 shows the required stages. Figure 6 (, ) (, ) (0, ) (0, 0) (, 0) (, ) ( 0, ) (, ) (, 0) (a) 4 Replace b ; shift right unit (b) ( ) 4 Multipl b ; compression b a factor of (c) ( ) 4 CHECK: Verif the result of Eample 3 b graphing Y = f = - 4 NOW WORK PROBLEMS 3 AND 7. 3 Identif the Zeros of a Polnomial Function and Their Multiplicit Figure 7 shows the graph of a polnomial function with four -intercepts. Notice that at the -intercepts the graph must either cross the -ais or touch the -ais. Consequentl, between consecutive -intercepts the graph is either above the -ais or below the -ais. We will make use of this propert of the graph of a polnomial shortl.

27 SECTION 3. Polnomial Functions and Models 75 Figure 7 Above -ais Above -ais Below -ais Crosses -ais Touches -ais Below -ais Crosses -ais If a polnomial function f is factored completel, it is eas to solve the equation f = 0 and locate the -intercepts of the graph. For eample, if f = - + 3, then the solutions of the equation f = = 0 are identified as and -3. Based on this result, we make the following observations: If f is a polnomial function and r is a real number for which fr = 0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then (a) r is an -intercept of the graph of f. (b) - r is a factor of f. So the real zeros of a function are the -intercepts of its graph, and the are found b solving the equation f = 0. EXAMPLE 4 Finding a Polnomial from Its Zeros (a) Find a polnomial of degree 3 whose zeros are -3,, and 5. (b) Graph the polnomial found in part (a) to verif our result. Solution (a) If r is a zero of a polnomial f, then - r is a factor of f. This means that Figure = + 3, -, and - 5 are factors of f. As a result, an 40 polnomial of the form f = a Seeing the Concept Graph the function found in Eample 4 for a = and a = -. Does the value of a affect the zeros of f? How does the value of a affect the graph of f? where a is an nonzero real number, qualifies. (b) The value of a causes a stretch, compression, or reflection, but does not affect the -intercepts. We choose to graph f with a =. f = = Figure 8 shows the graph of f. Notice that the -intercepts are -3,, and 5. NOW WORK PROBLEM 37. If the same factor - r occurs more than once, then r is called a repeated, or multiple, zero of f. More precisel, we have the following definition.

28 76 CHAPTER 3 Polnomial and Rational Functions If - r m is a factor of a polnomial f and - r m + is not a factor of f, then r is called a zero of multiplicit m of f. EXAMPLE 5 Identifing Zeros and Their Multiplicities For the polnomial f = a - b 4 is a zero of multiplicit because the eponent on the factor - is. -3 is a zero of multiplicit because the eponent on the factor + 3 is. is a zero of multiplicit 4 because the eponent on the factor - is 4. NOW WORK PROBLEM 45(a). In Eample 5 notice that, if ou add the multiplicities = 7, ou obtain the degree of the polnomial. Suppose that it is possible to factor completel a polnomial function and, as a result, locate all the -intercepts of its graph (the real zeros of the function). The following eample illustrates the role that the multiplicit of an -intercept plas. EXAMPLE 6 Investigating the Role of Multiplicit For the polnomial f = - : (a) Find the - and -intercepts of the graph of f. (b) Using a graphing utilit, graph the polnomial. (c) For each -intercept, determine whether it is of odd or even multiplicit. Figure 9 9 Solution (a) The -intercept is f0 = = 0. The -intercepts satisf the equation f = - = 0 from which we find that 4 Table 6 3 The -intercepts are 0 and. = 0 or - = 0 = 0 or = (b) See Figure 9 for the graph of f. (c) We can see from the factored form of f that 0 is a zero or root of multiplicit, and is a zero or root of multiplicit ; so 0 is of even multiplicit and is of odd multiplicit. We can use a TABLE to further analze the graph. See Table 6.The sign of f is the same on each side of = 0 and the graph of f just touches the -ais at = 0 (a zero of even multiplicit). The sign of f changes from one side of = to the other and the graph of f crosses the -ais at = (a zero of odd multiplicit). These observations suggest the following result:

29 SECTION 3. Polnomial Functions and Models 77 If r Is a Zero of Even Multiplicit Sign of f does not change from one side Graph touches of r to the other side of r. -ais at r. If r Is a Zero of Odd Multiplicit Sign of f changes from one side Graph crosses of r to the other side of r. -ais at r. NOW WORK PROBLEM 45(b). Figure 30 9 Points on the graph where the graph changes from an increasing function to a decreasing function, or vice versa, are called turning points. * Look at Figure 30. The graph of f = - = 3 -, has a turning point at 0, 0. After utilizing MINIMUM, we find that the graph also has a turning point at.33, -.9, rounded to two decimal places. 4 3 Eploration Graph Y and Y 3 = = 3, Y = 3 -, + 4. How man turning points do ou see? How does the number of turning points relate to the degree? Graph Y = 4, Y = 4-4 and 3 3, Y 3 = 4 -. How man turning points do ou see? How does the number of turning points compare to the degree? The following theorem from calculus supplies the answer. Theorem If f is a polnomial function of degree n, then f has at most n - turning points. One last remark about Figure 30. Notice that the graph of f = - looks somewhat like the graph of = 3. In fact, for ver large values of, either positive or negative, there is little difference. Eploration For each pair of functions Y and Y given in parts (a), (b), and (c), graph Y and Y on the same viewing window. Create a TABLE or TRACE for large positive and large negative values of. What do ou notice about the graphs of Y and Y as becomes ver large and positive or ver large and negative? (a) Y = ( - ); Y = 3 (b) Y = ; Y = 4 (c) Y = ; Y = - 3 * Graphing utilities can be used to approimate turning points. Calculus is needed to find the eact location of turning points.

30 78 CHAPTER 3 Polnomial and Rational Functions The behavior of the graph of a function for large values of, either positive or negative, is referred to as its end behavior. Theorem End Behavior For large values of, either positive or negative, that is, for large ƒƒ, the graph of the polnomial f = a n n + a n - n - + Á + a + a 0 resembles the graph of the power function = a n n Figure 3 End Behavior Look back at Figures 3 and 4. Based on this theorem and the previous discussion on power functions, the end behavior of a polnomial can be of onl four tpes. See Figure 3. f() = a n n + a n n a + a 0 (a) n even; a n > 0 (b) n even; a n < 0 (c) n 3 odd; a n > 0 (d) n 3 odd; a n < 0 For eample, consider the polnomial function f = The graph of f will resemble the graph of the power function = - 4 for large ƒƒ. The graph of f will look like Figure 3(b) for large ƒƒ. NOW WORK PROBLEM 45(c). Summar Graph of a Polnomial Function f a n n a n n Á a a 0, a n 0 Degree of the polnomial f: n Maimum number of turning points: n - At a zero of even multiplicit: The graph of f touches the -ais. At a zero of odd multiplicit: The graph of f crosses the -ais. Between zeros, the graph of f is either above or below the -ais. End behavior: For large ƒƒ, the graph of f behaves like the graph of = a n n.

31 SECTION 3. Polnomial Functions and Models 79 4 Analze the Graph of a Polnomial Function EXAMPLE 7 Analzing the Graph of a Polnomial Function Figure Figure 33 End behavior: Resembles = For the polnomial: (a) Find the degree of the polnomial. Determine the end behavior; that is, find the power function that the graph of f resembles for large values of ƒƒ. (b) Find the - and -intercepts of the graph of f. (c) Determine whether the graph crosses or touches the -ais at each -intercept. (d) Use a graphing utilit to graph f. (e) Determine the number of turning points on the graph of f. Approimate the turning points, if an eist, rounded to two decimal places. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f b hand. (g) Find the domain of f. Use the graph to find the range of f. (h) Use the graph to determine where f is increasing and where f is decreasing. Solution (a) The polnomial is of degree 4. End behavior: the graph of f resembles that of the power function = 4 for large values of ƒƒ. (b) The -intercept is f0 = 0. To find the -intercepts, if an, we first factor f. f = = = We find the -intercepts b solving the equation End behavior: Resembles = (.39, 6.35) 40 (.89, 45.33) So f = The -intercepts are -, 0, and 4. f = = 0 = 0 or - 4 = 0 or + = 0 = 0 or = 4 or = - (c) The -intercept 0 is a zero of multiplicit, so the graph of f will touch the -ais at 0; - and 4 are zeros of multiplicit, so the graph of f will cross the -ais at - and 4. (d) See Figure 3 for the graph of f. (e) From the graph of f shown in Figure 3, we can see that f has three turning points. Using MAXIMUM, one turning point is at 0, 0. Using MINIMUM, the two remaining turning points are at -.39, and.89, , rounded to two decimal places. (f) Figure 33 shows a graph of f using the information obtained in parts (a) to (e). (g) The domain of f is the set of all real numbers. The range of f is the interval , q. (h) Based on the graph, f is decreasing on the intervals - q, -.39 and 0,.89 and is increasing on the intervals -.39, 0 and.89, q. NOW WORK PROBLEM 69. For polnomial functions that have noninteger coefficients and for polnomials that are not easil factored, we utilize the graphing utilit earl in the analsis of the graph. This is because the amount of information that can be obtained from algebraic analsis is limited.

32 80 CHAPTER 3 Polnomial and Rational Functions Table 7 EXAMPLE 8 0 Using a Graphing Utilit to Analze the Graph of a Polnomial Function For the polnomial f = : (a) Find the degree of the polnomial. Determine the end behavior: that is, find the power function that the graph of f resembles for large values of ƒƒ. (b) Graph f using a graphing utilit. (c) Find the - and -intercepts of the graph. (d) Use a TABLE to find points on the graph around each -intercept. (e) Determine the local maima and local minima, if an eist, rounded to two decimal places. That is, locate an turning points. (f) Use the information obtained in parts (a) (e) to draw a complete graph of f b hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of f. Use the graph to find the range of f. (h) Use the graph to determine where f is increasing and where f is decreasing. Solution (a) The degree of the polnomial is 3. End behavior: the graph of f resembles that Figure 34 of the power function = 3 for large values of ƒƒ. 5 (b) See Figure 34 for the graph of f. (c) The -intercept is f0 = In Eample 7 we could easil factor f to find the -intercepts. However, it is not readil apparent how f factors in this 5 3 eample. Therefore, we use a graphing utilit s ZERO (or ROOT) feature and find the lone -intercept to be -3.79, rounded to two decimal places. (d) Table 7 shows values of around the -intercept. The points -4, and -, 3.04 are on the graph. (e) From the graph we see that it has two turning points: one between -3 and - the other between 0 and. Rounded to two decimal places, the local maimum is 3.36 and occurs at = -.8; the local minimum is and occurs at = The turning points are -.8, 3.36 and 0.63,. (f) Figure 35 shows a graph of f drawn b hand using the information obtained in parts (a) to (e). Figure 35 (.8, 3.36) (, 3.04) 6 End behavior: Resembles = 3 (0, ) 5 ( 3.79, 0) End behavior: ( 4, 4.57) (0.63, ) 6 Resembles = 3 (g) The domain and the range of f are the set of all real numbers. (h) Based on the graph, f is decreasing on the interval -.8, 0.63 and is increasing on the intervals - q, -.8 and 0.63, q. NOW WORK PROBLEM 8.

33 SECTION 3. Polnomial Functions and Models 8 Summar Steps for Graphing a Polnomial b Hand To analze the graph of a polnomial function = f, follow these steps: STEP : End behavior: find the power function that the graph of f resembles for large values of. STEP : (a) Find the -intercepts, if an, b solving the equation f = 0. (b) Find the -intercept b letting = 0 and finding the value of f0. STEP 3: Determine whether the graph of f crosses or touches the -ais at each -intercept. STEP 4: Use a graphing utilit to graph f. Determine the number of turning points on the graph of f. Approimate an turning points rounded to two decimal places. STEP 5: Use the information obtained in Steps to 4 to draw a complete graph of f b hand. 5 Find the Cubic Function of Best Fit to Data In Section.4 we found the line of best fit from data and in Section 3. we found the quadratic function of best fit. It is also possible to find polnomial functions of best fit. However, most statisticians do not recommend finding polnomials of best fit of degree higher than 3. Data that follow a cubic relation should look like Figure 36(a) or (b). Figure 36 a 3 b c d, a 0 (a) a 3 b c d, a 0 (b) Table 8 Year, EXAMPLE 9 Average Miles per Gallon, M 99, , , , , , , , , , , 7.6 SOURCE: United States Federal Highwa Administration A Cubic Function of Best Fit The data in Table 8 represent the average miles per gallon of vans, pickups, and sport utilit vehicles (SUVs) in the United States for 99 00, where represents 99, represents 99, and so on. (a) Draw a scatter diagram of the data using the ear as the independent variable and average miles per gallon M as the dependent variable. Comment on the tpe of relation that ma eist between the two variables and M. (b) Using a graphing utilit, find the cubic function of best fit M = M to these data. (c) Graph the cubic function of best fit on our scatter diagram. (d) Use the function found in part (b) to predict the average miles per gallon in 00 =.

34 8 CHAPTER 3 Polnomial and Rational Functions Solution (a) Figure 37 shows the scatter diagram. A cubic relation ma eist between the two variables. (b) Upon eecuting the CUBIC REGression program, we obtain the results shown in Figure 38. The output the utilit provides shows us the equation = a 3 + b + c + d. The cubic function of best fit to the data is M = (c) Figure 39 shows the graph of the cubic function of best fit on the scatter diagram. The function fits the data reasonabl well. Figure 37 Figure Figure (d) We evaluate the function M at =. M = L 8. The model predicts that the average miles per gallon of vans, pickups, and SUVs will be 8. miles per gallon in Assess Your Understanding Are You Prepared? Answers are given at the end of these eercises. If ou get a wrong answer, read the pages listed in red.. The intercepts of the equation = 36 are. 3. To graph = - 4, ou would shift the graph of = (pp. 5 7) a distance of units. (pp. 8 0). True or False: The epression is a 4. Use a graphing utilit to approimate (rounded to two decimal polnomial. (pp ) places) the local maima and local minima of f = , for (pp ) Concepts and Vocabular 5. The graph of ever polnomial function is both and. 6. A number r for which fr = 0 is called a(n) of the function f. 7. If r is a zero of even multiplicit for a function f, the graph of f the -ais at r. Skill Building 8. True or False: The graph of f = has eactl three -intercepts. 9. True or False: The -intercepts of the graph of a polnomial function are called turning points. 0. True or False: End behavior: the graph of the function f = resembles = 4 for large values of ƒƒ. In Problems, determine which functions are polnomial functions. For those that are, state the degree. For those that are not, tell wh not.. f = f = g = - 4. h = 3-5. f = - 6. f = -

35 SECTION 3. Polnomial Functions and Models g = 3> h = F = - 5. G = F = p 3 + G = In Problems 3 36, use transformations of the graph of = 4 or = to graph each function. 3. f = f = f = f = 4 8. f = f = f = f = f = f = 4 + f = - 4 f = f = f = In Problems 37 44, form a polnomial whose zeros and degree are given. 37. Zeros: -,, 3; degree Zeros: -,, 3; degree Zeros: -3, 0, 4; degree Zeros: -4, 0, ; degree 3 4. Zeros: -4, -,, 3; degree 4 4. Zeros: -3, -,, 5; degree Zeros: -, multiplicit ; 3, multiplicit ; degree Zeros: -, multiplicit ; 4, multiplicit ; degree 3 In Problems 45 56, for each polnomial function: (a) List each real zero and its multiplicit; (b) Determine whether the graph crosses or touches the -ais at each -intercept. (c) Find the power function that the graph of f resembles for large values of ƒƒ. 45. f = f = f = f = -a + b f = f = A + 3B f = f = f = f = a - 3 b - 3 f = f = 4-3 In Problems 57 80, for each polnomial function f: (a) Find the degree of the polnomial. Determine the end behavior: that is, find the power function that the graph of f resembles for large values of ƒƒ. (b) Find the - and -intercepts of the graph. (c) Determine whether the graph crosses or touches the -ais at each -intercept. (d) Graph f using a graphing utilit. (e) Determine the local maima and local minima, if an eist, rounded to two decimal places. That is, locate an turning points. (f) Use the information obtained in parts (a) (e) to draw a complete graph of f b hand. Be sure to label the intercepts and turning points. (g) Find the domain of f. Use the graph to find the range of f. (h) Use the graph to determine where f is increasing and where f is decreasing. 57. f = f = f = f = + 6. f = f = f = f = f = f = f = f = f = f = f = f = f = f = f = + - f = f = f = f = f =

36 84 CHAPTER 3 Polnomial and Rational Functions In Problems 8 90, for each polnomial function f: (a) Find the degree of the polnomial. Determine the end behavior: that is, find the power function that the graph of f resembles for large values of ƒƒ. (b) Graph f using a graphing utilit. (c) Find the - and -intercepts of the graph. (d) Use a TABLE to find points on the graph around each -intercept. (e) Determine the local maima and local minima, if an eist, rounded to two decimal places. That is, locate an turning points. (f) Use the information obtained in parts (a) (e) to draw a complete graph of f b hand. Be sure to label the intercepts, turning points, and the points obtained in part (d). (g) Find the domain of f. Use the graph to find the range of f. (h) Use the graph to determine where f is increasing and where f is decreasing. 8. f = f = f = f = f = f = f = 4 - p f = f = f = p 5 + p Applications and Etensions In Problems 9 94, construct a polnomial function that might have this graph. (More than one answer ma be possible.) Cost of Manufacturing The following data represent the cost C (in thousands of dollars) of manufacturing Chev Cavaliers and the number of Cavaliers produced. Number of Cavaliers Produced, Cost, C (a) Draw a scatter diagram of the data using as the independent variable and C as the dependent variable. Comment on the tpe of relation that ma eist between the two variables C and. (b) Find the average rate of change of cost from four to five Cavaliers. (c) What is the average rate of change in cost from eight to nine Cavaliers? (d) Use a graphing utilit to find the cubic function of best fit C = C. (e) Graph the cubic function of best fit on the scatter diagram. (f) Use the function found in part (d) to predict the cost of manufacturing Cavaliers. (g) Interpret the -intercept. 96. Cost of Printing The following data represent the weekl cost C (in thousands of dollars) of printing tetbooks and the number (in thousands of units) of tets printed.

37 SECTION 3. Polnomial Functions and Models 85 Number of Tet Books, (a) Draw a scatter diagram of the data using as the independent variable and C as the dependent variable. Comment on the tpe of relation that ma eist between the two variables C and. (b) Find the average rate of change in cost from 0,000 to 3,000 tetbooks. (c) What is the average rate of change in the cost of producing from 8,000 to 0,000 tetbooks? (d) Use a graphing utilit to find the cubic function of best fit C = C. (e) Graph the cubic function of best fit on the scatter diagram. (f) Use the function found in part (d) to predict the cost of printing,000 tets per week. (g) Interpret the -intercept. 97. Motor Vehicle Thefts The following data represent the number T (in thousands) of motor vehicle thefts in the United States for the ears , where = represents 989, = represents 990, and so on. Year, Cost, C , , , , , , , , , , , 5 000, 60 00, 3 6 SOURCE: U.S. Federal Bureau of Investigation Motor Vehicle Thefts, T (a) Draw a scatter diagram of the data using as the independent variable and T as the dependent variable. Comment on the tpe of relation that ma eist between the two variables T and. (b) Use a graphing utilit to find the cubic function of best fit T = T. (c) Graph the cubic function of best fit on the scatter diagram. (d) Use the function found in part (b) to predict the number of motor vehicle thefts in 005. (e) Do ou think the function found in part (b) will be useful in predicting motor vehicle thefts in 00? Wh? 98. Average Annual Miles The following data represent the average number of miles driven (in thousands) annuall b vans, pickups, and sport utilit vehicles (SUVs) for the ears , where = represents 993, = represents 994, and so on. Year, Average Miles Driven, M 993,.4 994,. 995, , , , , , , 9. SOURCE: United States Federal Highwa Administration (a) Draw a scatter diagram of the data using as the independent variable and M as the dependent variable. Comment on the tpe of relation that ma eist between the two variables M and. (b) Use a graphing utilit to find the cubic function of best fit M = M. (c) Graph the cubic function of best fit on the scatter diagram. (d) Use the function found in part (b) to predict the average miles driven b vans, pickups, and SUVs in 005. (e) Do ou think the function found in part (b) will be useful in predicting the average miles driven b vans, pickups, and SUVs in 00? Wh?

38 86 CHAPTER 3 Polnomial and Rational Functions Discussion and Writing 99. Can the graph of a polnomial function have no -intercept? Can it have no -intercepts? Eplain. 00. Write a few paragraphs that provide a general strateg for graphing a polnomial function. Be sure to mention the following: degree, intercepts, end behavior, and turning points. 0. Make up a polnomial that has the following characteristics: crosses the -ais at - and 4, touches the -ais at 0 and, and is above the -ais between 0 and. Give our polnomial to a fellow classmate and ask for a written critique of our polnomial. 0. Make up two polnomials, not of the same degree, with the following characteristics: crosses the -ais at -, touches the -ais at, and is above the -ais between - and. Give our polnomials to a fellow classmate and ask for a written critique of our polnomials. 03. The graph of a polnomial function is alwas smooth and continuous. Name a function studied earlier that is smooth and not continuous. Name one that is continuous, but not smooth. 04. Which of the following statements are true regarding the graph of the cubic polnomial f = 3 + b + c + d? (Give reasons for our conclusions.) (a) It intersects the -ais in one and onl one point. (b) It intersects the -ais in at most three points. (c) It intersects the -ais at least once. (d) For ƒƒ ver large, it behaves like the graph of = 3. (e) It is smmetric with respect to the origin. (f) It passes through the origin. Are You Prepared? Answers. -, 0,, 0, 0, 9. True 3. down; 4 4. Local maimum of 6.48 at = -0.67; local minimum of -3 at = 3.3 Properties of Rational Functions PREPARING FOR THIS SECTION Before getting started, review the following: Rational Epressions (Appendi, Section A.3, pp ) Graph of f = (Section., Eample 3, p. ) Polnomial Division (Appendi, Section A.4, pp ) Graphing Techniques: Transformations (Section.6, pp. 8 6) Now work the Are You Prepared? problems on page 95. OBJECTIVES Find the Domain of a Rational Function Find the Vertical Asmptotes of a Rational Function 3 Find the Horizontal or Oblique Asmptotes of a Rational Function Ratios of integers are called rational numbers. Similarl, ratios of polnomial functions are called rational functions. A rational function is a function of the form R = p q where p and q are polnomial functions and q is not the zero polnomial. The domain is the set of all real numbers ecept those for which the denominator q is 0.

39 SECTION 3.3 Properties of Rational Functions 87 Find the Domain of a Rational Function EXAMPLE Finding the Domain of a Rational Function (a) The domain of R = - 4 is the set of all real numbers ecept -5; that is, 5 ƒ Z (b) The domain of R = is the set of all real numbers ecept - and, - 4 that is, 5 ƒ Z -, Z 6. (c) The domain of R = (d) The domain of (e) The domain of 5 ƒ Z 6. 3 is the set of all real numbers. + R = - + is the set of all real numbers. 3 R = - is the set of all real numbers ecept, that is, - It is important to observe that the functions are not equal, since the domain of R is 5 ƒ Z 6 and the domain of f is the set of all real numbers. NOW WORK PROBLEM 3. If R = p is a rational function and if p and q have no common factors, q then the rational function R is said to be in lowest terms. For a rational function R = p in lowest terms, the zeros, if an, of the numerator are the -intercepts q of the graph of R and so will pla a major role in the graph of R. The zeros of the denominator of R [that is, the numbers, if an, for which q = 0], although not in the domain of R, also pla a major role in the graph of R. We will discuss this role shortl. We have alread discussed the properties of the rational function f =. (Refer to Eample 3, page.) The net rational function that we take up is H =. R = - - and f = + EXAMPLE Graphing Analze the graph of H =. Solution The domain of H = consists of all real numbers ecept 0. The graph has no -intercept, because can never equal 0. The graph has no -intercept because the equation H = 0 has no solution. Therefore, the graph of H will not cross either coordinate ais.

40 88 CHAPTER 3 Polnomial and Rational Functions Figure 40 7 Y 3 3 Table 9 Because H- = - = = H H is an even function, so its graph is smmetric with respect to the -ais. See Figure 40. Notice that the graph confirms the conclusions just reached. But what happens to the graph as the values of get closer and closer to 0? We use a TABLE to answer the question. See Table 9. The first four rows show that as approaches 0 the values of H become larger and larger positive numbers. When this happens, we sa that H is unbounded in the positive direction. We smbolize this b writing H : q [read as H approaches infinit ]. In calculus, limits are used to conve these ideas.there we use the smbolism lim H = q, read as :0 the limit of H as approaches 0 is infinit to mean that H : q as : 0. Look at the last three rows of Table 9. As : q, the values of H approach 0 (the end behavior of the graph). This is smbolized in calculus b writing H = 0. lim : q Figure 4 shows the graph of H = drawn b hand. Figure 4 5 (, 4) (, 4) (, ) (, ) (,, 4) ( ) Sometimes transformations (shifting, compressing, stretching, and reflection) can be used to graph a rational function. EXAMPLE 3 Solution Using Transformations to Graph a Rational Function Graph the rational function: R = - + First, notice that the domain of R is the set of all real numbers ecept =. To graph R, we start with the graph of = See Figure 4 for the stages.. Figure 4 (, ) (, ) (3, ) (, ) (3, ) (, ) 0 5 Replace b ; Add ; shift right shift up units unit (a) (b) (c) ( ) ( )

41 SECTION 3.3 Properties of Rational Functions 89 CHECK: Graph Y = using a graphing utilit to verif the graph - + obtained in Figure 4(c). NOW WORK PROBLEM 3. Asmptotes Notice that the -ais in Figure 4(a) is transformed into the vertical line = in Figure 4(c), and the -ais in Figure 4(a) is transformed into the horizontal line = in Figure 4(c). The Eploration that follows will help us analze the role of these lines. Eploration (a) Using a graphing utilit and the TABLE feature, evaluate the function H() = at ( - ) + = 0, 00, 000, and 0,000. What happens to the values of H as becomes unbounded in the positive direction, smbolized b (b) Evaluate H at = -0, -00, -000, and -0,000. What happens to the values of H as becomes unbounded in the negative direction, smbolized b lim H()? : -q (c) Evaluate H at =.5,.9,.99,.999, and What happens to the values of H as approaches, 6, smbolized b lim H()? : - (d) Evaluate H at =.5,.,.0,.00, and.000. What happens to the values of H as approaches, 7, smbolized b lim H()? : + Result lim : q H()? (a) Table 0 shows the values of Y = H() as approaches q. Notice that the values of H are approaching, so lim H() =. : q (b) Table shows the values of Y = H() as approaches - q. Again the values of H are approaching, so lim H() =. : -q (c) From Table we see that, as approaches, 6, the values of H are increasing without bound, so lim H() = q. : - (d) Finall, Table 3 reveals that, as approaches, 7, the values of H are increasing without bound, so lim H() = q. : + Table 0 Table Table Table 3 The results of the Eploration reveal an important propert of rational functions. The vertical line = and the horizontal line = are called asmptotes of the graph of H, which we define as follows:

42 90 CHAPTER 3 Polnomial and Rational Functions Let R denote a function: If, as : - q or as : q, the values of R approach some fied number L, then the line = L is a horizontal asmptote of the graph of R. If, as approaches some number c, the values ƒrƒ : q, then the line = c is a vertical asmptote of the graph of R. The graph of R never intersects a vertical asmptote. Figure 43 Even though the asmptotes of a function are not part of the graph of the function, the provide information about how the graph looks. Figure 43 illustrates some of the possibilities. c c R( ) L L R( ) (a) End behavior: As, the values of R( ) approach L [ lim R() L]. That is, the points on the graph of R are getting closer to the line L; L is a horizontal asmptote. (b) End behavior: As, the values of R() approach L [ lim R() L]. That is, the points on the graph of R are getting closer to the line L; L is a horizontal asmptote. (c) As approaches c, the values of R() [ lim c R() ; lim R() ]. That is, c the points on the graph of R are getting closer to the line c; c is a vertical asmptote. (d) As approaches c, the values of R() [ lim c R() ; lim R() ]. That is, c the points on the graph of R are getting closer to the line c; c is a vertical asmptote. Figure 44 Oblique asmptote A horizontal asmptote, when it occurs, describes a certain behavior of the graph as : q or as : - q, that is, its end behavior. The graph of a function ma intersect a horizontal asmptote. A vertical asmptote, when it occurs, describes a certain behavior of the graph when is close to some number c. The graph of the function will never intersect a vertical asmptote. If an asmptote is neither horizontal nor vertical, it is called oblique. Figure 44 shows an oblique asmptote. An oblique asmptote, when it occurs, describes the end behavior of the graph. The graph of a function ma intersect an oblique asmptote. Find the Vertical Asmptotes of a Rational Function The vertical asmptotes, if an, of a rational function R = p in lowest terms, q, are located at the zeros of the denominator of q. Suppose that r is a zero, so - r is a factor. Now, as approaches r, smbolized as : r, the values of - r approach 0, causing the ratio to become unbounded, that is, causing ƒrƒ : q. Based on the definition, we conclude that the line = r is a vertical asmptote. Theorem WARNING If a rational function is not in lowest terms, an application of this theorem will result in an incorrect listing of vertical asmptotes. Locating Vertical Asmptotes A rational function R = p in lowest terms, will have a vertical asmptote = r if r is a real zero of the denominator q. That is, if - r is a factor of q, the denominator q of a rational function R = p in lowest terms, then R q, will have the vertical asmptote = r.

43 SECTION 3.3 Properties of Rational Functions 9 EXAMPLE 4 Finding Vertical Asmptotes Find the vertical asmptotes, if an, of the graph of each rational function. (a) R = (b) F = (c) - 9 H = (d) G = Solution (a) R is in lowest terms and the zeros of the denominator - 4 are - and. Hence, the lines = - and = are the vertical asmptotes of the graph of R. (b) F is in lowest terms and the onl zero of the denominator is. The line = is the onl vertical asmptote of the graph of F. WARNING In Eample 4(a), the vertical (c) H is in lowest terms and the denominator has no real zeros because the discriminant of + is b - 4ac = 0-4 # # asmptotes are = - and = -4. The graph of H has no =. Do not sa that the vertical vertical asmptotes. asmptotes are and. (d) Factor G to determine if it is in lowest terms. G = = = , Z 3 The onl zero of the denominator of G in lowest terms is -7. Hence, the line = -7 is the onl vertical asmptote of the graph of G. As Eample 4 points out, rational functions can have no vertical asmptotes, one vertical asmptote, or more than one vertical asmptote. However, the graph of a rational function will never intersect an of its vertical asmptotes. (Do ou know wh?) Eploration Graph each of the following rational functions: R() = - R() = ( - ) R() = ( - ) 3 R() = ( - ) 4 Each has the vertical asmptote =. What happens to the value of R() as approaches from the right side of the vertical asmptote; that is, what is lim R()? What happens to the value of R() : + as approaches from the left side of the vertical asmptote; that is, what is lim R()? How does : the multiplicit of the zero in the denominator affect the graph of R? - NOW WORK PROBLEM 45. (Find the vertical asmptotes, if an.) 3 Find the Horizontal or Oblique Asmptotes of a Rational Function The procedure for finding horizontal and oblique asmptotes is somewhat more involved. To find such asmptotes, we need to know how the values of a function behave as : - q or as : q. If a rational function R is proper, that is, if the degree of the numerator is less than the degree of the denominator, then as : - q or as : q the value of R approaches 0. Consequentl, the line = 0 (the -ais) is a horizontal asmptote of the graph.

44 9 CHAPTER 3 Polnomial and Rational Functions Theorem If a rational function is proper, the line = 0 is a horizontal asmptote of its graph. EXAMPLE 5 Finding Horizontal Asmptotes Find the horizontal asmptotes, if an, of the graph of - R = Solution Since the degree of the numerator,, is less than the degree of the denominator,, the rational function R is proper. We conclude that the line = 0 is a horizontal asmptote of the graph of R. To see wh = 0 is a horizontal asmptote of the function R in Eample 5, we need to investigate the behavior of R as : - q and : q. When ƒƒ is unbounded, the numerator of R, which is -, can be approimated b the power function =, while the denominator of R, which is 4 + +, can be approimated b the power function = 4. Appling these ideas to R, we find This shows that the line = 0 is a horizontal asmptote of the graph of R. We verif these results in Tables 4(a) and (b). Notice, as : - q [Table 4(a)] or : q [Table 4(b)], respectivel, that the values of R approach 0. Table 4 - R = L q 4 = 4 : 0 q For ƒƒ unbounded As : - q or : q (a) (b) If a rational function R = p is improper, that is, if the degree of the q numerator is greater than or equal to the degree of the denominator, we must use long division to write the rational function as the sum of a polnomial f plus a r proper rational function That is, we write q. R = p q = f + r q r r where f is a polnomial and is a proper rational function. Since is q q r proper, then as : - q or as : q. As a result, q : 0 R = p q : f, as : - q or as : q

45 SECTION 3.3 Properties of Rational Functions 93 The possibilities are listed net.. If f = b, a constant, then the line = b is a horizontal asmptote of the graph of R.. If f = a + b, a Z 0, then the line = a + b is an oblique asmptote of the graph of R. 3. In all other cases, the graph of R approaches the graph of f, and there are no horizontal or oblique asmptotes. The following eamples demonstrate these conclusions. EXAMPLE 6 Finding Horizontal or Oblique Asmptotes Find the horizontal or oblique asmptotes, if an, of the graph of H = Solution Since the degree of the numerator, 4, is larger than the degree of the denominator, 3, the rational function H is improper. To find an horizontal or oblique asmptotes, we use long division. As a result, H = As : - q or as : q, = L 3 = : 0 As : - q or as : q, we have H : We conclude that the graph of the rational function H has an oblique asmptote = We verif these results in Tables 5(a) and (b) with Y = H and Y = As : - q or : q, the difference in the values between Y and Y is indistinguishable. Table 5 (a) (b)

46 94 CHAPTER 3 Polnomial and Rational Functions EXAMPLE 7 Solution Finding Horizontal or Oblique Asmptotes Find the horizontal or oblique asmptotes, if an, of the graph of Since the degree of the numerator,, equals the degree of the denominator,, the rational function R is improper. To find an horizontal or oblique asmptotes, we use long division As a result, R = R = Then, as : - q or as : q, = L - 4 = - 4 : 0 As : - q or as : q, we have R :. We conclude that = is a horizontal asmptote of the graph. CHECK: Verif the results of Eample 7 b creating a TABLE with Y = R and Y =. In Eample 7, we note that the quotient obtained b long division is the quotient of the leading coefficients of the numerator polnomial and the denominator polnomial a 8 This means that we can avoid the long division process for rational 4 b. functions whose numerator and denominator are of the same degree and conclude that the quotient of the leading coefficients will give us the horizontal asmptote. NOW WORK PROBLEM 4. EXAMPLE 8 Finding Horizontal or Oblique Asmptotes Find the horizontal or oblique asmptotes, if an, of the graph of G = Solution Since the degree of the numerator, 5, is larger than the degree of the denominator, 3, the rational function G is improper. To find an horizontal or oblique asmptotes, we use long division

47 SECTION 3.3 Properties of Rational Functions 95 As a result, G = Then, as : - q or as : q, = L 3 = : 0 As : - q or as : q, we have G : -. We conclude that, for large values of ƒƒ, the graph of G approaches the graph of = -. That is, the graph of G will look like the graph of = - as : - q or : q. Since = - is not a linear function, G has no horizontal or oblique asmptotes. We now summarize the procedure for finding horizontal and oblique asmptotes. Summar Finding Horizontal and Oblique Asmptotes of a Rational Function R Consider the rational function R = p q = a n n + a n - n - + Á + a + a 0 b m m + b m - m - + Á + b + b 0 in which the degree of the numerator is n and the degree of the denominator is m.. If n 6 m, then R is a proper rational function, and the graph of R will have the horizontal asmptote = 0 (the -ais).. If n Ú m, then R is improper. Here long division is used. (a) If the quotient obtained will be a number and the line = a n n = m,, is a horizontal asmptote. b m b m (b) If n = m +, the quotient obtained is of the form a + b (a polnomial of degree ), and the line = a + b is an oblique asmptote. (c) If n 7 m +, the quotient obtained is a polnomial of degree or higher, and R has neither a horizontal nor an oblique asmptote. In this case, for ƒƒ unbounded, the graph of R will behave like the graph of the quotient. NOTE The graph of a rational function either has one horizontal or one oblique asmptote or else has no horizontal and no oblique asmptote. a n 3.3 Assess Your Understanding Are You Prepared? Answers are given at the end of these eercises. If ou get a wrong answer, read the pages listed in red.. True or False: The quotient of two polnomial epressions is a rational epression. (p. 97). What is the quotient and remainder when is divided b -? (pp ) 3. Graph = (p. ). 4. Graph = using transformations. (pp. 8 6)

48 96 CHAPTER 3 Polnomial and Rational Functions Concepts and Vocabular 5. The line is a horizontal asmptote of 6. The line is a vertical asmptote of R = R = For a rational function R, if the degree of the numerator is less than the degree of the denominator, then R is. Skill Building 8. True or False: The domain of ever rational function is the set of all real numbers. 9. True or False: If an asmptote is neither horizontal nor vertical, it is called oblique. 0. True or False: If the degree of the numerator of a rational function equals the degree of the denominator, then the ratio of the leading coefficients gives rise to the horizontal asmptote. In Problems, find the domain of each rational function R =. R = 5 3. H = 4. G = F = 6. Q = 7. R = 8. R = R = H = 3 + G = - 3. F = In Problems 3 8, use the graph shown to find: (a) The domain and range of each function (b) The intercepts, if an (c) Horizontal asmptotes, if an (d) Vertical asmptotes, if an (e) Oblique asmptotes, if an (0, ) (, 0) (, 0) (, ) (, ) 3 3 In Problems 9 40, graph each rational function using transformations. Verif our result using a graphing utilit. 9. F = Q = R = 3. R = 3 -

49 SECTION 3.3 Properties of Rational Functions R = - H = G = R = - 4 G = + F = In Problems 4 5, find the vertical, horizontal, and oblique asmptotes, if an, of each rational function R = 4. R = H = T = 46. P = 47. Q = G = 3 - R = R = R = R = - 4 G = F = F = Applications and Etensions 54. Population Model A rare species of insect was discovered 53. Gravit In phsics, it is established that the acceleration due to gravit, g (in meters>sec ), at a height h meters above sea level is given b in the Amazon Rain Forest. To protect the species, environmentalists declare the insect endangered and transplant the gh = 3.99 * 0 4 insect into a protected area. The population P of the insect * h t months after being transplanted is given below. where * 0 6 is the radius of Earth in meters. (a) What is the acceleration due to gravit at sea level? (b) The Sears Tower in Chicago, Illinois, is 443 meters tall. What is the acceleration due to gravit at the top of the Sears Tower? (c) The peak of Mount Everest is 8848 meters above sea level. What is the acceleration due to gravit on the peak of Mount Everest? (d) Find the horizontal asmptote of gh. (e) Solve gh = 0. How do ou interpret our answer? t Pt = + 0.0t (a) How man insects were discovered? In other words, what was the population when t = 0? (b) What will the population be after 5 ears? (c) Determine the horizontal asmptote of Pt. What is the largest population that the protected area can sustain? Discussion and Writing 55. If the graph of a rational function R has the vertical asmptote = 4, then the factor - 4 must be present in the denominator of R. Eplain wh. 56. If the graph of a rational function R has the horizontal asmptote =, then the degree of the numerator of R equals the degree of the denominator of R. Eplain wh. 57. Can the graph of a rational function have both a horizontal and an oblique asmptote? Eplain. 58. Make up a rational function that has = + as an oblique asmptote. Eplain the methodolog that ou used. Are You Prepared? Answers. True. Quotient: 3-6; Remainder: (, ) 3 (, ) 3 (0, ) 3 (, 3) 3

50 98 CHAPTER 3 Polnomial and Rational Functions 3.4 The Graph of a Rational Function; Inverse and Joint Variation PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section., pp. 5 7) Even and Odd Functions (Section.3, pp. 80 8) Now work the Are You Prepared? problems on page 07. OBJECTIVES Analze the Graph of a Rational Function Solve Applied Problems Involving Rational Functions 3 Construct a Model Using Inverse Variation 4 Construct a Model Using Joint or Combined Variation Analze the Graph of a Rational Function Graphing utilities make the task of graphing rational functions less time consuming. However, the results of algebraic analsis must be taken into account before drawing conclusions based on the graph provided b the utilit. We will use the information collected in the last section in conjunction with the graphing utilit to analze the graph of a rational function R = p The analsis will require the q. following steps: Analzing the Graph of a Rational Function STEP : Find the domain of the rational function. STEP : Write R in lowest terms. STEP 3: Locate the intercepts of the graph. The -intercepts, if an, of R = p in lowest terms, satisf the equation p = 0. q The -intercept, if there is one, is R0. STEP 4: Test for smmetr. Replace b - in R. If R- = R, there is smmetr with respect to the -ais; if R- = -R, there is smmetr with respect to the origin. STEP 5: Locate the vertical asmptotes. The vertical asmptotes, if an, of R = p in lowest terms are found b identifing the real zeros q of q. Each zero of the denominator gives rise to a vertical asmptote. STEP 6: Locate the horizontal or oblique asmptotes, if an, using the procedure given in Section 3.3. Determine points, if an, at which the graph of R intersects these asmptotes. STEP 7: Graph R using a graphing utilit. STEP 8: Use the results obtained in Steps through 7 to graph R b hand. EXAMPLE Analzing the Graph of a Rational Function Analze the graph of the rational function: R = - - 4

51 SECTION 3.4 The Graph of a Rational Function; Inverse and Joint Variation 99 Solution First, we factor both the numerator and the denominator of R. STEP : The domain of R is 5 ƒ Z -, Z 6. STEP : R is in lowest terms. R = STEP 3: We locate the -intercepts b finding the zeros of the numerator. Figure Figure Connected mode (a) 4 4 Dot mode (b) = = ( 0, ) (, 0) 3 B inspection, is the onl -intercept. The -intercept is R0 = 4. STEP 4: Because R- = we conclude that R is neither even nor odd. There is no smmetr with respect to the -ais or the origin. STEP 5: We locate the vertical asmptotes b finding the zeros of the denominator. Since R is in lowest terms, the graph of R has two vertical asmptotes: the lines = - and =. STEP 6: The degree of the numerator is less than the degree of the denominator, so R is proper and the line = 0 (the -ais) is a horizontal asmptote of the graph. To determine if the graph of R intersects the horizontal asmptote, we solve the equation R = 0: The onl solution is =, so the graph of R intersects the horizontal asmptote at, 0. STEP 7: The analsis just completed helps us to set the viewing window to obtain a complete graph. Figure 45(a) shows the graph of R = - in - 4 connected mode, and Figure 45(b) shows it in dot mode. Notice in Figure 45(a) that the graph has vertical lines at = - and =. This is due to the fact that, when the graphing utilit is in connected mode, it will connect the dots between consecutive piels. We know that the graph of R does not cross the lines = - and =, since R is not defined at = - or =. When graphing rational functions, dot mode should be used if etraneous vertical lines appear. You should confirm that all the algebraic conclusions that we arrived at in Steps through 6 are part of the graph. For eample, the graph has vertical asmptotes at = - and =, and the graph has a horizontal asmptote at = 0. The -intercept is and the -intercept is. 4 STEP 8: Using the information gathered in Steps through 7, we obtain the graph of R shown in Figure 46. NOW WORK PROBLEM = 0 - = 0 = =

52 00 CHAPTER 3 Polnomial and Rational Functions Figure 47 EXAMPLE Solution NOTE Because the determinator of the rational function is a monomial, we can also find the oblique asmptote as follows: - = - = - Since : 0 as : q, = is an oblique asmptote. Analzing the Graph of a Rational Function Analze the graph of the rational function: R = - STEP : The domain of R is 5 ƒ Z 06. STEP : R is in lowest terms. STEP 3: The graph has two -intercepts: - and. There is no -intercept, since cannot equal 0. STEP 4: Since R- = -R, the function is odd and the graph is smmetric with respect to the origin. STEP 5: Since R is in lowest terms, the graph of R has the line = 0 (the -ais) as a vertical asmptote. STEP 6: Since the degree of the numerator,, is larger than the degree of the denominator,, the rational function R is improper. To find an horizontal or oblique asmptotes, we use long division. - - The quotient is, so the line = is an oblique asmptote of the graph. To determine whether the graph of R intersects the asmptote =, we solve the equation R =. R = - = - = - = 0 Impossible. - We conclude that the equation = has no solution, so the graph of R does not intersect the line =. STEP 7: See Figure 47. We see from the graph that there is no -intercept and there are two -intercepts, - and. The smmetr with respect to the origin is also evident. We can also see that there is a vertical asmptote at = 0. Finall, it is not necessar to graph this function in dot mode since no etraneous vertical lines are present. STEP 8: Using the information gathered in Steps through 7, we obtain the graph of R shown in Figure 48. Notice how the oblique asmptote is used as a guide in graphing the rational function b hand. Figure 48 3 = (, 0) 3 (, 0) 3 3 NOW WORK PROBLEM 5.

53 SECTION 3.4 The Graph of a Rational Function; Inverse and Joint Variation 0 Figure 49 EXAMPLE 3 0 Solution Connected mode (a) Dot mode (b) Analzing the Graph of a Rational Function Analze the graph of the rational function: We factor R to get R = R = STEP : The domain of R is 5 ƒ Z -4, Z 36. STEP : R is in lowest terms. STEP 3: The graph has two -intercepts: 0 and. The -intercept is R0 = 0. STEP 4: Because R- = = we conclude that R is neither even nor odd. There is no smmetr with respect to the -ais or the origin. STEP 5: Since R is in lowest terms, the graph of R has two vertical asmptotes: = -4 and = 3. STEP 6: Since the degree of the numerator equals the degree of the denominator, the graph has a horizontal asmptote.to find it, we form the quotient of the leading coefficient of the numerator, 3, and the leading coefficient of the denominator,. The graph of R has the horizontal asmptote = 3. To find out whether the graph of R intersects the asmptote, we solve the equation R = R = + - = = = -36 = 6 The graph intersects the line = 3 at = 6, and 6, 3 is a point on the graph of R. STEP 7: Figure 49(a) shows the graph of R in connected mode. Notice the etraneous vertical lines at = -4 and = 3 (the vertical asmptotes). As a result, we also graph R in dot mode. See Figure 49(b). STEP 8: Figure 49 does not displa the graph between the two -intercepts, 0 and. However, because the zeros in the numerator, 0 and, are of odd multiplicit (both are multiplicit ), we know that the graph of R crosses the -ais at 0 and. Therefore, the graph of R is above the -ais for To see this part better, we graph R for - in Figure 50. Using MAX- IMUM, we approimate the turning point to be 0.5, 0.07, rounded to two decimal places. Figure

54 0 CHAPTER 3 Polnomial and Rational Functions Figure 49 also does not displa the graph of R crossing the horizontal asmptote at 6, 3. To see this part better, we graph R for 4 60 in Figure 5. Using MINIMUM, we approimate the turning point to be.48,.75, rounded to two decimal places. Figure Using this information along with the information gathered in Steps through 7, we obtain the graph of R shown in Figure 5. Figure 5 = 4 = 3 8 (6, 3) = 3 (0.5, 0.07) (.48,.75) EXAMPLE 4 Analzing the Graph of a Rational Function with a Hole Analze the graph of the rational function: R = Solution We factor R and obtain R = STEP : The domain of R is 5 ƒ Z -, Z 6. STEP : In lowest terms, STEP 3: The graph has one -intercept: 0.5. The -intercept is R0 = STEP 4: Because R = - +, Z R- = we conclude that R is neither even nor odd. There is no smmetr with respect to the -ais or the origin.

55 SECTION 3.4 The Graph of a Rational Function; Inverse and Joint Variation 03 STEP 5: Since + is the onl factor of the denominator of R in lowest terms the graph has one vertical asmptote, = -. However, the rational function is undefined at both = and = -. STEP 6: Since the degree of the numerator equals the degree of the denominator, the graph has a horizontal asmptote.to find it, we form the quotient of the leading coefficient of the numerator,, and the leading coefficient of the denominator,. The graph of R has the horizontal asmptote =. To find whether the graph of R intersects the asmptote, we solve the equation R =. R = - + = - = + - = = 4 Impossible Figure 53 0 The graph does not intersect the line =. STEP 7: Figure 53 shows the graph of R. Notice that the graph has one vertical asmptote at = -. Also, the function appears to be continuous at = STEP 8: The analsis presented thus far does not eplain the behavior of the graph at =. We use the TABLE feature of our graphing utilit to determine the behavior of the graph of R as approaches. See Table 6. From the table, we conclude that the value of R approaches 0.75 as approaches. This result is further verified b evaluating R in lowest terms at =. We conclude that there is a hole in the graph at, Using the information gathered in Steps through 7, we obtain the graph of R shown in Figure 54. Table 6 Figure 54 = 8 (0, 0.5) (, 0.75) = 4 4 (0.5, 0) As Eample 4 shows, the values ecluded from the domain of a rational function give rise to either vertical asmptotes or holes. NOW WORK PROBLEM 33.

56 04 CHAPTER 3 Polnomial and Rational Functions EXAMPLE 5 Constructing a Rational Function from Its Graph Make up a rational function that might have the graph shown in Figure 55. Figure Solution The numerator of a rational function R = p in lowest terms determines the q -intercepts of its graph. The graph shown in Figure 55 has -intercepts - (even multiplicit; graph touches the -ais) and 5 (odd multiplicit; graph crosses the -ais). So one possibilit for the numerator is p = The denominator of a rational function in lowest terms determines the vertical asmptotes of its graph.the vertical asmptotes of the graph are = -5 and =. Since R approaches q from the left of = -5 and R approaches - q from the right of = -5, we know that + 5 is a factor of odd multiplicit in q. Also, R approaches - q from both sides of =, so - is a factor of even Figure 56 multiplicit in q. A possibilit for the denominator is q = So far we have R = However, the horizontal asmptote of the graph given in Figure 55, is =, so we know that the degree of the numerator must equal the degree in the denominator and the quotient of leading 5 0 coefficients must be This leads to R = Figure 56 shows the graph of R drawn on a graphing utilit. Since Figure 56 looks similar to Figure 55, we 5 have found a rational function R for the graph in Figure 55. NOW WORK PROBLEM 5. EXAMPLE 6 Solve Applied Problems Involving Rational Functions Finding the Least Cost of a Can Renolds Metal Compan manufactures aluminum cans in the shape of a clinder with a capacit of 500 cubic centimeters a The top and bottom of the can literb. are made of a special aluminum allo that costs 0.05 per square centimeter. The sides of the can are made of material that costs 0.0 per square centimeter.

57 SECTION 3.4 The Graph of a Rational Function; Inverse and Joint Variation 05 Figure 57 r h h Figure Top r Lateral Surface Area rh Bottom Solution Area r Area r (a) Epress the cost of material for the can as a function of the radius r of the can. (b) Use a graphing utilit to graph the function C = Cr. (c) What value of r will result in the least cost? (d) What is this least cost? (a) Figure 57 illustrates the situation. Notice that the material required to produce a clindrical can of height h and radius r consists of a rectangle of area prh and two circles, each of area pr. The total cost C (in cents) of manufacturing the can is therefore C Cost of top and bottom Cost of side ( r cm ) (0.05 /cm ) ( rh cm ) (0.0 /cm ) Total area Cost/unit area of top and bottom 0.0 r 0.04 rh Total area of side But we have the additional restriction that the height h and radius r must be chosen so that the volume V of the can is 500 cubic centimeters. Since V = pr h, we have 500 = pr h or h = 500 pr Cost/unit area Substituting this epression for h, the cost C, in cents, as a function of the radius r,is Cr = 0.0pr pr pr = 0.0pr + 0 r = 0.0pr3 + 0 r (b) See Figure 58 for the graph of Cr. (c) Using the MINIMUM command, the cost is least for a radius of about 3.7 centimeters. (d) The least cost is C3.7 L NOW WORK PROBLEM 6. Figure 59 = k, k 7 0, Construct a Model Using Inverse Variation In Section.4, we discussed direct variation. We now stud inverse variation. Let and denote two quantities. Then varies inversel with, or is inversel proportional to, if there is a nonzero constant k such that = k The graph in Figure 59 illustrates the relationship between and if varies inversel with and k 7 0, 7 0.

58 06 CHAPTER 3 Polnomial and Rational Functions EXAMPLE 7 Maimum Weight That Can Be Supported b a Piece of Pine Figure 60 See Figure 60. The maimum weight W that can be safel supported b a -inch b 4-inch piece of pine varies inversel with its length l. Eperiments indicate that the maimum weight that a 0-foot-long pine -b-4 can support is 500 pounds. Write a general formula relating the maimum weight W (in pounds) to the length l (in feet). Find the maimum weight W that can be safel supported b a length of 5 feet. Solution Because W varies inversel with l, we know that W = k l for some constant k. Because W = 500 when l = 0, we have 500 = k 0 k = 5000 Thus, in all cases, Figure 6 W Wl = 5000 l In particular, the maimum weight W that can be safel supported b a piece of pine 5 feet in length is (0, 500) W5 = = 00 pounds (5, 00) l Figure 6 illustrates the relationship between the weight W and the length l. In direct or inverse variation, the quantities that var ma be raised to powers. For eample, in the earl seventeenth centur, Johannes Kepler (57 630) discovered that the square of the period of revolution T of a planet around the Sun varies directl with the cube of its mean distance a from the Sun. That is, T = ka 3, where k is the constant of proportionalit. NOW WORK PROBLEM Construct a Model Using Joint or Combined Variation When a variable quantit Q is proportional to the product of two or more other variables, we sa that Q varies jointl with these quantities. Finall, combinations of direct and/or inverse variation ma occur. This is usuall referred to as combined variation. Let s look at an eample. EXAMPLE 8 Loss of Heat Through a Wall The loss of heat through a wall varies jointl with the area of the wall and the difference between the inside and outside temperatures and varies inversel with the thickness of the wall. Write an equation that relates these quantities.

59 SECTION 3.4 The Graph of a Rational Function; Inverse and Joint Variation 07 Solution We begin b assigning smbols to represent the quantities: L = Heat loss T = Temperature difference A = Area of wall d = Thickness of wall Then AT L = k d where k is the constant of proportionalit. NOW WORK PROBLEM 69. Figure 6 EXAMPLE 9 Wind Force of the Wind on a Window See Figure 6. The force F of the wind on a flat surface positioned at a right angle to the direction of the wind varies jointl with the area A of the surface and the square of the speed v of the wind. A wind of 30 miles per hour blowing on a window measuring 4 feet b 5 feet has a force of 50 pounds. What is the force on a window measuring 3 feet b 4 feet caused b a wind of 50 miles per hour? Solution Since F varies jointl with A and v, we have F = kav where k is the constant of proportionalit. We are told that F = 50 when A = 4 # 5 = 0 and v = 30. Then we have 50 = k0900 F = kav, F = 50, A = 0, v = 30 k = 0 The general formula is therefore F = 0 Av For a wind of 50 miles per hour blowing on a window whose area is square feet, the force F is F = 500 = 50 pounds 0 A = 3 # 4 = 3.4 Assess Your Understanding Are You Prepared? Answers are given at the end of these eercises. If ou get a wrong answer, read the pages listed in red.. Find the intercepts of = 30. (pp. 5 7). Determine whether f = + is even, odd, or neither. 4 - Is the graph of f smmetric with respect to the -ais or origin? (pp. 80 8) Concepts and Vocabular 3. If the numerator and the denominator of a rational function have no common factors, the rational function is. 4. True or False: The graph of a polnomial function sometimes has a hole. 5. True or False: The graph of a rational function never intersects a horizontal asmptote. 6. True or False: The graph of a rational function sometimes has a hole.

60 08 CHAPTER 3 Polnomial and Rational Functions Skill Building In Problems 7 44, follow Steps through 8 on page 98 to analze the graph of each function R = 8. R = R = + 4. R = Q = 4 - P = G = R = G = 0. G = H = 4 - R = F = F = F = + - R = R = 3. R = R = R = R = R = f = f = + 4. R = R = H = R = R = H = R = G = R = R = f = + f = f = f = In Problems 45 50, graph each function and use MINIMUM to obtain the minimum value, rounded to two decimal places. 45. f = f = , 7 0, 7 0 f = +, f = 49. f = , 7 0 3, 7 0 f = + 9 3, 7 0 In Problems 5 54, find a rational function that might have this graph. (More than one answer might be possible.)

61 SECTION 3.4 The Graph of a Rational Function; Inverse and Joint Variation Applications and Etensions 55. Drug Concentration The concentration C of a certain drug in a patient s bloodstream t hours after injection is given b t Ct = t + (a) Find the horizontal asmptote of Ct. What happens to the concentration of the drug as t increases? (b) Using a graphing utilit, graph C = Ct. (c) Determine the time at which the concentration is highest. 56. Drug Concentration The concentration C of a certain drug in a patient s bloodstream t minutes after injection is given b 50t Ct = t + 5 (a) Find the horizontal asmptote of Ct. What happens to the concentration of the drug as t increases? (b) Using a graphing utilit, graph C = Ct. (c) Determine the time at which the concentration is highest. 57. Average Cost In Problem 95, Eercise 3., the cost function C (in thousands of dollars) for manufacturing Chev Cavaliers was determined to be C = Economists define the average cost function as C = C (a) Find the average cost function. (b) What is the average cost of producing si Cavaliers? (c) What is the average cost of producing nine Cavaliers? (d) Using a graphing utilit, graph the average cost function. (e) Using a graphing utilit, find the number of Cavaliers that should be produced to minimize the average cost. (f) What is the minimum average cost? Average Cost In Problem 96, Eercise 3., the cost function C (in thousands of dollars) for printing tetbooks (in thousands of units) was found to be C = (a) Find the average cost function (refer to Problem 57). (b) What is the average cost of printing 3,000 tetbooks per week? (c) What is the average cost of printing 5,000 tetbooks per week? (d) Using our graphing utilit, graph the average cost function. (e) Using our graphing utilit, find the number of tetbooks that should be printed to minimize the average cost. (f) What is the minimum average cost? 59. Minimizing Surface Area United Parcel Service has contracted ou to design a closed bo with a square base that has a volume of 0,000 cubic inches. See the illustration. (a) Find a function for the surface area of the bo. (b) Using a graphing utilit, graph the function found in part (a). (c) What is the minimum amount of cardboard that can be used to construct the bo? (d) What are the dimensions of the bo that minimize the surface area? (e) Wh might UPS be interested in designing a bo that minimizes the surface area? 60. Minimizing Surface Area United Parcel Service has contracted ou to design a closed bo with a square base that has a volume of 5000 cubic inches. See the illustration.

62 0 CHAPTER 3 Polnomial and Rational Functions (a) Find a function for the surface area of the bo. (b) Using a graphing utilit, graph the function found in part (a). (c) What is the minimum amount of cardboard that can be used to construct the bo? (d) What are the dimensions of the bo that minimize the surface area? (e) Wh might UPS be interested in designing a bo that minimizes the surface area? 6. Cost of a Can A can in the shape of a right circular clinder is required to have a volume of 500 cubic centimeters. The top and bottom are made of material that costs 6 per square centimeter, while the sides are made of material that costs 4 per square centimeter. (a) Epress the total cost C of the material as a function of the radius r of the clinder. (Refer to Figure 57.) (b) Using a graphing utilit graph C = Cr. For what value of r is the cost C a minimum? 6. Material Needed to Make a Drum A steel drum in the shape of a right circular clinder is required to have a volume of 00 cubic feet. 65. Pressure The volume of a gas V held at a constant temperature in a closed container varies inversel with its pressure P. If the volume of a gas is 600 cubic centimeters cm 3 when the pressure is 50 millimeters of mercur (mm Hg), find the volume when the pressure is 00 mm Hg. 66. Resistance The current i in a circuit is inversel proportional to its resistance R measured in ohms. Suppose that when the current in a circuit is 30 amperes the resistance is 8 ohms. Find the current in the same circuit when the resistance is 0 ohms. 67. Weight The weight of an object above the surface of Earth varies inversel with the square of the distance from the center of Earth. If Maria weighs 5 pounds when she is on the surface of Earth (3960 miles from the center), determine Maria s weight if she is at the top of Mount McKinle (3.8 miles from the surface of Earth). 68. Intensit of Light The intensit I of light (measured in foot-candles) varies inversel with the square of the distance from the bulb. Suppose that the intensit of a 00-watt light bulb at a distance of meters is foot-candle. Determine the intensit of the bulb at a distance of 5 meters. 69. Geometr The volume V of a right circular clinder varies jointl with the square of its radius r and its height h.the constant of proportionalit is p. (See the figure.) Write an equation for V. r h (a) Epress the amount A of material required to make the drum as a function of the radius r of the clinder. (b) How much material is required if the drum s radius is 3 feet? (c) How much material is required if the drum s radius is 4 feet? (d) How much material is required if the drum s radius is 5 feet? (e) Using a graphing utilit graph A = Ar. For what value of r is A smallest? 63. Demand Suppose that the demand D for cand at the movie theater is inversel related to the price p. (a) When the price of cand is $.75 per bag, the theater sells 56 bags of cand on a tpical Saturda. Epress the demand for cand as a function of its price. (b) Determine the number of bags of cand that will be sold if the price is raised to $3 a bag. 64. Driving to School The time t that it takes to get to school varies inversel with our average speed s. (a) Suppose that it takes ou 40 minutes to get to school when our average speed is 30 miles per hour. Epress the driving time to school as a function of average speed. (b) Suppose that our average speed to school is 40 miles per hour. How long will it take ou to get to school? 70. Geometr The volume V of a right circular cone varies jointl with the square of its radius r and its height h.the p constant of proportionalit is (See the figure.) Write an 3. equation for V. h 7. Horsepower The horsepower (hp) that a shaft can safel transmit varies jointl with its speed (in revolutions per minute, rpm) and the cube of its diameter. If a shaft of a certain material inches in diameter can transmit 36 hp at 75 rpm, what diameter must the shaft have to transmit 45 hp at 5 rpm? r

63 SECTION 3.4 The Graph of a Rational Function; Inverse and Joint Variation 7. Chemistr: Gas Laws The volume V of an ideal gas varies directl with the temperature T and inversel with the pressure P. Write an equation relating V, T, and P using k as the constant of proportionalit. If a clinder contains ogen at a temperature of 300 K and a pressure of 5 atmospheres in a volume of 00 liters, what is the constant of proportionalit k? If a piston is lowered into the clinder, decreasing the volume occupied b the gas to 80 liters and raising the temperature to 30 K, what is the gas pressure? 73. Phsics: Kinetic Energ The kinetic energ K of a moving object varies jointl with its mass m and the square of its velocit v. If an object weighing 5 pounds and moving with a velocit of 00 feet per second has a kinetic energ of 400 foot-pounds, find its kinetic energ when the velocit is 50 feet per second. 74. Electrical Resistance of a Wire The electrical resistance of a wire varies directl with the length of the wire and inversel with the square of the diameter of the wire. If a wire 43 feet long and 4 millimeters in diameter has a resistance of.4 ohms, find the length of a wire of the same material whose resistance is.44 ohms and whose diameter is 3 millimeters. 75. Measuring the Stress of Materials The stress in the material of a pipe subject to internal pressure varies jointl with the internal pressure and the internal diameter of the pipe and inversel with the thickness of the pipe. The stress is 00 pounds per square inch when the diameter is 5 inches, the thickness is 0.75 inch, and the internal pressure is 5 pounds per square inch. Find the stress when the internal pressure is 40 pounds per square inch if the diameter is 8 inches and the thickness is 0.50 inch. 76. Safe Load for a Beam The maimum safe load for a horizontal rectangular beam varies jointl with the width of the beam and the square of the thickness of the beam and inversel with its length. If an 8-foot beam will support up to 750 pounds when the beam is 4 inches wide and inches thick, what is the maimum safe load in a similar beam 0 feet long, 6 inches wide, and inches thick? Discussion and Writing 77. Graph each of the following functions: Is = a vertical asmptote? Wh not? What is happening for =? What do ou conjecture about = n - -, n Ú an integer, for =? 78. Graph each of the following functions: = - = = - = = 4 - = = 6 - = What similarities do ou see? What differences? Write a few paragraphs that provide a general strateg for graphing a rational function. Be sure to mention the following: proper, improper, intercepts, and asmptotes. 80. Create a rational function that has the following characteristics: crosses the -ais at ; touches the -ais at -; one vertical asmptote at = -5 and another at = 6; and one horizontal asmptote, = 3. Compare ours to a fellow classmate s. How do the differ? What are their similarities? 8. Create a rational function that has the following characteristics: crosses the -ais at 3; touches the -ais at -; one vertical asmptote, = ; and one horizontal asmptote, =. Give our rational function to a fellow classmate and ask for a written critique of our rational function. 8. The formula on page 06 attributed to Johannes Kepler is one of the famous three Keplerian Laws of Planetar Motion. Go to the librar and research these laws. Write a brief paper about these laws and Kepler s place in histor. 83. Using a situation that has not been discussed in the tet, write a real-world problem that ou think involves two variables that var directl. Echange our problem with another student s to solve and critique. 84. Using a situation that has not been discussed in the tet, write a real-world problem that ou think involves two variables that var inversel. Echange our problem with another student s to solve and critique. 85. Using a situation that has not been discussed in the tet, write a real-world problem that ou think involves three variables that var jointl. Echange our problem with another student s to solve and critique. Are You Prepared? Answers. -0, 0, 0, 6. Even; the graph of f is smmetric with respect to the -ais.

64 CHAPTER 3 Polnomial and Rational Functions 3.5 Polnomial and Rational Inequalities PREPARING FOR THIS SECTION Solving Inequalities (Appendi, Section A.8, pp ) Now work the Are You Prepared? problem on page 7. Before getting started, review the following: OBJECTIVES Solve Polnomial Inequalities Algebraicall and Graphicall Solve Rational Inequalities Algebraicall and Graphicall In this section we solve inequalities that involve polnomials of degree and higher, as well as some that involve rational epressions. To solve such inequalities, we use the information obtained in the previous three sections about the graph of polnomial and rational functions. The general idea follows: Suppose that the polnomial or rational inequalit is in one of the forms f 6 0 f 7 0 f 0 f Ú 0 Locate the zeros of f if f is a polnomial function, and locate the zeros of the numerator and the denominator if f is a rational function. If we use these zeros to divide the real number line into intervals, then we know that on each interval the graph of f is either above the -ais 3f 7 04 or below the -ais 3f In other words, we have found the solution of the inequalit. The following steps provide more detail. Steps for Solving Polnomial and Rational Inequalities Algebraicall STEP : Write the inequalit so that a polnomial or rational epression f is on the left side and zero is on the right side in one of the following forms: f 7 0 f Ú 0 f 6 0 f 0 For rational epressions, be sure that the left side is written as a single quotient. STEP : Determine the numbers at which the epression f on the left side equals zero and, if the epression is rational, the numbers at which the epression f on the left side is undefined. STEP 3: Use the numbers found in Step to separate the real number line into intervals. STEP 4: Select a number in each interval and evaluate f at the number. (a) If the value of f is positive, then f 7 0 for all numbers in the interval. (b) If the value of f is negative, then f 6 0 for all numbers in the interval. If the inequalit is not strict Ú or, include the solutions of f = 0 in the solution set, but be careful not to include values of where the epression is undefined.

65 SECTION 3.5 Polnomial and Rational Inequalities 3 Solve Polnomial Inequalities Algebraicall and Graphicall EXAMPLE Solving a Polnomial Inequalit Solve the inequalit 4 +, and graph the solution set. Algebraic Solution STEP : Rearrange the inequalit so that 0 is on the right side Subtract 4 + from both sides of the inequalit. This inequalit is equivalent to the one we wish to solve. STEP : Find the zeros of f = b solving the equation = = = 0 = - or = 6 Factor. STEP 3: We use the zeros of f to separate the real number line into three intervals. - q, - -, 6 6, q STEP 4: We select a number in each interval found in Step 3 and evaluate f = at each number to determine if f is positive or negative. See Table 7. Graphing Solution We graph Y = and Y = 4 + on the same screen. See Figure 63. Using the IN- TERSECT command, we find that Y and Y intersect at = - and at = 6. The graph of Y is below that of Y, Y 6 Y, between the points of intersection. Since the inequalit is not strict, the solution set is 5 ƒ - 66 or, using interval notation, 3-, 64. Figure Y Y Table 7 6 Interval (- q, -) (-, 6) (6, q) Number Chosen Value of f f(-3) = 9 f(0) = - f(7) = 9 Conclusion Positive Negative Positive Since we want to know where f is negative, we conclude that f() 6 0 for all such that However, because the original inequalit is not strict, numbers that satisf the equation = 4 + are also solutions of the inequalit 4 +. Thus, we include - and 6. The solution set of the given inequalit is 5 ƒ - 66 or, using interval notation, 3-, 64. Figure Figure 64 shows the graph of the solution set. NOW WORK PROBLEMS 5 AND 9.

66 4 CHAPTER 3 Polnomial and Rational Functions EXAMPLE Solving a Polnomial Inequalit Algebraic Solution STEP : Rearrange the inequalit so that 0 is on the right side. Table Solve the inequalit 4 7, and graph the solution set. Subtract from both sides of the inequalit. This inequalit is equivalent to the one we wish to solve. STEP : Find the zeros of f = 4 - b solving 4 - = = = 0 Factor out = 0 Factor the difference of two cubes. = 0 or - = 0 or + + = 0 Set each factor equal to zero and solve. = 0 or = The equation + + = 0 has no real solutions. (Do ou see wh?) STEP 3: We use the zeros to separate the real number line into three intervals: - q, 0 0,, q STEP 4: We select a test number in each interval found in Step 3 and evaluate f = 4 - at each number to determine if f is positive or negative. See Table 8. Graphing Solution We graph Y = 4 and Y = on the same screen. See Figure 65. Using the INTER- SECT command, we find that Y and Y intersect at = 0 and at =. The graph of Y is above that of Y, Y 7 Y, to the left of = 0 and to the right of =. Since the inequalit is strict, the solution set is 5 ƒ 6 0 or 7 6 or, using interval notation, - q, 0 or, q. Figure 65 3 Y 4 Y 0 Interval (- q, 0) (0, ) (, q) Number Chosen - Value of f f(-) = fa b = f() = 4 Conclusion Positive Negative Positive Since we want to know where f is positive, we conclude that f() 7 0 for all numbers for which 6 0 or 7. Because the original inequalit is strict, numbers that satisf the equation 4 = are not solutions. The solution set of the given inequalit is 5 ƒ 6 0 or 7 6 or, using interval notation, - q, 0 or, q. Figure 66 0 Figure 66 shows the graph of the solution set. NOW WORK PROBLEM.

67 SECTION 3.5 Polnomial and Rational Inequalities 5 Solve Rational Inequalities Algebraicall and Graphicall Table 9 EXAMPLE 3 Algebraic Solution Solving a Rational Inequalit Solve the inequalit Ú 3, STEP : We first note that the domain of the variable consists of all real numbers ecept -. We rearrange terms so that 0 is on the right side Ú 0 Subtract 3 from both sides of the inequalit. STEP : For f = , find the zeros of f and the values of at which f is undefined. To find these numbers, we must epress f as a single quotient. f = Least Common Denominator: + + = Multipl 3 b # = + = - + Write as a single quotient. Combine like terms. The zero of f is. Also, f is undefined for = -. STEP 3: We use the numbers found in Step to separate the real number line into three intervals. - q, - -,, q STEP 4: We select a number in each interval found in Step 3 and evaluate f = at each number to determine if f is positive or negative. See Table 9. Interval (q, -) (-, ) (, q) Number Chosen -3 0 Value of f f(-3) = 4 f(0) = - f() = 4 Conclusion Positive Negative Positive We want to know where f is positive. We conclude that f() 7 0 for all such that 6 - or 7. Because the original inequalit is not strict, numbers - that satisf the equation are also solutions of the inequalit. Since + = 0 - onl if =, we conclude that the solution set is 5 ƒ 6 - or Ú 6 + = 0 or, using interval notation, - q, - or 3, q. and graph the solution set. Graphing Solution We first note that the domain of the variable consists of all real numbers ecept -. We graph Y = and Y = 3 on the same screen. See Figure 67. Using the INTERSECT command, we find that Y and Y intersect at =. The graph of Y is above that of Y, Y 7 Y, to the left of = - and to the right of =. Since the inequalit is not strict, the solution set is 5 ƒ 6 - or Ú 6. Using interval notation, the solution set is - q, - or 3, q. Figure 67 Y Y 4 5 Y 3 Figure Figure 68 shows the graph of the solution set. NOW WORK PROBLEM 35.

68 6 CHAPTER 3 Polnomial and Rational Functions Table 0 EXAMPLE 4 Pounds of Cookies (in Hundreds), Profit, P 0, ,93 6,583 36,948 44,38 49,638 49,5 44,38 34,0 Minimum Sales Requirements Tami is considering leaving her $30,000 a ear job and buing a cookie compan. According to the financial records of the firm, the relationship between pounds of cookies sold and profit is as ehibited b Table 0. (a) Draw a scatter diagram of the data in Table 0 with the pounds of cookies sold as the independent variable. (b) Use a graphing utilit to find the quadratic function of best fit. (c) Use the function found in part (b) to determine the number of pounds of cookies that Tami must sell for the profits to eceed $30,000 a ear and therefore make it worthwhile for her to quit her job. (d) Using the function found in part (b), determine the number of pounds of cookies that Tami should sell to maimize profits. (e) Using the function found in part (b), determine the maimum profit that Tami can epect to earn. Solution (a) Figure 69 shows the scatter diagram. (b) Using a graphing utilit, the quadratic function of best fit is P = ,04.5 where P is the profit achieved from selling pounds (in hundreds) of cookies. See Figure 70. (c) Since we want profit to eceed $30,000 we need to solve the inequalit We graph Figure 69 Figure 70 Y = P = ,04.5 and Y = 30,000 on the same screen. See Figure , ,000 Figure 7 55,000 55,000 Y ,04.5 Y 30,000 00, ,000 5,000 Using the INTERSECT command, we find that Y and Y intersect at = 0 and at = 735. The graph of Y is above that of Y, Y 7 Y, between the points of intersection. Tami must sell between 0 hundred or,000 and 735 hundred or 73,500 pounds of cookies to make it worthwhile to quit her job. (d) The function P found in part (b) is a quadratic function whose graph opens down. The verte is therefore the highest point. The -coordinate of the verte is b = - a = = 477 To maimize profit, 477 hundred (47,700) pounds of cookies must be sold. (e) The maimum profit is P477 = ,04.5 = $50,549

69 SECTION 3.5 Polnomial and Rational Inequalities Assess Your Understanding Are You Prepared? Answer is given at the end of these eercises. If ou get the wrong answer, read the pages listed in red.. Solve the inequalit: Graph the solution set. (pp ) Concepts and Vocabular. True or False: A test number for the interval is 0. Skill Building In Problems 3 56, solve each inequalit algebraicall. Verif our results using a graphing utilit Ú Ú Ú Ú Ú Ú Ú 0-4 Ú Ú Ú Applications and Etensions 57. For what positive numbers will the cube of a number eceed four times its square? 58. For what positive numbers will the square of a number eceed twice the number? 59. What is the domain of the function f = 3-6? 60. What is the domain of the function f = 3 3-3? - 6. What is the domain of the function R = A + 4? 6. What is the domain of the function R = A - + 4?

70 8 CHAPTER 3 Polnomial and Rational Functions 63. Phsics A ball is thrown verticall upward with an initial velocit of 80 feet per second. The distance s (in feet) of the ball from the ground after t seconds is s = 80t - 6t. See the figure. s 80t 6t 96 ft (a) For what time interval is the ball more than 96 feet above the ground? (b) Using a graphing utilit, graph the relation between s and t. (c) What is the maimum height of the ball? (d) After how man seconds does the ball reach the maimum height? 64. Phsics A ball is thrown verticall upward with an initial velocit of 96 feet per second. The distance s (in feet) of the ball from the ground after t seconds is s = 96t - 6t. (a) For what time interval is the ball more than feet above the ground? (b) Using a graphing utilit, graph the relation between s and t. (c) What is the maimum height of the ball? (d) After how man seconds does the ball reach the maimum height? 65. Business The monthl revenue achieved b selling wristwatches is figured to be dollars. The wholesale cost of each watch is $3. (a) How man watches must be sold each month to achieve a profit ( revenue - cost) of at least $50? (b) Using a graphing utilit, graph the revenue function. (c) What is the maimum revenue that this firm could earn? (d) How man wristwatches should the firm sell to maimize revenue? (e) Using a graphing utilit, graph the profit function. (f) What is the maimum profit that this firm can earn? (g) How man watches should the firm sell to maimize profit? (h) Provide a reasonable eplanation as to wh the answers found in parts (d) and (g) differ. Is the shape of the revenue function reasonable in our opinion? Wh? 66. Business The monthl revenue achieved b selling boes of cand is figured to be dollars. The wholesale cost of each bo of cand is $.50. (a) How man boes must be sold each month to achieve a profit of at least $60? (b) Using a graphing utilit, graph the revenue function. (c) What is the maimum revenue that this firm could earn? (d) How man boes of cand should the firm sell to maimize revenue? (e) Using a graphing utilit, graph the profit function. (f) What is the maimum profit that this firm can earn? (g) How man boes of cand should the firm sell to maimize profit? (h) Provide a reasonable eplanation as to wh the answers found in parts (d) and (g) differ. Is the shape of the revenue function reasonable in our opinion? Wh? 67. Cost of Manufacturing In Problem 95 of Section 3., a cubic function of best fit relating the cost C of manufacturing Chev Cavaliers was found. Budget constraints will not allow Chev to spend more than $97,000. Determine the number of Cavaliers that could be produced. 68. Cost of Printing In Problem 96 of Section 3., a cubic function of best fit relating the cost C of printing tetbooks in a week was found. Budget constraints will not allow the printer to spend more than $70,000 per week. Determine the number of tetbooks that could be printed in a week. 69. Prove that, if a, b are real numbers and a Ú 0, b Ú 0, then a b is equivalent to a b [Hint: b - a = A b - abab + ab. ] Discussion and Writing 70. Make up an inequalit that has no solution. Make up one that has eactl one solution. 7. The inequalit has no solution. Eplain wh. Are You Prepared? Answer. e ƒ 6 - f 0

71 SECTION 3.6 The Real Zeros of a Polnomial Function The Real Zeros of a Polnomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendi, Section A., p. 95) Polnomial Division (Appendi, Section A.4, pp ) Factoring Polnomials (Appendi, Section A.3, pp ) Quadratic Formula (Appendi, Section A.5, pp ) Snthetic Division (Appendi, Section A.4, pp ) Now work the Are You Prepared? problems on page 30. OBJECTIVES Use the Remainder and Factor Theorems Use the Rational Zeros Theorem 3 Find the Real Zeros of a Polnomial Function 4 Solve Polnomial Equations 5 Use the Theorem for Bounds on Zeros 6 Use the Intermediate Value Theorem Theorem In this section, we discuss techniques that can be used to find the real zeros of a polnomial function. Recall that if r is a real zero of a polnomial function f then fr = 0, r is an -intercept of the graph of f, and r is a solution of the equation f = 0. For polnomial and rational functions, we have seen the importance of the zeros for graphing. In most cases, however, the zeros of a polnomial function are difficult to find using algebraic methods. No nice formulas like the quadratic formula are available to help us find zeros for polnomials of degree 3 or higher. Formulas do eist for solving an third- or fourth-degree polnomial equation, but the are somewhat complicated. No general formulas eist for polnomial equations of degree 5 or higher. Refer to the Historical Feature at the end of this section for more information. Use the Remainder and Factor Theorems When we divide one polnomial (the dividend) b another (the divisor), we obtain a quotient polnomial and a remainder, the remainder being either the zero polnomial or a polnomial whose degree is less than the degree of the divisor. To check our work, we verif that QuotientDivisor + Remainder = Dividend This checking routine is the basis for a famous theorem called the division algorithm * for polnomials, which we now state without proof. Division Algorithm for Polnomials If f and g denote polnomial functions and if g is a polnomial whose degree is greater than zero, then there are unique polnomial functions q and r such that f g = q + r g or f = qg + r q q q q dividend quotient divisor remainder () where r is either the zero polnomial or a polnomial of degree less than that of g. * A sstematic process in which certain steps are repeated a finite number of times is called an algorithm. For eample, long division is an algorithm.

72 0 CHAPTER 3 Polnomial and Rational Functions In equation (), f is the dividend, g is the divisor, q is the quotient, and r is the remainder. If the divisor g is a first-degree polnomial of the form g = - c, c a real number then the remainder r is either the zero polnomial or a polnomial of degree 0. As a result, for such divisors, the remainder is some number, sa R, and we ma write f = - cq + R () This equation is an identit in and is true for all real numbers. Suppose that = c. Then equation () becomes fc = c - cqc + R fc = R Substitute fc for R in equation () to obtain f = - cq + fc (3) We have now proved the Remainder Theorem. Remainder Theorem Let f be a polnomial function. If f is divided b - c, then the remainder is fc. EXAMPLE Using the Remainder Theorem Find the remainder if f = is divided b (a) - 3 (b) + Solution Figure (a) We could use long division or snthetic division, but it is easier to use the Remainder Theorem, which sas that the remainder is f3. f3 = = = -4 The remainder is -4. (b) To find the remainder when f is divided b + = - -, we evaluate f-. f- = = = -9 The remainder is -9. Compare the method used in Eample (a) with the method used in Eample 4 on page 98 (snthetic division). Which method do ou prefer? Give reasons. NOTE A graphing utilit provides another wa to find the value of a function, using the evalueate feature. Consult our manual for details. See Figure 7 for the results of Eample (a). 50 An important and useful consequence of the Remainder Theorem is the Factor Theorem. Factor Theorem Let f be a polnomial function. Then - c is a factor of f if and onl if fc = 0.

73 SECTION 3.6 The Real Zeros of a Polnomial Function EXAMPLE The Factor Theorem actuall consists of two separate statements:. If fc = 0, then - c is a factor of f.. If - c is a factor of f, then fc = 0. The proof requires two parts. Proof. Suppose that fc = 0. Then, b equation (3), we have f = - cq + fc = - cq + 0 = - cq for some polnomial q. That is, - c is a factor of f.. Suppose that - c is a factor of f. Then there is a polnomial function q such that f = - cq Replacing b c, we find that fc = c - cqc = 0 # qc = 0 This completes the proof. One use of the Factor Theorem is to determine whether a polnomial has a particular factor. Using the Factor Theorem Use the Factor Theorem to determine whether the function f = has the factor (a) - (b) + Solution The Factor Theorem states that if fc = 0 then - c is a factor. (a) Because - is of the form - c with c =, we find the value of f. We choose to use substitution. f = = = 0 See also Figure 73(a). B the Factor Theorem, - is a factor of f. (b) We first need to write + in the form - c. Since + = - -, we find the value of f-. See Figure 73(b). Because f- =-7 Z 0, we conclude from the Factor Theorem that - - = + is not a factor of f. Figure (a) 40 (b)

74 CHAPTER 3 Polnomial and Rational Functions In Eample, we found that - was a factor of f. To write f in factored form, we can use long division or snthetic division. Using snthetic division, we find that The quotient is q = with a remainder of 0, as epected. We can write f in factored form as f = = NOW WORK PROBLEM. The net theorem concerns the number of real zeros that a polnomial function ma have. In counting the zeros of a polnomial, we count each zero as man times as its multiplicit. Theorem Number of Real Zeros A polnomial function of degree n, n Ú, has at most n real zeros. Proof The proof is based on the Factor Theorem. If r is a zero of a polnomial function f, then fr = 0 and, hence, - r is a factor of f. Each zero corresponds to a factor of degree. Because f cannot have more first-degree factors than its degree, the result follows. Theorem Use the Rational Zeros Theorem The net result, called the Rational Zeros Theorem, provides information about the rational zeros of a polnomial with integer coefficients. Rational Zeros Theorem Let f be a polnomial function of degree or higher of the form f = a n n + a n - n - + Á + a + a 0, a n Z 0, a 0 Z 0 p where each coefficient is an integer. If in lowest terms, is a rational zero of q, f, then p must be a factor of a 0, and q must be a factor of a n. EXAMPLE 3 Listing Potential Rational Zeros List the potential rational zeros of f = Solution Because f has integer coefficients, we ma use the Rational Zeros Theorem. First, we list all the integers p that are factors of a 0 = -6 and all the integers q that are factors of the leading coefficient a 3 =. p: ;, ;, ;3, ;6 q: ;, ;

75 EXAMPLE 4 3 Now we form all possible ratios SECTION 3.6 The Real Zeros of a Polnomial Function 3 p q. p q : ;, ;, ;3, ;6, ;, ; 3 If f has a rational zero, it will be found in this list, which contains possibilities. NOW WORK PROBLEM. Be sure that ou understand what the Rational Zeros Theorem sas: For a polnomial with integer coefficients, if there is a rational zero, it is one of those listed. The Rational Zeros Theorem does not sa that if a rational number is in the list of potential rational zeros, then it is a zero. It ma be the case that the function does not have an rational zeros. The Rational Zeros Theorem provides a list of potential rational zeros of a function f. If we graph f, we can get a better sense of the location of the -intercepts and test to see if the are rational.we can also use the potential rational zeros to select our initial viewing window to graph f and then adjust the window based on the results. The graphs shown throughout the tet will be those obtained after setting the final viewing window. Find the Real Zeros of a Polnomial Function Finding the Rational Zeros of a Polnomial Function Continue working with Eample 3 to find the rational zeros of f = Solution Figure We gather all the information that we can about the zeros. STEP : Since f is a polnomial of degree 3, there are at most three real zeros. STEP : We list the potential rational zeros obtained in Eample 3: ;, ;, ;3, ;6, ;, ; 3. STEP 3: We could, of course, use the Factor Theorem to test each potential rational zero to see if the value of f there is zero. This is not ver efficient. The graph of f will tell us approimatel where the real zeros are. So we onl need to test those rational zeros that are nearb. Figure 74 shows the graph of f. We see that f has three zeros: one near -6, one between - and 0, and one near. From our original list of potential rational zeros, we will test -6, using evalueate. STEP 4: Because f-6 = 0, we know that -6 is a zero and + 6 is a factor of f. We can use long division or snthetic division to factor f. f = = Now an solution of the equation - - = 0 will be a zero of f. Because of this, we call the equation - - = 0 a depressed equation of f. Since the degree of the depressed equation of f is less than that of the original polnomial, we work with the depressed equation to find the zeros of f.

76 4 CHAPTER 3 Polnomial and Rational Functions The depressed equation - - = 0 is a quadratic equation with discriminant b - 4ac = = The equation has two real solutions, which can be found b factoring. - - = + - = 0 + = 0 or - = 0 = - The zeros of are -6, - and., Because f = , we completel factor f as follows: f = = = or = Notice that the three zeros of f are among those given in the list of potential rational zeros in Eample 3. Steps for Finding the Real Zeros of a Polnomial Function STEP : Use the degree of the polnomial to determine the maimum number of zeros. STEP : If the polnomial has integer coefficients, use the Rational Zeros Theorem to identif those rational numbers that potentiall can be zeros. STEP 3: Using a graphing utilit, graph the polnomial function. STEP 4: (a) Use evalueate, substitution, snthetic division, or long division to test a potential rational zero based on the graph. (b) Each time that a zero (and thus a factor) is found, repeat Step 4 on the depressed equation. In attempting to find the zeros, remember to use (if possible) the factoring techniques that ou alread know (special products, factoring b grouping, and so on). EXAMPLE 5 Finding the Real Zeros of a Polnomial Function Find the real zeros of f = Write f in factored form. Solution STEP : There are at most five real zeros. STEP : To obtain the list of potential rational zeros, we write the factors p of a 0 = 48 and the factors q of the leading coefficient a 5 =. p: ;, ;, ;3, ;4, ;6, ;8, ;, ;6, ;4, ;48 q: ; The potential rational zeros consist of all possible quotients p q : p : ;, ;, ;3, ;4, ;6, ;8, ;, ;6, ;4, ;48 q

77 SECTION 3.6 The Real Zeros of a Polnomial Function 5 Figure STEP 3: Figure 75 shows the graph of f. The graph has the characteristics that we epect of the given polnomial of degree 5: no more than four turning points, -intercept 48, and it behaves like = 5 for large ƒƒ. STEP 4: Since -3 appears to be a zero and -3 is a potential rational zero, we evalueate f at -3 and find that f-3 = 0. We use snthetic division to factor f. We can factor f as f = = We now work with the first depressed equation: Repeat Step 4: In looking back at Figure 75, it appears that might be a zero of even multiplicit. We check the potential rational zero and find that f = 0. Using snthetic division, Now we can factor f as q = = f = Repeat Step 4: The depressed equation q = = 0 can be factored b grouping = = = = 0 - = 0 or + 4 = 0 = Since + 4 = 0 has no real solutions, the real zeros of f are -3 and, with being a zero of multiplicit. The factored form of f is f = = NOW WORK PROBLEM Solve Polnomial Equations EXAMPLE 6 Solving a Polnomial Equation Solve the equation: = 0 Solution The solutions of this equation are the zeros of the polnomial function f = Using the result of Eample 5, the real zeros are -3 and.these are the real solutions of the equation = 0. NOW WORK PROBLEM 63.

78 6 CHAPTER 3 Polnomial and Rational Functions In Eample 5, the quadratic factor + 4 that appears in the factored form of f is called irreducible, because the polnomial + 4 cannot be factored over the real numbers. In general, we sa that a quadratic factor a + b + c is irreducible if it cannot be factored over the real numbers, that is, if it is prime over the real numbers. Refer back to Eamples 4 and 5. The polnomial function of Eample 4 has three real zeros, and its factored form contains three linear factors. The polnomial function of Eample 5 has two distinct real zeros, and its factored form contains two distinct linear factors and one irreducible quadratic factor. Theorem Ever polnomial function (with real coefficients) can be uniquel factored into a product of linear factors and/or irreducible quadratic factors. We shall prove this result in Section 3.7, and, in fact, we shall draw several additional conclusions about the zeros of a polnomial function. One conclusion is worth noting now. If a polnomial (with real coefficients) is of odd degree, then it must contain at least one linear factor. (Do ou see wh?) This means that it must have at least one real zero. COROLLARY A polnomial function (with real coefficients) of odd degree has at least one real zero. 5 Use the Theorem for Bounds on Zeros One challenge in using a graphing utilit is to set the viewing window so that a complete graph is obtained. The net theorem is a tool that can be used to find bounds on the zeros. This will assure that the function does not have an zeros outside these bounds. Then using these bounds to set Xmin and Xma assures that all the -intercepts appear in the viewing window. A number M is a bound on the zeros of a polnomial if ever zero r lies between -M and M, inclusive. That is, M is a bound to the zeros of a polnomial f if -M an zero of f M Theorem Bounds on Zeros Let f denote a polnomial function whose leading coefficient is. f = n + a n - n - + Á + a + a 0 A bound M on the zeros of f is the smaller of the two numbers Ma5, ƒa 0 ƒ + ƒa ƒ + Á + ƒa n - ƒ6, + Ma5ƒa 0 ƒ, ƒa ƒ, Á, ƒa n - ƒ6 (4) where Ma 5 6 means choose the largest entr in 5 6. An eample will help to make the theorem clear.

79 SECTION 3.6 The Real Zeros of a Polnomial Function 7 EXAMPLE 7 Using the Theorem for Finding Bounds on Zeros Find a bound to the zeros of each polnomial. (a) f = (b) g = Solution (a) The leading coefficient of f is. f = a 4 = 0, a 3 = 3, a = -9, a = 0, a 0 = 5 We evaluate the epressions in formula (4). Ma5, ƒa 0 ƒ + ƒa ƒ + Á + ƒa n - ƒ6 = Ma5, ƒ5ƒ + ƒ0ƒ + ƒ -9ƒ + ƒ3ƒ + ƒ0ƒ6 = Ma5, 76 = 7 + Ma5ƒa 0 ƒ, ƒa ƒ, Á, ƒa n - ƒ6 = + Ma5ƒ5ƒ, ƒ0ƒ, ƒ -9ƒ, ƒ3ƒ, ƒ0ƒ6 = + 9 = 0 The smaller of the two numbers, 0, is the bound. Ever zero of f lies between -0 and 0. (b) First we write g so that its leading coefficient is. g = = 4a b Net we evaluate the two epressions in formula (4) with a 4 = 0, a 3 = - a and a 0 = =, a = 0, 4., Ma5, ƒa 0 ƒ + ƒa ƒ + Á + ƒa n - ƒ6 = Mae, ` 4 ` + ƒ0ƒ + ` ` + ` - ` + ƒ0ƒ f = Mae, 5 4 f = Ma5ƒa 0 ƒ, ƒa ƒ, Á, ƒa n - ƒ6 = + Mae ` 4 `, ƒ0ƒ, ` `, ` - `, ƒ0ƒ f 5 The smaller of the two numbers, , and 4 4. = + = 3 is the bound. Ever zero of g lies between EXAMPLE 8 Obtaining Graphs Using Bounds on Zeros Obtain a graph for each polnomial. (a) f = (b) g =

80 8 CHAPTER 3 Polnomial and Rational Functions Solution (a) Based on Eample 7(a), ever zero lies between -0 and 0. Using Xmin = -0 and Xma = 0, we graph Y = f = Figure 76(a) shows the graph obtained using ZOOM-FIT. Figure 76(b) shows the graph after adjusting the viewing window to improve the graph. Figure 76 0, Figure ,895 (a) (b) Based on Eample 7(b), ever zero lies between and Using Xmin = and Xma = 5 we graph Y = g = Figure 77 shows 4, the graph after using ZOOM-FIT. Here no adjustment of the viewing window is needed. NOW WORK PROBLEM 33. The net eample shows how to proceed when some of the coefficients of the polnomial are not integers. 4 (b) EXAMPLE 9 Finding the Zeros of a Polnomial Find all the real zeros of the polnomial function f = Figure 78 Solution (a) (b) STEP : There are at most five real zeros. STEP : Since there are noninteger coefficients, the Rational Zeros Theorem does not appl. STEP 3: We determine the bounds of f. The leading coefficient of f is with a 4 = -.8, a 3 = -7.79, a = 3.67, a = 37.95, and a 0 = We evaluate the epressions using formula (4). Ma5, ƒ -8.7ƒ + ƒ37.95ƒ + ƒ3.67 ƒ + ƒ -7.79ƒ + ƒ -.8ƒ6 = Ma5, = Ma5ƒ -8.7ƒ, ƒ37.95ƒ, ƒ3.67ƒ, ƒ -7.79ƒ, ƒ -.8ƒ6 = = The smaller of the two numbers, 38.95, is the bound. Ever real zero of f lies between and Figure 78(a) shows the graph of f with Xmin = and Xma = Figure 78(b) shows a graph of f after adjusting the viewing window to improve the graph. STEP 4: From Figure 78(b), we see that f appears to have four -intercepts: one near -4, one near -, one between 0 and, and one near 3.The -intercept near 3 might be a zero of even multiplicit since the graph seems to touch the -ais at that point.

81 SECTION 3.6 The Real Zeros of a Polnomial Function 9 We use the Factor Theorem to determine if -4 and - are zeros. Since f-4 = f- = 0, we know that -4 and - are zeros. Using ZERO (or ROOT), we find that the remaining zeros are 0.0 and 3.30, rounded to two decimal places. There are no real zeros on the graph that have not alread been identified. So, either 3.30 is a zero of multiplicit or there are two distinct zeros, each of which is 3.30, rounded to two decimal places. (Eample 0 will eplain how to determine which is true.) 6 Use the Intermediate Value Theorem The Intermediate Value Theorem requires that the function be continuous. Although calculus is needed to eplain the meaning precisel, the idea of a continuous function is eas to understand. Ver basicall, a function f is continuous when its graph can be drawn without lifting pencil from paper, that is, when the graph contains no holes or jumps or gaps. For eample, ever polnomial function is continuous. Intermediate Value Theorem Let f denote a continuous function. If a 6 b and if fa and fb are of opposite sign, then f has at least one zero between a and b. Although the proof of this result requires advanced methods in calculus, it is eas to see wh the result is true. Look at Figure 79. Figure 79 If f(a) 6 0 and f(b) 7 0 and if f is continuous, there is at least one zero between a and b. f(b) a f (a) Zero f() b f (b) f(a) The Intermediate Value Theorem together with the TABLE feature of a graphing utilit provides a basis for finding zeros. EXAMPLE 0 Using the Intermediate Value Theorem and a Graphing Utilit to Locate Zeros Continue working with Eample 9 to determine whether there is a repeated zero or two distinct zeros near Table Solution We use the TABLE feature of a graphing utilit. See Table. Since f3.9 = and f3.30 = , b the Intermediate Value Theorem there is a zero between 3.9 and Similarl, since f3.30 = and f3.3 = , there is another zero between 3.30 and 3.3. Now we know that the five zeros of f are distinct. NOW WORK PROBLEM 75.

82 30 CHAPTER 3 Polnomial and Rational Functions HISTORICAL FEATURE Tartaglia ( ) Formulas for the solution of third- and fourth-degree polnomial equations eist, and, while not ver practical, the do have an interesting histor. In the 500s in Ital, mathematical contests were a popular pastime, and persons possessing methods for solving problems kept them secret. (Solutions that were published were alread common knowledge.) Niccolo of Brescia ( ), commonl referred to as Tartaglia ( the stammerer ), had the secret for solving cubic (third-degree) equations, which gave him a decided advantage in the contests. See the Historical Problems. Girolamo Cardano (50 576) found out that Tartaglia had the secret, and, being interested in cubics, he requested it from Tartaglia. The reluctant Tartaglia hesitated for some time, but finall, swearing Cardano to secrec with midnight oaths b candlelight, told him the secret. Cardano then published the solution in his book Ars Magna (545), giving Tartaglia the credit but rather compromising the secrec. Tartaglia eploded into bitter recriminations, and each wrote pamphlets that reflected on the other s mathematics, moral character, and ancestr. See the Historical Problems. The quartic (fourth-degree) equation was solved b Cardano s student Lodovico Ferrari, and this solution also was included, with credit and this time with permission, in the Ars Magna. Attempts were made to solve the fifth-degree equation in similar was, all of which failed. In the earl 800s, P. Ruffini, Niels Abel, and Evariste Galois all found was to show that it is not possible to solve fifth-degree equations b formula, but the proofs required the introduction of new methods. Galois s methods eventuall developed into a large part of modern algebra. Historical Problems Problems 8 develop the Tartaglia Cardano solution of the cubic equation and show wh it is not altogether practical.. Show that the general cubic equation 3 + b + c + d = 0 can be transformed into an equation of the form b using the substitution = - b 3 + p + q = In the equation 3 + p + q = 0, replace b H + K. Let 3HK = -p, and show that H 3 + K 3 = -q. [Hint: 3H K + 3HK = 3HK. ] 3. Based on Problem, we have the two equations 3HK = -p and H 3 + K 3 = -q Solve for K in 3HK = -p and substitute into H 3 + K 3 = -q. Then show that H = C 3 -q + A [Hint: Look for an equation that is quadratic in form.] q 4 + p Use the solution for H from Problem 3 and the equation H 3 + K 3 = -q to show that 5. Use the results from Problems 4 to show that the solution of 3 + p + q = 0 is = C 3 K = C 3 -q + A q -q - A 4 + p3 7 + C 3 -q - A 6. Use the result of Problem 5 to solve the equation = Use a calculator and the result of Problem 5 to solve the equation = Use the methods of this chapter to solve the equation = 0. q 4 + p3 7 q 4 + p Assess Your Understanding Are You Prepared? Answers are given at the end of these eercises. If ou get a wrong answer, read the pages listed in red.. In the set e -, -, 0, name the numbers that, 4.5, p f, 3. Find the quotient and remainder if is divided b - 3. (pp and pp ) are integers. Which are rational numbers? (p. 95) 4. Solve the equation = 0. (pp ). Factor the epression (pp )

83 SECTION 3.6 The Real Zeros of a Polnomial Function 3 Concepts and Vocabular 5. In the process of polnomial division, (Divisor)(Quotient) + =. 6. When a polnomial function f is divided b - c, the remainder is. 7. If a function f, whose domain is all real numbers, is even and if 4 is a zero of f, then is also a zero. 8. True or False: Ever polnomial function of degree 3 with real coefficients has eactl three real zeros. 9. True or False: The onl potential rational zeros of f = are ;, ;. 0. True or False: If f is a polnomial function of degree 4 and if f = 5, then f - = p where p is a polnomial of degree 3. Skill Building In Problems 0, use the Factor Theorem to determine whether - c is a factor of f. If it is, write f in factored form, that is, write f in the form f = - c (quotient).. f = ; c = 3. f = ; c = - 3. f = ; c = 4. f = ; c = 5. f = ; c = - 6. f = ; c = f = ; c = 4 8. f = ; c = f = f = ; c = ; c = - 3 In Problems 3, tell the maimum number of real zeros that each polnomial function ma have. Then list the potential rational zeros of each polnomial function. Do not attempt to find the zeros.. f = f = f = f = f = f = f = f = f = f = f = f = In Problems 33 38, find the bounds to the zeros of each polnomial function. Use the bounds to obtain a complete graph of f. 33. f = f = f = f = f = f = In Problems 39 56, find the real zeros of f. Use the real zeros to factor f. 39. f = f = f = f = f = f = f = f = f = f = f = f = f = f = f = f = f = f = In Problems 57 6, find the real zeros of f rounded to two decimal places. 57. f = f = f = f = f = f =

84 3 CHAPTER 3 Polnomial and Rational Functions In Problems 63 7, find the real solutions of each equation = = = = = = = = = = 0 In Problems 73 78, use the Intermediate Value Theorem to show that each function has a zero in the given interval. Approimate the zero rounded to two decimal places. 73. f = ; 30, f = ; 3-, f = ; 3-5, f = ; 3.4, f = ; 3-3, -4 f = ; 3.7,.84 Applications and Etensions 79. Cost of Manufacturing In Problem 95 of Section 3., ou found the cost function for manufacturing Chev Cavaliers. Use the methods learned in this section to determine how man Cavaliers can be manufactured at a total cost of $3,000,000. In other words, solve the equation C = Cost of Printing In Problem 96 of Section 3., ou found the cost function for printing tetbooks. Use the methods learned in this section to determine how man tetbooks can be printed at a total cost of $00,000. In other words, solve the equation C = Find k such that f = 3 - k + k + has the factor Find k such that f = 4 - k 3 + k + has the factor What is the remainder when f = is divided b -? 84. What is the remainder when f = is divided b +? 85. One solution of the equation = 0 is 3. Find the sum of the remaining solutions. 86. One solution of the equation = 0 is -. Find the sum of the remaining solutions. 87. What is the length of the edge of a cube if, after a slice inch thick is cut from one side, the volume remaining is 94 cubic inches? Discussion and Writing 93. Is a zero of f = ? Eplain Is a zero of f = ? Eplain. 3 Are You Prepared? Answers 88. What is the length of the edge of a cube if its volume could be doubled b an increase of 6 centimeters in one edge, an increase of centimeters in a second edge, and a decrease of 4 centimeters in the third edge? 89. Use the Factor Theorem to prove that - c is a factor of n - c n for an positive integer n. 90. Use the Factor Theorem to prove that + c is a factor of n + c n if n Ú is an odd integer. 9. Let f be a polnomial function whose coefficients are integers. Suppose that r is a real zero of f and that the leading coefficient of f is. Use the Rational Zeros Theorem to show that r is either an integer or an irrational number. 9. Prove the Rational Zeros Theorem. p [Hint: Let where p and q have no common factors q, ecept and -, be a zero of the polnomial function f = a n n + a n - n - + Á + a + a 0, whose coefficients are all integers. Show that a n p n + a n - p n - q + Á + a pq n - + a 0 q n = 0. Now, because p is a factor of the first n terms of this equation, p must also be a factor of the term a 0 q n. Since p is not a factor of q (wh?), p must be a factor of a 0. Similarl, q must be a factor of a n.] Is a zero of f = ? Eplain Is a zero of f = ? Eplain. 3. Integers: 5-, 06; Rational e -, 0,, 4.5 f 3. Quotient: ; Remainder: e - - 3, f

85 SECTION 3.7 Comple Zeros; Fundamental Theorem of Algebra Comple Zeros; Fundamental Theorem of Algebra PREPARING FOR THIS SECTION Before getting started, review the following: Comple Numbers (Appendi, Section A.6, pp ) Now work the Are You Prepared? problems on page 37. Quadratic Equations with a Negative Discriminant (Appendi, Section A.6, pp ) OBJECTIVES Use the Conjugate Pairs Theorem Find a Polnomial Function with Specified Zeros 3 Find the Comple Zeros of a Polnomial In Section 3.6 we found the real zeros of a polnomial function. In this section we will find the comple zeros of a polnomial function. Finding the comple zeros of a function requires finding all zeros of the form a + bi. These zeros will be real if b = 0. A variable in the comple number sstem is referred to as a comple variable. A comple polnomial function f of degree n is a function of the form f = a n n + a n - n - + Á + a + a 0 () where a n, a n -, Á, a, a 0 are comple numbers, a n Z 0, n is a nonnegative integer, and is a comple variable. As before, a n is called the leading coefficient of f. A comple number r is called a (comple) zero of f if fr = 0. We have learned that some quadratic equations have no real solutions, but that in the comple number sstem ever quadratic equation has a solution, either real or comple. The net result, proved b Karl Friedrich Gauss ( ) when he was ears old, * gives an etension to comple polnomials. In fact, this result is so important and useful that it has become known as the Fundamental Theorem of Algebra. Fundamental Theorem of Algebra Ever comple polnomial function f of degree n Ú has at least one comple zero. We shall not prove this result, as the proof is beond the scope of this book. However, using the Fundamental Theorem of Algebra and the Factor Theorem, we can prove the following result: Theorem Ever comple polnomial function f of degree n Ú can be factored into n linear factors (not necessaril distinct) of the form f = a n - r - r # Á # - rn () where a n, r, r, Á, r n are comple numbers. That is, ever comple polnomial function of degree n Ú has eactl n (not necessaril distinct) zeros. * In all, Gauss gave four different proofs of this theorem, the first one in 799 being the subject of his doctoral dissertation.

86 34 CHAPTER 3 Polnomial and Rational Functions Proof Let f = a n n + a n - n - + Á + a + a 0 B the Fundamental Theorem of Algebra, f has at least one zero, sa r. Then, b the Factor Theorem, - r is a factor, and f = - r q where q is a comple polnomial of degree n - whose leading coefficient is a n. Again b the Fundamental Theorem of Algebra, the comple polnomial q has at least one zero, sa r. B the Factor Theorem, q has the factor - r, so q = - r q where q is a comple polnomial of degree n - whose leading coefficient is a n. Consequentl, f = - r - r q Repeating this argument n times, we finall arrive at f = - r - r # Á # - rn q n where q n is a comple polnomial of degree n - n = 0 whose leading coefficient is a Thus, q n = a n 0 n. = a n, and so f = a n - r - r # Á # - rn We conclude that ever comple polnomial function f of degree n Ú has eactl n (not necessaril distinct) zeros. Conjugate Pairs Theorem Use the Conjugate Pairs Theorem We can use the Fundamental Theorem of Algebra to obtain valuable information about the comple zeros of polnomials whose coefficients are real numbers. Let f be a polnomial whose coefficients are real numbers. If r = a + bi is a zero of f, then the comple conjugate r = a - bi is also a zero of f. In other words, for polnomials whose coefficients are real numbers, the zeros occur in conjugate pairs. Proof Let f = a n n + a n - n - + Á + a + a 0 where a n, a n -, Á, a, a 0 are real numbers and a n Z 0. If r = a + bi is a zero of f, then fr = fa + bi = 0, so a n r n + a n - r n - + Á + a r + a 0 = 0 We take the conjugate of both sides to get a n r n + a n - r n - + Á + a r + a 0 = 0 a n r n + a n - r n - + Á + a r + a 0 = 0 The conjugate of a sum equals the sum of the conjugates (see the Appendi, Section A.6). a n r n + a n - r n - + Á + a r + a 0 = 0 The conjugate of a product equals the product of the conjugates. a n r n + a n - r n - + Á + a r + a 0 = 0 The conjugate of a real number equals the real number. This last equation states that fr = 0; that is, r = a - bi is a zero of f.

87 SECTION 3.7 Comple Zeros; Fundamental Theorem of Algebra 35 The value of this result should be clear. If we know that, 3 + 4i is a zero of a polnomial with real coefficients, then we know that 3-4i is also a zero. This result has an important corollar. COROLLARY A polnomial f of odd degree with real coefficients has at least one real zero. Proof Because comple zeros occur as conjugate pairs in a polnomial with real coefficients, there will alwas be an even number of zeros that are not real numbers. Consequentl, since f is of odd degree, one of its zeros has to be a real number. For eample, the polnomial f = has at least one zero that is a real number, since f is of degree 5 (odd) and has real coefficients. EXAMPLE Using the Conjugate Pairs Theorem A polnomial f of degree 5 whose coefficients are real numbers has the zeros, 5i, and + i. Find the remaining two zeros. Solution Since comple zeros appear as conjugate pairs, it follows that -5i, the conjugate of 5i, and - i, the conjugate of + i, are the two remaining zeros. NOW WORK PROBLEM 7. Find a Polnomial Function with Specified Zeros EXAMPLE Finding a Polnomial Function Whose Zeros Are Given Figure (a) Find a polnomial f of degree 4 whose coefficients are real numbers and that has the zeros,, and -4 + i. (b) Graph the polnomial found in part (a) to verif our result. Solution (a) Since -4 + i is a zero, b the Conjugate Pairs Theorem, -4 - i must also be a zero of f. Because of the Factor Theorem, if fc = 0, then - c is a factor of f. So we can now write f as f = a i i4 where a is an real number. If we let a =, we obtain f = i i4 = i i i-4 - i4 = i i i - 4i - i = = = (b) A quick analsis of the polnomial f tells us what to epect: At most three turning points. For large ƒƒ, the graph will behave like = 4. A repeated real zero at so that the graph will touch the -ais at. The onl -intercept is at. Figure 80 shows the complete graph. (Do ou see wh? The graph has eactl three turning points and the degree of the polnomial is 4.)

88 36 CHAPTER 3 Polnomial and Rational Functions Eploration Graph the function found in Eample for a = and a = -. Does the value of a affect the zeros of f? How does the value of a affect the graph of f? Now we can prove the theorem we conjectured earlier in Section 3.6. Theorem Ever polnomial function with real coefficients can be uniquel factored over the real numbers into a product of linear factors and/or irreducible quadratic factors. Proof Ever comple polnomial f of degree n has eactl n zeros and can be factored into a product of n linear factors. If its coefficients are real, then those zeros that are comple numbers will alwas occur as conjugate pairs. As a result, if r = a + bi is a comple zero, then so is r = a - bi. Consequentl, when the linear factors - r and - r of f are multiplied, we have - r - r = - r + r + rr = - a + a + b This second-degree polnomial has real coefficients and is irreducible (over the real numbers). Thus, the factors of f are either linear or irreducible quadratic factors. 3 Find the Comple Zeros of a Polnomial EXAMPLE 3 Finding the Comple Zeros of a Polnomial Find the comple zeros of the polnomial function f = Solution STEP : The degree of f is 4. So f will have four comple zeros. STEP : The Rational Zeros Theorem provides information about the potential rational zeros of polnomials with integer coefficients. For this polnomial (which has integer coefficients), the potential rational zeros are ; 3, ;, ;, ;, ;3, ;6, ;9, ;8 3 Figure STEP 3: Figure 8 shows the graph of f. The graph has the characteristics that we epect of this polnomial of degree 4: It behaves like = 3 4 for large ƒƒ and has -intercept -8. There are -intercepts near - and between 0 and. STEP 4: Because f- = 0, we known that - is a zero and + is a factor of f. We can use long division or snthetic division to factor f: f = From the graph of f and the list of potential rational zeros, it appears that ma also be a zero of Since fa we know that is a zero of f. 3 f. 3 b = 0, 3 We use snthetic division on the depressed equation of f to factor

89 SECTION 3.7 Comple Zeros; Fundamental Theorem of Algebra 37 Using the bottom row of the snthetic division, we find f = + a - 3 b3 + 7 = 3 + a - 3 b + 9 The factor + 9 does not have an real zeros; its comple zeros are ;3i. The comple zeros of f = are -, 3, 3i, -3i. NOW WORK PROBLEM Assess Your Understanding Are You Prepared? Answers are given at the end of these eercises. If ou get a wrong answer, read the pages listed in red.. Find the sum and the product of the comple numbers 3 - i and i. (pp ). In the comple number sstem, solve the equation (pp ) + + = 0. Concepts and Vocabular 3. Ever polnomial function of odd degree with real coefficients will have at least real zero(s). 4. If 3 + 4i is a zero of a polnomial function of degree 5 with real coefficients, then so is. 5. True or False: A polnomial function of degree n with real coefficients has eactl n comple zeros. At most n of them are real zeros. 6. True or False: A polnomial function of degree 4 with real coefficients could have -3, + i, - i, and i as its zeros. Skill Building In Problems 7 6, information is given about a polnomial f whose coefficients are real numbers. Find the remaining zeros of f. 7. Degree 3; zeros: 3, 4 - i 8. Degree 3; zeros: 4, 3 + i 9. Degree 4; zeros: i, + i 0. Degree 4; zeros:,, + i. Degree 5; zeros:, i, i. Degree 5; zeros: 0,,, i 3. Degree 4; zeros: i,, - 4. Degree 4; zeros: - i, -i 5. Degree 6; zeros:, + i, -3 - i, 0 6. Degree 6; zeros: i, 3 - i, - + i In Problems 7, form a polnomial f with real coefficients having the given degree and zeros. 7. Degree 4; zeros: 3 + i; 4, multiplicit 8. Degree 4; zeros: i, + i 9. Degree 5; zeros: ; -i; + i 0. Degree 6; zeros: i, 4 - i; + i. Degree 4; zeros: 3, multiplicit ; -i. Degree 5; zeros:, multiplicit 3; + i In Problems 3 30, use the given zero to find the remaining zeros of each function. 3. f = ; zero: i 4. g = ; zero: -5i 5. f = ; zero: -i 6. h = ; zero: 3i 7. h = ; zero: 3 - i 8. f = ; zero: + 3i 9. h = ; zero: -4i 30. g = ; zero: 3i In Problems 3 40, find the comple zeros of each polnomial function. Write f in factored form. 3. f = 3-3. f = f = f =

90 38 CHAPTER 3 Polnomial and Rational Functions 35. f = f = f = f = f = f = Discussion and Writing In Problems 4 and 4, eplain wh the facts given are contradictor. 4. f is a polnomial of degree 3 whose coefficients are real 44. f is a polnomial of degree 4 whose coefficients are real numbers; its zeros are 4 + i, 4 - i, and + i. numbers; two of its zeros are -3 and 4 - i. Eplain wh one 4. f is a polnomial of degree 3 whose coefficients are real of the remaining zeros must be a real number. Write down numbers; its zeros are, i, and 3 + i. one of the missing zeros. 43. f is a polnomial of degree 4 whose coefficients are real numbers; three of its zeros are, + i, and - i. Eplain wh the remaining zero must be a real number. Are You Prepared? Answers. Sum: 3i; product: + i i, - + i6 Chapter Review Things to Know Quadratic function (pp ) f = a + b + c Graph is a parabola that opens up if a 7 0 and opens down if a 6 0. Verte: b a - a, fa - b a bb b Ais of smmetr: = - a -intercept: f0 -intercept(s): If an, found b finding the real solutions of the equation a + b + c = 0. Power function (pp. 7 74) f = n, n Ú even Domain: all real numbers Range: nonnegative real numbers Passes through -,, 0, 0,, Even function Decreasing on - q, 0, increasing on 0, q f = n, n Ú 3 odd Domain: all real numbers Range: all real numbers Passes through -, -, 0, 0,, Odd function Increasing on - q, q Polnomial function (p. 70 and pp ) f = a n n + a n - n - + Á + a + a 0, a n Z 0 Domain: all real numbers At most n - turning points End behavior: Behaves like = a n n for large ƒƒ Zeros of a polnomial function f (p. 75) Numbers for which f = 0; the real zeros of f are the -intercepts of the graph of f.

91 Chapter Review 39 Rational function (p. 86) R = p q p, q are polnomial functions. Inverse Variation (p. 05) Domain: 5 ƒ q Z 06 Vertical asmptotes: With R in lowest terms, if qr = 0, then = r is a vertical asmptote. Horizontal or oblique asmptotes: See the summar on page 98. Let and denote two quantities. Then varies inversel with, or is inversel proportional to, if there is a nonzero constant k such that = k. Remainder Theorem (p. 0) If a polnomial f is divided b - c, then the remainder is fc. Factor Theorem (p. 0) - c is a factor of a polnomial f if and onl if fc = 0. Rational Zeros Theorem (p. ) Let f be a polnomial function of degree or higher of the form f = a n n + a n - n - + Á + a + a 0, a n Z 0, a 0 Z 0 p where each coefficient is an integer. If in lowest terms, is a rational zero of f, then q, p must be a factor of a 0 and q must be a factor of a n. Intermediate Value Theorem (p. 9) Let f denote a continuous function. If a 6 b and fa and fb are of opposite sign, then f has at least one zero between a and b. Fundamental Theorem of Algebra (p. 33) Ever comple polnomial function f of degree n Ú has at least one comple zero. Conjugate Pairs Theorem (p. 34) Let f be a polnomial whose coefficients are real numbers. If r = a + bi is a zero of f, then its comple conjugate r = a - bi is also a zero of f. Objectives Section You should be able to Á Review Eercises 3. Graph a quadratic function using transformations (p. 5) 6 Identif the verte and ais of smmetr of a quadratic function (p. 53) Graph a quadratic function using its verte, ais, and intercepts (p. 54) Use the maimum or minimum value of a quadratic function to solve applied problems (p. 58) 5 5 Use a graphing utilit to find the quadratic function of best fit to data (p. 6) 5 3. Identif polnomial functions and their degree (p. 70) 3 6 Graph polnomial functions using transformations (p. 7) 6, Identif the zeros of a polnomial function and their multiplicit (p. 74) Analze the graph of a polnomial function (p. 79) Find the cubic function of best fit to data (p. 8) Find the domain of a rational function (p. 87) 4 44 Find the vertical asmptotes of a rational function (p. 90) Find the horizontal or oblique asmptotes of a rational function (p. 9) Analze the graph of a rational function (p. 98) Solve applied problems involving rational functions (p. 04) 7 3 Construct a model using inverse variation (p. 05) 3

92 40 CHAPTER 3 Polnomial and Rational Functions 4 Construct a model using joint or combined variation (p. 06) Solve polnomial inequalities algebraicall and graphicall (p. 3) Solve rational inequalities algebraicall and graphicall (p. 5) Use the Remainder and Factor Theorems (p. 9) 67 7 Use the Rational Zeros Theorem (p. ) Find the real zeros of a polnomial function (p. 3) Solve polnomial equations (p. 5) Use the Theorem for Bounds on Zeros (p. 6) Use the Intermediate Value Theorem (p. 9) Use the Conjugate Pairs Theorem (p. 34) Find a polnomial function with specified zeros (p. 35) Find the comple zeros of a polnomial (p. 36) 0 4 Review Eercises In Problems 6, graph each function using transformations (shifting, compressing, stretching, and reflection). Verif our result using a graphing utilit.. f = - +. f = f = f = f = f = In Problems 7 6, graph each quadratic function b determining whether its graph opens up or down and b finding its verte, ais of smmetr, -intercept, and -intercepts, if an. 7. f = f = f = 0. f = f = f = f = f = f = f = In Problems 7, determine whether the given quadratic function has a maimum value or a minimum value, and then find the value. 7. f = f = f = f = f = f = In Problems 3 6, determine which functions are polnomial functions. For those that are, state the degree. For those that are not, tell wh not. 3. f = f = + 5. f = > - 6. f = 3 In Problems 7 3, graph each function using transformations (shifting, compressing, stretching, and reflection). Show all the stages. Verif our result using a graphing utilit. 7. f = f = f = f = f = f = - 3 In Problems 33 40, for each polnomial function f: (a) Find the - and -intercepts of the graph of f. (b) Determine whether the graph crosses or touches the -ais at each -intercept. (c) End behavior: Find the power function that the graph of f resembles for large values of ƒƒ. (d) Use a graphing utilit to graph f. (e) Determine the number of turning points on the graph of f. Approimate the turning points if an eist, rounded to two decimal places. (f) Use the information obtained in parts (a) to (e) to draw a complete graph of f b hand. (g) Find the domain of f. Use the graph to find the range of f. (h) Use the graph to determine where f is increasing and where f is decreasing. 33. f = f = f = - + 4

93 Chapter Review f = f = f = f = f = In Problems 4 44, find the domain of each rational function. Find an horizontal, vertical, or oblique asmptotes R = R = + 4 R = R = In Problems 45 56, discuss each rational function following the eight steps on page R = R = H = R = R = F = R = 54. R = 55. G = H = F = - - F = - In Problems 57 66, solve each inequalit (a) algebraicall and (b) graphicall Ú Ú Ú In Problems 67 70, find the remainder R when f is divided b g. Is g a factor of f? 67. f = ; g = f = ; g = f = ; g = f = ; g = + 7. Find the value of f = at = Find the value of f = at = -. In Problems 73 and 74, tell the maimum number of real zeros that each polnomial function ma have. Then list the potential rational zeros of each polnomial function. Do not attempt to find the zeros. 73. f = f = In Problems 75 80, find all the real zeros of each polnomial function. 75. f = f = f = f = f = f = In Problems 8 84, determine the real zeros of the polnomial function. Approimate all irrational zeros rounded to two decimal places. 8. f = f = g = g = In Problems 85 88, find the real solutions of each equation = = = = 0 In Problems 89 9, find bounds to the zeros of each polnomial function. Obtain a complete graph of f. 89. f = f = f = f =

94 4 CHAPTER 3 Polnomial and Rational Functions In Problems 93 96, use the Intermediate Value Theorem to show that each polnomial has a zero in the given interval. Approimate the zero rounded to two decimal places. 93. f = ; 30, f = ; 3, f = ; 30, f = ; 3, 4 In Problems 97 00, information is given about a comple polnomial f whose coefficients are real numbers. Find the remaining zeros of f. Write a polnomial function whose zeros are given. 97. Degree 3; zeros: 4 + i, Degree 3; zeros: 3 + 4i, Degree 4; zeros: i, + i 00. Degree 4; zeros:,, + i In Problems 0 4, solve each equation in the comple number sstem = = = = = = = = = = = 0 + = = = 0 5. Find the point on the line = that is closest to the point 3,. [Hint: Find the minimum value of the function f = d, where d is the distance from 3, to a point on the line.] 6. Landscaping A landscape engineer has 00 feet of border to enclose a rectangular pond. What dimensions will result in the largest pond? 7. Enclosing the Most Area with a Fence A farmer with 0,000 meters of fencing wants to enclose a rectangular field and then divide it into two plots with a fence parallel to one of the sides (see the figure).what is the largest area that can be enclosed? 0. Parabolic Arch Bridges A horizontal bridge is in the shape of a parabolic arch. Given the information shown in the figure, what is the height h of the arch feet from shore? 0 ft 0 ft h ft 8. A rectangle has one verte on the line = 8 -, 7 0, another at the origin, one on the positive -ais, and one on the positive -ais. Find the largest area A that can be enclosed b the rectangle. 9. Architecture A special window in the shape of a rectangle with semicircles at each end is to be constructed so that the outside dimensions are 00 feet in length. See the illustration. Find the dimensions that maimizes the area of the rectangle.. Minimizing Marginal Cost The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For eample, if the marginal cost of producing the 50th product is $6.0, then it cost $6.0 to increase production from 49 to 50 units of output. Callawa Golf Compan has determined that the marginal cost C of manufacturing Big Bertha golf clubs ma be epressed b the quadratic function C = ,600 (a) How man clubs should be manufactured to minimize the marginal cost? (b) At this level of production, what is the marginal cost?. Violent Crimes The function Vt = -0.0t + 39.t

95 Chapter Review 43 models the number V (in thousands) of violent crimes committed in the United States t ears after 990. So t = 0 represents 990, t = represents 99, and so on. (a) Determine the ear in which the most violent crimes were committed. (b) Approimatel how man violent crimes were committed during this ear? (c) Using a graphing utilit, graph V = Vt. Were the number of violent crimes increasing or decreasing during the ears 994 to 998? SOURCE: Based on data obtained from the Federal Bureau of Investigation. 3. Weight of a Bod The weight of a bod varies inversel with the square of its distance from the center of Earth. Assuming that the radius of Earth is 3960 miles, how much would a man weigh at an altitude of mile above Earth s surface if he weighs 00 pounds on Earth s surface? 4. Resistance due to a Conductor The resistance (in ohms) of a circular conductor varies directl with the length of the conductor and inversel with the square of the radius of the conductor. If 50 feet of wire with a radius of 6 * 0-3 inch has a resistance of 0 ohms, what would be the resistance of 00 feet of the same wire if the radius is increased to 7 * 0-3 inch? 5. Advertising A small manufacturing firm collected the following data on advertising ependitures A (in thousands of dollars) and total revenue R (in thousands of dollars). Advertising Total Revenue $60 $6 $6350 $6378 $6453 $643 $6360 $63 (a) Draw a scatter diagram of the data. Comment on the tpe of relation that ma eist between the two variables. (b) Use a graphing utilit to find the quadratic function of best fit to these data. (c) Use the function found in part (b) to determine the optimal level of advertising for this firm. (d) Use the function found in part (b) to find the revenue that the firm can epect if it uses the optimal level of advertising. (e) With a graphing utilit, graph the quadratic function of best fit on the scatter diagram. 6. AIDS Cases in the United States The following data represent the cumulative number of reported AIDS cases in the United States for Year, t Number of AIDS Cases, A 990, 93,878 99, 5,638 99, 3 36, , 4 399,63 994, 5 457,80 995, 6 58,5 996, 7 594, , 8 653,53 SOURCE: U.S. Center for Disease Control and Prevention (a) Draw a scatter diagram of the data. (b) The cubic function of best fit to these data is At = -t t + 59,569t + 30,003 Use this function to predict the cumulative number of AIDS cases reported in the United States in 000. (c) Use a graphing utilit to verif that the function given in part (b) is the cubic function of best fit. (d) With a graphing utilit, draw a scatter diagram of the data and then graph the cubic function of best fit on the scatter diagram. (e) Do ou think the function given in part (b) will be useful in predicting the number of AIDS cases in 005? 7. Making a Can A can in the shape of a right circular clinder is required to have a volume of 50 cubic centimeters. (a) Epress the amount A of material to make the can as a function of the radius r of the clinder. (b) How much material is required if the can is of radius 3 centimeters? (c) How much material is required if the can is of radius 5 centimeters? (d) Graph A = Ar. For what value of r is A smallest? 8. Design a polnomial function with the following characteristics: degree 6; four real zeros, one of multiplicit 3; -intercept 3; behaves like = -5 6 for large values of ƒƒ. Is this polnomial unique? Compare our polnomial with those of other students. What terms will be the same as everone else s? Add some more characteristics, such as smmetr or naming the real zeros. How does this modif the polnomial? 9. Design a rational function with the following characteristics: three real zeros, one of multiplicit ; -intercept ; vertical asmptotes = - and = 3; oblique asmptote = +. Is this rational function unique? Compare ours with those of other students. What will be the same as everone else s? Add some more characteristics, such as smmetr or naming the real zeros. How does this modif the rational function?

96 44 CHAPTER 3 Polnomial and Rational Functions 30. The illustration shows the graph of a polnomial function. (a) Is the degree of the polnomial even or odd? (b) Is the leading coefficient positive or negative? (c) Is the function even, odd, or neither? (d) Wh is necessaril a factor of the polnomial? (e) What is the minimum degree of the polnomial? (f) Formulate five different polnomials whose graphs could look like the one shown. Compare ours to those of other students. What similarities do ou see? What differences? Chapter Test. Graph f = f = (a) Determine if the function has a maimum or a minimum and eplain how ou know. (b) Algebraicall, determine the verte of the graph. (c) Determine the ais of smmetr of the graph. (d) Algebraicall, determine the intercepts of the graph. (e) Graph the function b hand. 3. For the polnomial function g = , (a) Determine the maimum number of real zeros that the function ma have. (b) Find bounds to the zeros of the function. (c) List the potential rational zeros. (d) Determine the real zeros of g. Factor g over the reals. 4. Find the comple zeros of 5. Solve = 8-4 in the comple number sstem. In Problems 6 and 7, find the domain of each function. Find an horizontal, vertical, or oblique asmptotes. 6. g = f = r = Sketch the graph of the function in Problem 7. Label all intercepts, vertical asmptotes, horizontal asmptotes, and oblique asmptotes. Verif our results using a graphing utilit. In Problems 9 and 0, write a function that meets the given conditions. 9. Fourth-degree polnomial with real coefficients; zeros: -, 0, 3 + i ƒ 0. Rational function; asmptotes: =, = 4; domain: 5 Z 4, Z 96. Use the Intermediate Value Theorem to show that the function f = has at least one real zero on the interval 30, 44. In Problems and 3, solve each inequalit algebraicall. Write the solution set in interval notation. Verif our results using a graphing utilit Ú Given the polnomial function f = - - +, do the following: (a) Find the - and -intercepts of the graph of f. (b) Determine whether the graph crosses or touches the -ais at each -intercept. (c) Find the power function that the graph of f resembles for large values of ƒƒ. (d) Using results of parts (a) (c), describe what the graph of f should look like. Verif this with a graphing utilit. (e) Approimate the turning points of f rounded to two decimal places. (f) B hand, use the information in parts (a) (e) to draw a complete graph of f. 5. The table gives the average home game attendance A for the St. Louis Cardinals (in thousands) during the ears , where is the number of ears since 995. SOURCE: A (a) Use a graphing utilit to find the quadratic function of best fit to the data. Round coefficients to three decimal places. (b) Use the function found in part (a) to estimate the average home game attendance for the St. Louis Cardinals in 004.

97 Chapter Projects 45 Chapter Projects. Weed Pollen Given in the table below is the Weed pollen count for Leington, Kentuck, from August 4, 004, to October 6, 004, the prime season for weed pollen. Date Grains per Date Grains per cubic meter cubic meter 8>4 8 9>8 45 8>9 4 9>0 93 8> 9>3 4 8>3 6 9>5 0 8>7 37 9>7 8>8 30 9>0 5 8>3 53 9> 3 8>4 9>7 4 8>5 47 9>9 6 8>6 7 0> 4 8>3 4 0>4 4 9> 3 0>5 7 9>3 44 0>6 7 9>6 50 SOURCE: National Allerg Bureau at the American Academ of Allerg, Asthma and Immunolog website ( (a) (b) (c) (d) (e) (f) (g) Let the independent variable D represent the date, where D = on 8>4, D = 6 on 8>9, D = 9 on 8> Á and D = 64 on 0>6. Let the dependent variable P represent the number of grains per cubic meter of weed pollen. Draw a scatter diagram of the data using our graphing utilit, and b hand on graph paper. In the hand-drawn scatter diagram, sketch a smooth curve that fits the data. Tr to have the curve pass through as man of the points as closel as possible. How man turning points are there? What does this tell ou about the degree of the polnomial that could be fit to the data? If the ends of the graph on the left and on the right are etended downward, how man real zeros are indicated b the graph? How man comple? Wh? Use a graphing utilit to determine the quartic function of the best fit. Graph it on our graphing utilit. Does it look like what ou epected from our handdrawn graph? Eplain. Use a graphing utilit to determine the cubic function of best fit. Graph it on our graphing utilit. Does it look like what ou epected from our hand-drawn graph? Eplain. Use a graphing utilit to determine the quadratic functon of best fit. Graph it on our graphing utilit. Does it look like what ou epected from our hand-drawn graph? Eplain. Which of the three functions seems to fit the best? Eplain our reasoning. Investigate weed pollen counts at other times of the ear. (Go to the National Allerg Bureau at Would the polnomial ou found in part (c) work for other -month periods? Wh or wh not? What about for the same period of time in other ears? The following projects are available at the Instructor s Resource Center (IRC):. Project at Motorola How Man Cellphones Can I Make? 3. First and Second Differences 4. Cannons 5. Maclaurin Series 6. Theor of Equations 7. CBL Eperiment

98 46 CHAPTER 3 Polnomial and Rational Functions Cumulative Review. Find the distance between the points P =, 3 and Q = -4,.. Solve the inequalit Ú and graph the solution set. 3. Solve the inequalit and graph the solution set. 4. Find a linear function with slope -3 that contains the point -, 4. Graph the function. 5. Find the equation of the line parallel to the line = + and containing the point 3, 5. Epress our answer in slope intercept form and graph the line. 6. Graph the equation = Does the relation 53, 6,, 3,, 5, 3, 86 represent a function? Wh or wh not? 8. Solve the equation = Solve the inequalit and graph the solution set. 0. Find the center and radius of the circle = 0. Graph the circle.. For the equation = 3-9, determine the intercepts and test for smmetr.. Find an equation of the line perpendicular to 3 - = 7 that contains the point, Is the following graph the graph of a function? Wh or wh not? 4. For the function f = + 5 -, find (a) f3 (b) f- (c) -f (d) f3 f + h - f (e), h Z 0 h 5. Answer the following questions regarding the function f = (a) What is the domain of f? (b) Is the point, 6 on the graph of f? (c) If = 3, what is f? What point is on the graph of f? (d) If f = 9, what is? What point is on the graph of f? 6. Graph the function f = Graph f = b determining whether its graph opens up or down and b finding its verte, ais of smmetr, -intercept, and -intercepts, if an. 8. Find the average rate of change of f = from to. Use this result to find the slope of the secant line containing, f and, f. 9. In parts (a) to (f) use the following graph. ( 3, 5) (0, 3) 7 (, 6) (a) Determine the intercepts. (b) Based on the graph, tell whether the graph is smmetric with respect to the -ais, the -ais, and/or the origin. (c) Based on the graph, tell whether the function is even, odd, or neither. (d) List the intervals on which f is increasing. List the intervals on which f is decreasing. (e) List the numbers, if an, at which f has a local maimum. What are these local maima? (f) List the numbers, if an, at which f has a local minimum. What are these local minima? 0. Determine algebraicall whether the function 5 f = is even, odd, or neither if For the function f = b if Ú (a) Find the domain of f. (b) Locate an intercepts. (c) Graph the function. (d) Based on the graph, find the range.. Graph the function f = using transformations. 3. Suppose that f = and g = (a) Find f + g and state its domain. f (b) Find and state its domain. g 4. Demand Equation The price p (in dollars) and the quantit sold of a certain product obe the demand equation p = , (a) Epress the revenue R as a function of. (b) What is the revenue if 00 units are sold? (c) What quantit maimizes revenue? What is the maimum revenue? (d) What price should the compan charge to maimize revenue?