Summar, Review, and Test 79 56. Galileo s telescope brought about revolutionar changes in astronom. A comparable leap in our abilit to observe the universe took place as a result of the Hubble Space Telescope. The space telescope was able to see stars and galaies whose brightness is of the faintest objects observable using ground-based telescopes. Use the fact that the brightness of a point source, such as a star, varies inversel as the square of its distance from an observer to show that the space telescope was able to see about seven times farther than a ground-based telescope. Group Eercise 50 57. Begin b deciding on a product that interests the group because ou are now in charge of advertising this product. Members were told that the demand for the product varies directl as the amount spent on advertising and inversel as the price of the product. However, as more mone is spent on advertising, the price of our product rises. Under what conditions would members recommend an increased epense in advertising? Once ou ve determined what our product is, write formulas for the given conditions and eperiment with hpothetical numbers. What other factors might ou take into consideration in terms of our recommendation? How do these factor affect the demand for our product? Preview Eercises Eercises 58 60 will help ou prepare for the material covered in the first section of the net chapter. 58. Use point plotting to graph f =. Begin b setting up a partial table of coordinates, selecting integers from - to, inclusive, for. Because = 0 is a horizontal asmptote, our graph should approach, but never touch, the negative portion of the -ais. In Eercises 59 60, use transformations of our graph from Eercise 58 to graph each function. 59. g = f- = - 60. h = f + = + Chapter Summar Summar, Review, and Test DEFINITIONS AND CONCEPTS. Comple Numbers EXAMPLES a. The imaginar unit i is defined as Figure., i = -, where i p. 78 = -. The set of numbers in the form a + bi is called the set of comple numbers; a is the real part and b is the imaginar part. If b = 0, the comple number is a real number. If b Z 0, the comple number is an imaginar number. Comple numbers in the form bi are called pure imaginar numbers. b. Rules for adding and subtracting comple numbers are given in the bo on page 79. E., p. 79 c. To multipl comple numbers, multipl as if the are polnomials. After completing the multiplication, E., p. 80 replace i with - and simplif. d. The comple conjugate of a + bi is a - bi and vice versa. The multiplication of comple conjugates gives a real number: a + bia - bi = a + b. e. To divide comple numbers, multipl the numerator and the denominator b the comple conjugate of the denominator. f. When performing operations with square roots of negative numbers, begin b epressing all square roots in terms of i. The principal square root of -b is defined b -b = ib. E., p. 8 E. 4, p. 8 g. Quadratic equations a + b + c = 0, a Z 0 with negative discriminants b - 4ac 6 0 have imaginar E. 5, p. 8 solutions that are comple conjugates.. Quadratic Functions a. A quadratic function is of the form f = a + b + c, a Z 0. b. The standard form of a quadratic function is f = a - h + k, a Z 0.
80 Chapter Polnomial and Rational Functions DEFINITIONS AND CONCEPTS EXAMPLES c. The graph of a quadratic function is a parabola. The verte is h, k b or a - A procedure a, fa - b a bb. E., p. 88; E., p. 89; for graphing a quadratic function in standard form is given in the bo on page 88.A procedure for graphing E., p. 9 a quadratic function in the form f = a + b + c is given in the bo on page 90. d. See the bo on page 9 for minimum or maimum values of quadratic functions. E. 4, p. 9; E. 5, p. 9 e. A strateg for solving problems involving maimizing or minimizing quadratic functions is given in the bo on page 95.. Polnomial Functions and Their Graphs a. Polnomial Function of Degree n: f = a n n + a n - n - + Á + a + a + a 0, a n Z 0 g. If f is a polnomial of degree n, the graph of f has at most n - turning points. Figure.4, p. 0 h. A strateg for graphing a polnomial function is given in the bo on page 0. E. 9, p. 0.4 Dividing Polnomials; Remainder and Factor Theorems a. Long division of polnomials is performed b dividing, multipling, subtracting, bringing down the net term, and repeating this process until the degree of the remainder is less than the degree of the divisor. The details are given in the bo on page 7. b. The Division Algorithm: f = dq + r. The dividend is the product of the divisor and the quotient plus the remainder. E., p. 6; E., p. 8; E., p. 9 c. Snthetic division is used to divide a polnomial b - c. The details are given in the bo on page 0. E. 4, p. d. The Remainder Theorem: If a polnomial f is divided b - c, then the remainder is fc. E. 5, p. e. The Factor Theorem: If - c is a factor of a polnomial function f, then c is a zero of f and a root of E. 6, p. f = 0. If c is a zero of f or a root of f = 0, then - c is a factor of f..5 Zeros of Polnomial Functions a. The Rational Zero Theorem states that the possible rational zeros of a polnomial Factors of the constant term function = The theorem is stated in the bo on page 7. Factors of the leading coefficient. b. Number of roots: If f is a polnomial of degree n Ú, then, counting multiple roots separatel, the equation f = 0 has n roots. E. 6, p. 96; E. 7, p. 97 b. The graphs of polnomial functions are smooth and continuous. Figure., p. 0 c. The end behavior of the graph of a polnomial function depends on the leading term, given b the Leading Coefficient Test in the bo on page 04. E., p. 04; E., p. 05; E., p. 05; E. 4, p. 06 d. The values of for which f is equal to 0 are the zeros of the polnomial function f. These values are the E. 5, p. 06; roots, or solutions, of the polnomial equation f = 0. E. 6, p. 07 e. If - r occurs k times in a polnomial function s factorization, r is a repeated zero with multiplicit k. If k E. 7, p. 08 is even, the graph touches the -ais and turns around at r. If k is odd, the graph crosses the -ais at r. f. The Intermediate Value Theorem: If f is a polnomial function and fa and fb have opposite signs, there E. 8, p. 09 is at least one value of c between a and b for which fc = 0. E., p. 7; E., p. 8; E., p. 8; E. 4, p. 9; E. 5, p. 0 c. If a + bi is a root of f = 0 with real coefficients, then a - bi is also a root. d. The Linear Factorization Theorem: An nth-degree polnomial can be epressed as the product of linear factors. Thus, f = a n - c - c Á n E. 6, p. - c n. e. Descartes s Rule of Signs: The number of positive real zeros of f equals the number of sign changes of f Table., or is less than that number b an even integer. The number of negative real zeros of f applies a similar p. 4; statement to f-. E. 7, p. 4
Summar, Review, and Test 8 DEFINITIONS AND CONCEPTS EXAMPLES.6 Rational Functions and Their Graphs a. Rational function: f = p ; p and q are polnomial functions and q Z 0. The domain of f is the E., p. 40 q set of all real numbers ecluding values of that make q zero. b. Arrow notation is summarized in the bo on page 4. c. The line = a is a vertical asmptote of the graph of f if f increases or decreases without bound as E., p. 4 approaches a. Vertical asmptotes are located using the theorem in the bo on page 4. d. The line = b is a horizontal asmptote of the graph of f if f approaches b as increases or decreases E., p. 46 without bound. Horizontal asmptotes are located using the theorem in the bo on page 45. e. Table. on page 46 shows the graphs of f = and f = Some rational functions can be graphed. using transformations of these common graphs. E. 4, p. 47 f. A strateg for graphing rational functions is given in the bo on page 47. E. 5, p. 48; E. 6, p. 49; E. 7, p. 50 g. The graph of a rational function has a slant asmptote when the degree of the numerator is one more than E. 8, p. 5 the degree of the denominator. The equation of the slant asmptote is found using division and dropping the remainder term..7 Polnomial and Rational Inequalities a. A polnomial inequalit can be epressed as f 6 0, f 7 0, f 0, or f Ú 0, where f is a E., p. 60; polnomial function. A procedure for solving polnomial inequalities is given in the bo on page 60. E., p. 6 b. A rational inequalit can be epressed as f 6 0, f 7 0, f 0, or f Ú 0, where f is a rational E., p. 6 function. The procedure for solving such inequalities begins with epressing them so that one side is zero and the other side is a single quotient. Find boundar points b setting the numerator and denominator equal to zero. Then follow a procedure similar to that for solving polnomial inequalities..8 Modeling Using Variation a. A procedure for solving variation problems is given in the bo on page 70. b. English Statement Equation varies directl as. = k E., p. 70 is directl proportional to. varies directl as n. = k n E., p. 7 is directl proportional to n. varies inversel as. = k E., p. 7; is inversel proportional to. E. 4, p. 74 varies inversel as n. is inversel proportional to n. = k n varies jointl as and z. = kz E. 5, p. 76 Review Eercises. In Eercises 0 perform the indicated operations and write the result in standard form.. 8 - i - 7-7i. 4ii -. 7 - i + i 4. - 4i 5. 7 + 8i7-8i 6. 6 5 + i + 4i 7. 8. - - -8 4 - i 9. A - + -00B 0. 4 + -8 In Eercises, solve each quadratic equation using the quadratic formula. Epress solutions in standard form.. - + 4 = 0. - 6 + 5 = 0
8 Chapter Polnomial and Rational Functions. In Eercises 6, use the verte and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola s ais of smmetr. Use the graph to determine the function s domain and range.. f = - + + 4 4. f = + 4-5. f = - + + 6. f = - 4-6 In Eercises 7 8, use the function s equation, and not its graph, to find a. the minimum or maimum value and where it occurs. b. the function s domain and its range. 7. f = - + 4-06 8. f = + + 70 9. A quarterback tosses a football to a receiver 40 ards downfield. The height of the football, f, in feet, can be modeled b f = -0.05 + + 6, where is the ball s horizontal distance, in ards, from the quarterback. a. What is the ball s maimum height and how far from the quarterback does this occur? b. From what height did the quarterback toss the football? c. If the football is not blocked b a defensive plaer nor caught b the receiver, how far down the field will it go before hitting the ground? Round to the nearest tenth of a ard. d. Graph the function that models the football s parabolic path. 0. A field bordering a straight stream is to be enclosed. The side bordering the stream is not to be fenced. If 000 ards of fencing material is to be used, what are the dimensions of the largest rectangular field that can be fenced? What is the maimum area?. Among all pairs of numbers whose difference is 4, find a pair whose product is as small as possible. What is the minimum product?. You have 000 feet of fencing to construct si corrals, as shown in the figure. Find the dimensions that maimize the enclosed area. What is the maimum area?. The annual ield per fruit tree is fairl constant at 50 pounds per tree when the number of trees per acre is 5 or fewer. For each additional tree over 5, the annual ield per tree for all trees on the acre decreases b 4 pounds due to overcrowding. How man fruit trees should be planted per acre to maimize the annual ield for the acre? What is the maimum number of pounds of fruit per acre?. In Eercises 4 7, use the Leading Coefficient Test to determine the end behavior of the graph of the given polnomial function. Then use this end behavior to match the polnomial function with its graph. [The graphs are labeled (a) through (d).] 4. f = - + + 5. f = 6-6 4 + 9 6. f = 5-5 + 4 7. f = - 4 + a. b. c. d. 8. The polnomial function f = -0.87 + 0.5 + 8.6 + 7684.94 models the number of thefts, f, in thousands, in the United States ears after 987. Will this function be useful in modeling the number of thefts over an etended period of time? Eplain our answer. 9. A herd of 00 elk is introduced to a small island.the number of elk, f, after ears is modeled b the polnomial function f = - 4 + + 00. Use the Leading Coefficient Test to determine the graph s end behavior to the right. What does this mean about what will eventuall happen to the elk population? In Eercises 0, find the zeros for each polnomial function and give the multiplicit of each zero. State whether the graph crosses the -ais, or touches the -ais and turns around, at each zero. 0. f = - - + + 5. f = - 5-5 + 5. Show that f = - - has a real zero between and. In Eercises 8, a. Use the Leading Coefficient Test to determine the graph s end behavior. b. Determine whether the graph has -ais smmetr, origin smmetr, or neither. c. Graph the function.. f = - - 9 + 9 4. f = 4-5. f = + - 8-6. f = - + 5 7. f = - 4 + 6-9 8. f = 4-5 In Eercises 9 40, graph each polnomial function. 9. f = - + 40. f = - + 4 -
Summar, Review, and Test 8.4 In Eercises 4 4, divide using long division. 4. 4 - - +, + 4. 0-6 + 7 -, 5-4. 4 4 + 6 + -, + In Eercises 65 68, graphs of fifth-degree polnomial functions are shown. In each case, specif the number of real zeros and the number of imaginar zeros. Indicate whether there are an real zeros with multiplicit other than. 65. 66. In Eercises 44 45, divide using snthetic division. 44. 45. 4 + - 0 + 7 + 5, + 5 4 - - 0, - 46. Given f = - 7 + 9 -, use the Remainder Theorem to find f-. 47. Use snthetic division to divide f = + - + 6 b -. Use the result to find all zeros of f. 48. Solve the equation - 7 + 4 = 0 given that 4 is a root. 67. 68..5 In Eercises 49 50, use the Rational Zero Theorem to list all possible rational zeros for each given function. 49. f = 4-6 + 4-4 + 5 50. f = 5-4 - 5 + 0 + - 8 In Eercises 5 5, use Descartes s Rule of Signs to determine the possible number of positive and negative real zeros for each given function. 5. 5. f = 4 - - 8 + 5 f = 5 - - 5 + - 5. Use Descartes s Rule of Signs to eplain wh 4 + 6 + 8 = 0 has no real roots. For Eercises 54 60, a. List all possible rational roots or rational zeros. b. Use Descartes s Rule of Signs to determine the possible number of positive and negative real roots or real zeros. c. Use snthetic division to test the possible rational roots or zeros and find an actual root or zero. d. Use the quotient from part (c) to find all the remaining roots or zeros. 54. f = + - 4 55. f = 6 + - 4 + 56. 8-6 + 46-5 = 0 57. + 9-7 + = 0 58. 4 - - 7 + + 6 = 0 59. 4 4 + 7 - = 0 60. f = 4 + - 9-4 + 4 In Eercises 6 6, find an nth-degree polnomial function with real coefficients satisfing the given conditions. If ou are using a graphing utilit, graph the function and verif the real zeros and the given function value. 6. n = ; and - i are zeros; f = -0 6. n = 4; i is a zero; - is a zero of multiplicit ; f- = 6 In Eercises 6 64, find all the zeros of each polnomial function and write the polnomial as a product of linear factors. 6. f = 4 + + - 64. g = 4-6 + + 4 + 6.6 In Eercises 69 70, use transformations of f = or f = to graph each rational function. 69. g = 70. h = + - - + In Eercises 7 78, find the vertical asmptotes, if an, the horizontal asmptote, if one eists, and the slant asmptote, if there is one, of the graph of each rational function. Then graph the rational function. 7. f = 7. g = - 4-9 + 7. 74. r = + 4 + h = - - 4 - - 6 + 75. 76. = + - = + - 77. 78. g = 4-6 + 6 f = - + - 79. A compan is planning to manufacture affordable graphing calculators. The fied monthl cost will be $50,000 and it will cost $5 to produce each calculator. a. Write the cost function, C, of producing graphing calculators. b. Write the average cost function, C, of producing graphing calculators. c. Find and interpret C50, C00, C000, and C00,000. d. What is the horizontal asmptote for the graph of this function and what does it represent? Eercises 80 8 involve rational functions that model the given situations. In each case, find the horizontal asmptote as : q and then describe what this means in practical terms. 80. 50 + 0 f = 0.05 + ; the number of bass, f, after months in a lake that was stocked with 0 bass 8. P = 7,900 the percentage, P, of people in the 00 + 79 ; United States with ears of education who are unemploed
84 Chapter Polnomial and Rational Functions 8. The bar graph shows the population of the United States, in millions, for five selected ears. 9. The graph shows stopping distances for motorccles at various speeds on dr roads and on wet roads. Population (millions) Population of the United States Male Female 60 50 40 0 7 9 4 8 4 47 5 0 6 0 00 985 990 995 000 005 Year Source: U.S. Census Bureau Stopping Distance (feet) Stopping Distances for Motorccles at Selected Speeds Dr Pavement Wet Pavement 800 700 705 600 50 575 500 45 85 400 5 00 60 5 00 00 5 45 55 65 Speed (miles per hour) Source: National Highwa Traffic Safet Administration The functions Here are two functions that model the data: M()=.58+4.4 Male U.S. population, M(), in millions, ears after 985 and f()=0.5-0.8+99 Dr pavement Wet pavement g()=0.5 +.+7 a. Write a function that models the total U.S. population, P, in millions, ears after 985. b. Write a rational function that models the fraction of men in the U.S. population, R, ears after 985. c. What is the equation of the horizontal asmptote associated with the function in part (b)? Round to two decimal places. What does this mean about the percentage of men in the U.S. population over time? 8. A jogger ran 4 miles and then walked miles. The average velocit running was miles per hour faster than the average velocit walking. Epress the total time for running and walking, T, as a function of the average velocit walking,. 84. The area of a rectangular floor is 000 square feet. Epress the perimeter of the floor, P, as a function of the width of the rectangle,..7 F()=.48+0.6. In Eercises 85 90, solve each inequalit and graph the solution set on a real number line. 85. + 5-6 0 86. + 9 + 4 Ú 0 87. + 7 88. + - 89. - Ú 0 90. Female U.S. population, F(), in millions, ears after 985-6 + 7 0 + - 4 5 model a motorccle s stopping distance, f or g, in feet, traveling at miles per hour. Function f models stopping distance on dr pavement and function g models stopping distance on wet pavement. a. Use function g to find the stopping distance on wet pavement for a motorccle traveling at 5 miles per hour. Round to the nearest foot. Does our rounded answer overestimate or underestimate the stopping distance shown b the graph? B how man feet? b. Use function f to determine speeds on dr pavement requiring stopping distances that eceed 67 feet. 9. Use the position function st = -6t + v 0 t + s 0.8 to solve this problem. A projectile is fired verticall upward from ground level with an initial velocit of 48 feet per second. During which time period will the projectile s height eceed feet? Solve the variation problems in Eercises 9 98. 9. Man areas of Northern California depend on the snowpack of the Sierra Nevada mountain range for their water suppl. The volume of water produced from melting snow varies directl as the volume of snow. Meteorologists have determined that 50 cubic centimeters of snow will melt to 8 cubic centimeters of water. How much water does 00 cubic centimeters of melting snow produce? 94. The distance that a bod falls from rest is directl proportional to the square of the time of the fall. If skdivers fall 44 feet in seconds, how far will the fall in 0 seconds?
Summar, Review, and Test 85 95. The pitch of a musical tone varies inversel as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of.6 feet. What is the pitch of a tone that has a wavelength of.4 feet? 96. The loudness of a stereo speaker, measured in decibels, varies inversel as the square of our distance from the speaker. When ou are 8 feet from the speaker, the loudness is 8 decibels. What is the loudness when ou are 4 feet from the speaker? 97. The time required to assemble computers varies directl as the number of computers assembled and inversel as the number of workers. If 0 computers can be assembled b 6 workers in 0 hours, how long would it take 5 workers to assemble 40 computers? 98. The volume of a pramid varies jointl as its height and the area of its base.a pramid with a height of 5 feet and a base with an area of 5 square feet has a volume of 75 cubic feet. Find the volume of a pramid with a height of 0 feet and a base with an area of 0 square feet. 99. Heart rates and life spans of most mammals can be modeled using inverse variation. The bar graph shows the average heart rate and the average life span of five mammals. Average Heart Rate (beats per minute) 00 75 50 5 00 75 50 5 Heart Rate and Life Span Average Heart Rate 90 0 58 Squirrels Rabbits 6 5 Cats Animal Average Life Span 76 5 Lions Source: The Hand Science Answer Book, Visible Ink Press, 00 6 0 Horses a. A mammal s average life span, L, in ears, varies inversel as its average heart rate, R, in beats per minute. Use the data shown for horses to write the equation that models this relationship. b. Is the inverse variation equation in part (a) an eact model or an approimate model for the data shown for lions? c. Elephants have an average heart rate of 7 beats per minute. Determine their average life span. 40 5 0 5 0 5 0 5 Average Life Span (ears) Chapter Test In Eercises, perform the indicated operations and write the result in standard form. 5. 6-7i + 5i. - i. -49 + -64 4. Solve and epress solutions in standard form: = 4-8. In Eercises 5 6, use the verte and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola s ais of smmetr. Use the graph to determine the function s domain and range. 5. f = + + 4 6. f = - - 7. Determine, without graphing, whether the quadratic function f = - + - 6 has a minimum value or a maimum value. Then find a. the minimum or maimum value and where it occurs. b. the function s domain and its range. 8. The function f = - + 46-60 models the dail profit, f, in hundreds of dollars, for a compan that manufactures computers dail. How man computers should be manufactured each da to maimize profit? What is the maimum dail profit? 9. Among all pairs of numbers whose sum is 4, find a pair whose product is as large as possible. What is the maimum product? 0. Consider the function f = - 5-4 + 0. a. Use factoring to find all zeros of f. b. Use the Leading Coefficient Test and the zeros of f to graph the function.. Use end behavior to eplain wh the graph at the top of the net column cannot be the graph of f = 5 -. Then use intercepts to eplain wh the graph cannot represent f = 5 -.. The graph of f = 6-9 + 6-4 is shown in the figure. f() = 6 9 + 6 4 4 4 4 4 a. Based on the graph of f, find the root of the equation 6-9 + 6-4 = 0 that is an integer. b. Use snthetic division to find the other two roots of 6-9 + 6-4 = 0. 4
86 Chapter Polnomial and Rational Functions. Use the Rational Zero Theorem to list all possible rational zeros of f = + - 7-6. 4. Use Descartes s Rule of Signs to determine the possible number of positive and negative real zeros of f = 5-4 - + -. 5. Solve: + 9 + 6-6 = 0. 6. Consider the function whose equation is given b f = 4 - - + 5 + 5. a. List all possible rational zeros. b. Use the graph of f in the figure shown and snthetic division to find all zeros of the function. f() = 4 + 5 + 5 4 5 0 5 0 7. Use the graph of f = + - 4 in the figure shown to factor + - 4. 4 4 5 0 5 0 5 4 4 f() = + 4 8. Find a fourth-degree polnomial function f with real coefficients that has -,, and i as zeros and such that f = 60. 9. The figure shows an incomplete graph of f = - - 4 + +. Find all the zeros of the function. Then draw a complete graph. 5 4 5 4 4 5 4 5 In Eercises 0 5, find the domain of each rational function and graph the function. 0. f =. f = + - +.. f = - 9 f = - 6-4. f = + 4 5. f = + - + 6. A compan is planning to manufacture portable satellite radio plaers. The fied monthl cost will be $00,000 and it will cost $0 to produce each plaer. a. Write the average cost function, C, of producing plaers. b. What is the horizontal asmptote for the graph of this function and what does it represent? Solve each inequalit in Eercises 7 8 and graph the solution set on a real number line. Epress each solution set in interval notation. + 7. 6 + 8. - 9. The intensit of light received at a source varies inversel as the square of the distance from the source. A particular light has an intensit of 0 foot-candles at 5 feet. What is the light s intensit at 0 feet? Cumulative Review Eercises (Chapters P ) Use the graph of = f to solve Eercises 6. 4 = f(). Find the domain and the range of f.. Find the zeros and the least possible multiplicit of each zero.. Where does the relative maimum occur? 4. Find f f-. 5. Use arrow notation to complete this statement: f : q as or as. 6. Graph g = f + +. In Eercises 7, solve each equation or inequalit. 7. ƒ - ƒ = 8. - 5 + = 0 ƒ ƒ 9. 9 + = 0. + - 5-6 = 0. - 5 7. 7 + 5 ƒ ƒ ƒ ƒ In Eercises 8, graph each equation in a rectangular coordinate sstem. If two functions are given, graph both in the same sstem.. f = - 4 - + 4 4. f = + - 8 5. f = - 6. f = - - 7. f = and g = - - 8. + - + 4-4 = 0 In Eercises 9 0, let f = - - and g = 4 -. f + h - f 9. Find f g. 0. Find. h