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**Michael Sullivan, Precalculus 9th Edition, Pearson/Prentice Hall**

**Section 1.1 **

43. If the point (2, 5) is shifted 3 units to the right and 2 units down, what are its new coordinates?

44. If the point (-1, 6) is shifted 2 units to the left and 4 units up, what are its new coordinates?

45. Find all points having an x-coordinate of 3 whose distance from the point (-2,-1) is 13. (a) By using the Pythagorean Theorem. (b) By using the distance formula.

46. Find all points having a y-coordinate of -6 whose distance from the point (1, 2) is 17. (a) By using the Pythagorean Theorem. (b) By using the distance formula.

47. Find all points on the x-axis that are 6 units from the point (4, -3).

48. Find all points on the y-axis that are 5 units from the point (4, 4).

49. The midpoint of the line segment from P1 to P2 is (-1, 4). If P1 = (-3, 6), what is P2?

50. The midpoint of the line segment from P1 to P2 is (5, -4). If P2 = (7, -2), what is P1?

51. Geometry. The medians of a triangle are the line segments from each vertex to the midpoint of the opposite side (see the figure). Find the lengths of the medians of the triangle with vertices at A = (0, 0), B = (6, 0), and C = (4, 4).

52. Geometry. An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are (0, 4) and (0, 0) find the third vertex. How many of these triangles are possible?

53. Geometry. Find the midpoint of each diagonal of a square with side of length s. Draw the conclusion that the diagonals of a square intersect at their midpoints.

54. Geometry. Verify that the points (0, 0), (a, 0), and (a/2, √3a/2) are the vertices of an equilateral triangle. Then show that the midpoints of the three sides are the vertices of a second equilateral triangle.

In Problems 55–58, find the length of each side of the triangle determined by the three points P1, P2, and P3. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length.)

55. P1 = (2, 1); P2 = (-4, 1); P3 = (-4, -3)

59. Baseball. A major league baseball “diamond” is actually a square, 90 feet on a side (see the figure). What is the distance directly from home plate to second base (the diagonal of the square)?

60. Little League Baseball. The layout of a Little League playing field is a square, 60 feet on a side. How far is it directly from home plate to second base (the diagonal of the square)?

61. Baseball. Refer to Problem 59. Overlay a rectangular coordinate system on a major league baseball diamond so that the origin is at home plate, the positive x-axis lies in the direction from home plate to first base, and the positive y-axis lies in the direction from home plate to third base. (a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement. (b) If the right fielder is located at (310, 15) how far is it from the right fielder to second base? (c) If the center fielder is located at (300, 300) how far is it from the center fielder to third base?

62. Little League Baseball. Refer to Problem 60. Overlay a rectangular coordinate system on a Little League baseball diamond so that the origin is at home plate, the positive x-axis lies in the direction from home plate to first base, and the positive y-axis lies in the direction from home plate to third base. (a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement. (b) If the right fielder is located at (180, 20) how far is it from the right fielder to second base? (c) If the center fielder is located at (220, 220), how far is it from the center fielder to third base?

63. Distance between Moving Objects. A Dodge Neon and a Mack truck leave an intersection at the same time. The Neon heads east at an average speed of 30 miles per hour, while the truck heads south at an average speed of 40 miles per hour. Find an expression for their distance apart d (in miles) at the end of t hours.

64. Distance of a Moving Object from a Fixed Point. A hot-air balloon, headed due east at an average speed of 15 miles per hour and at a constant altitude of 100 feet, passes over an intersection (see the figure). Find an expression for the distance d (measured in feet) from the balloon to the intersection t seconds later.

65. Drafting Error. When a draftsman draws three lines that are to intersect at one point, the lines may not intersect as intended and subsequently will form an error triangle. If this error triangle is long and thin, one estimate for the location of the desired point is the midpoint of the shortest side. The figure shows one such error triangle. (a) Find an estimate for the desired intersection point. (b) Find the length of the median for the midpoint found in part (a). See Problem 51.

66. Net Sales. The figure illustrates how net sales of Wal-Mart Stores, Inc., have grown from 2002 through 2008. Use the midpoint formula to estimate the net sales of Wal-Mart Stores, Inc., in 2005. How does your result compare to the reported value of $282 billion?

67. Poverty Threshold. Poverty thresholds are determined by the U.S. Census Bureau. A poverty threshold represents the minimum annual household income for a family not to be considered poor. In 1998, the poverty threshold for a family of four with two children under the age of 18 years was

$16530. In 2008, the poverty threshold for a family of four with two children under the age of 18 years was $21834. Assuming poverty thresholds increase in a straight-line fashion, use the midpoint formula to estimate the poverty threshold of a family of four with two children under the age of 18 in 2003. How does your result compare to the actual poverty threshold in 2003 of $18660?

Section 1.2

In Problems 11–16, determine which of the given points are on the graph of the equation.

11. Equation: y = x4 - √x; Points: (0, 0); (1, 1); (-1, 0)

In Problems 17–28, find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

17. y = x + 2

In Problems 29–38, plot each point. Then plot the point that is symmetric to it with respect to (a) the x-axis; (b) the y-axis; (c) the origin.

29. (3, 4)

In Problems 39–50, the graph of an equation is given. (a) Find the intercepts. (b) Indicate whether the graph is symmetric with respect to the x-axis, the y-axis, or the origin.

In Problems 51–54, draw a complete graph so that it has the type of symmetry indicated.

In Problems 55–70, list the intercepts and test for symmetry.

55. y2 = x + 4

79. Given that the point (1, 2) is on the graph of an equation that is symmetric with respect to the origin, what other point is on the graph?

80. If the graph of an equation is symmetric with respect to the y-axis and 6 is an x-intercept of this graph, name another x-intercept.

81. If the graph of an equation is symmetric with respect to the origin and -4 is an x-intercept of this graph, name another x-intercept.

82. If the graph of an equation is symmetric with respect to the x-axis and 2 is a y-intercept, name another y-intercept.

83. Microphones. In studios and on stages, cardioid microphones are often preferred for the richness they add to voices and for their ability to reduce the level of sound from the sides and rear of the microphone. Suppose one such cardioid pattern is given by the equation (x2 + y2 – x)2 = x2 + y2 (a) Find the intercepts of the graph of the equation. (b) Test for symmetry with respect to the x-axis, y-axis, and origin.

84. Solar Energy. The solar electric generating systems at Kramer Junction, California, use parabolic troughs to heat a heat-transfer fluid to a high temperature. This fluid is used to generate steam that drives a power conversion system to produce electricity. For troughs 7.5 feet wide, an equation for the cross-section is 16y2 = 120x - 225. (a) Find the intercepts of the graph of the equation. (b) Test for symmetry with respect to the x-axis, y-axis, and origin.

Section 1.3

In Problems 11–14, (a) find the slope of the line and (b) interpret the slope.

In Problems 15–22, plot each pair of points and determine the slope of the line containing them. Graph the line.

15. (2, 3); (4, 0)

In Problems 23–30, graph the line containing the point P and having slope m.

23. P = (1, 2); m = 3

29. P = (0, 3); slope undefined

In Problems 31–36, the slope and a point on a line are given. Use this information to locate three additional points on the line. Answers may vary.

31. Slope 4; point (1, 2)

In Problems 37–44, find an equation of the line L.

In Problems 45–70, find an equation for the line with the given properties. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer.

45. Slope = 3; containing the point (-2, 3)

57. Horizontal; containing the point (-3, 2)

61. Parallel to the line 2x – y = -2; containing the point (0, 0)

In Problems 71–90, find the slope and y-intercept of each line. Graph the line.

71. y = 2x + 3

In Problems 91–100, (a) find the intercepts of the graph of each equation and (b) graph the equation.

91. 2x + 3y = 6

In Problems 103–106, the equations of two lines are given. Determine if the lines are parallel, perpendicular, or neither.

103. y = 2x – 3, y = 2x + 4

In Problems 107–110, write an equation of each line. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer.

111. Geometry. Use slopes to show that the triangle whose vertices are (-2, 5), (1, 3), and (-1, 0) is a right triangle.

112. Geometry. Use slopes to show that the quadrilateral whose vertices are (1, -1), (4, 1), (2, 2), and (5, 4) is a parallelogram.

113. Geometry. Use slopes to show that the quadrilateral whose vertices are (-1, 0), (2, 3), (1, -2), and (4, 1) is a rectangle.

114. Geometry. Use slopes and the distance formula to show that the quadrilateral whose vertices are (0, 0), (1, 3), (4, 2), and (3, -1) is a square.

115. Truck Rentals. A truck rental company rents a moving truck for one day by charging $29 plus $0.20 per mile. Write a linear equation that relates the cost C, in dollars, of renting the truck to the number x of miles driven. What is the cost of renting the truck if the truck is driven 110 miles? 230 miles?

116. Cost Equation. The fixed costs of operating a business are the costs incurred regardless of the level of production. Fixed costs include rent, fixed salaries, and costs of leasing machinery. The variable costs of operating a business are the costs that change with the level of output. Variable costs include raw materials, hourly wages, and electricity. Suppose that a manufacturer of jeans has fixed daily costs of $500 and variable costs of $8 for each pair of jeans manufactured. Write a linear equation that relates the daily cost C, in dollars, of manufacturing the jeans to the number x of jeans manufactured. What is the cost of manufacturing 400 pairs of jeans? 740 pairs?

117. Cost of Driving a Car. The annual fixed costs for owning a small sedan are $1289, assuming the car is completely paid for. The cost to drive the car is approximately $0.15 per mile. Write a linear equation that relates the cost C and the number x of miles driven annually.

118. Wages of a Car Salesperson. Dan receives $375 per week for selling new and used cars at a car dealership in Oak Lawn, Illinois. In addition, he receives 5% of the profit on any sales that he generates. Write a linear equation that represents Dan’s weekly salary S when he has sales that generate a profit of x dollars.

119. Electricity Rates in Illinois. Commonwealth Edison Company supplies electricity to residential customers for a monthly customer charge of $10.55 plus 9.44 cents per kilowatt-hour for up to 600 kilowatt-hours. (a) Write a linear equation that relates the monthly charge C, in dollars, to the number x of kilowatt-hours used in a month, 0 ≤ x ≤ 600. (b) Graph this equation. (c) What is the monthly charge for using 200 kilowatt-hours? (d) What is the monthly charge for using 500 kilowatt-hours? (e) Interpret the slope of the line.

120. Electricity Rates in Florida. Florida Power & Light Company supplies electricity to residential customers for a monthly customer charge of $5.69 plus 8.48 cents per kilowatt-hour for up to 1000 kilowatt-hours. (a) Write a linear equation that relates the monthly charge C, in dollars, to the number x of kilowatt-hours used in a month, 0 ≤ x ≤ 1000. (b) Graph this equation. (c) What is the monthly charge for using 200 kilowatt-hours? (d) What is the monthly charge for using 500 kilowatt-hours?(e) Interpret the slope of the line.

121. Measuring Temperature. The relationship between Celsius (°C) and Fahrenheit (°F) degrees of measuring temperature is linear. Find a linear equation relating °C and °F if 0°C corresponds to 32 °F and 100 °C corresponds to 212°F. Use the equation to find the Celsius measure of 70°F.

122. The Kelvin (K) scale for measuring temperature is obtained by adding 273 to the Celsius temperature. (a) Write a linear equation relating K and °C. (b) Write a linear equation relating K and °F (see Problem 121).

123. Access Ramp. A wooden access ramp is being built to reach a platform that sits 30 inches above the floor. The ramp drops 2 inches for every 25-inch run. (a) Write a linear equation that relates the height y of the ramp above the floor to the horizontal distance x from the platform. (b) Find and interpret the x-intercept of the graph of your equation. (c) Design requirements stipulate that the maximum run be 30 feet and that the maximum slope be a drop of 1 inch for each 12 inches of run. Will this ramp meet the requirements? Explain. (d) What slopes could be used to obtain the 30-inch rise and still meet design requirements?

124. Cigarette Use. A report in the Child Trends DataBase indicated that, in 1996, 22.2% of twelfth grade students reported daily use of cigarettes. In 2006,12.2% of twelfth grade students reported daily use of cigarettes. (a) Write a linear equation that relates the percent y of twelfth grade students who smoke cigarettes daily to the number x of years after 1996. (b) Find the intercepts of the graph of your equation. (c) Do the intercepts have any meaningful interpretation? (d) Use your equation to predict the percent for the year 2016. Is this result reasonable?

126. Product Promotion. A cereal company finds that the number of people who will buy one of its products in the first month that it is introduced is linearly related to the amount of money it spends on advertising. If it spends $40,000 on advertising, then 100,000 boxes of cereal will be sold, and if it spends $60,000, then 200,000 boxes will be sold. (a) Write a linear equation that relates the amount A spent on advertising to the number x of boxes the company aims to sell. (b) How much advertising is needed to sell 300,000 boxes of cereal? (c) Interpret the slope.

130. Which of the following equations might have the graph shown? (More than one answer is possible.)

131. The figure shows the graph of two parallel lines. Which of the following pairs of equations might have such a graph?

132. The figure shows the graph of two perpendicular lines. Which of the following pairs of equations might have such a graph?

133. m is for Slope. The accepted symbol used to denote the slope of a line is the letter m. Investigate the origin of this symbolism. Begin by consulting a French dictionary and looking up the French word monter. Write a brief essay on your findings.

134. Grade of a Road. The term grade is used to describe the inclination of a road. How does this term relate to the notion of slope of a line? Is a 4% grade very steep? Investigate the grades of some mountainous roads and determine their slopes. Write a brief essay on your findings.

135. Carpentry. Carpenters use the term pitch to describe the steepness of staircases and roofs. How does pitch relate to slope? Investigate typical pitches used for stairs and for roofs. Write a brief essay on your findings.

136. Can the equation of every line be written in slope–intercept form? Why?

137. Does every line have exactly one x-intercept and one y-intercept? Are there any lines that have no intercepts?

138. What can you say about two lines that have equal slopes and equal y-intercepts?

139. What can you say about two lines with the same x-intercept and the same y-intercept? Assume that the x-intercept is not 0.

140. If two distinct lines have the same slope, but different x-intercepts, can they have the same y-intercept?

141. If two distinct lines have the same y-intercept, but different slopes, can they have the same x-intercept?

Section 1.4

In Problems 7–10, find the center and radius of each circle. Write the standard form of the equation.

In Problems 11–20, write the standard form of the equation and the general form of the equation of each circle of radius r and center (h, k). Graph each circle.

1. r = 2; (h, k) =(0, 0)

In Problems 21–34, (a) find the center (h, k) and radius r of each circle; (b) graph each circle; (c) find the intercepts, if any.

21. x2 + y2 = 4

In Problems 35–42, find the standard form of the equation of each circle.

35. Center at the origin and containing the point (-2, 3)

36. Center (1, 0) and containing the point (-3, 2)

37. Center (2, 3) and tangent to the x-axis

38. With endpoints of a diameter at (1, 4) and (-3, 2)

47. Find the area of the square in the figure.

48. Find the area of the blue shaded region in the figure, assuming the quadrilateral inside the circle is a square.

49. Ferris Wheel. The original Ferris wheel was built in 1893 by Pittsburgh, Pennsylvania, bridge builder George W. Ferris. The Ferris wheel was originally built for the 1893 World’s Fair in Chicago, but was also later reconstructed for the 1904 World’s Fair in St. Louis. It had a maximum height of 64 feet and a wheel diameter of 250 feet. Find an equation for the wheel if the center of the wheel is on the y-axis.

50. Ferris Wheel. In 2008, the Singapore Flyer opened as the world’s largest Ferris wheel. It has a maximum height of 165 meters and a diameter of 150 meters, with one full rotation taking approximately 30 minutes. Find an equation for the wheel if the center of the wheel is on the y-axis.

51. Weather Satellites. Earth is represented on a map of a portion of the solar system so that its surface is the circle with equation x2 + y2 + 2x +4y – 4091 = 0. A weather satellite circles 0.6 unit above Earth with the center of its circular orbit at the center of Earth. Find the equation for the orbit of the satellite on this map.

53. The Greek Method. The Greek method for finding the equation of the tangent line to a circle uses the fact that at any point on a circle the lines containing the center and the tangent line are perpendicular (see Problem 52). Use this method to find an equation of the tangent line to the circle x2 + y2 = 9 at the point (1, 2√2)

54. Use the Greek method described in Problem 53 to find an equation of the tangent line to the circle x2 + y2 – 4x + 6y + 4 = 0 at the point (3, 2√2 – 3)

55. Refer to Problem 52. The line x – 2y + 4 = 0 is tangent to a circle at (0, 2). The line y = 2x - 7 is tangent to the same circle at (3, 1). Find the center of the circle.

56. Find an equation of the line containing the centers of the two circles x2 + y2 – 4x + 6y + 4 = 0 and x2 + y2 + 6x + 4y + 9 = 0.

57. If a circle of radius 2 is made to roll along the x-axis, what is an equation for the path of the center of the circle?

58. If the circumference of a circle is 6π, what is its radius?

59. Which of the following equations might have the graph shown? (More than one answer is possible.)

**Section 2.1**

In Problems 15–26, determine whether each relation represents a function. For each function, state the domain and range.

In Problems 27–38, determine whether the equation defines y as a function of x.

27. y = x2

89. Geometry. Express the area A of a rectangle as a function of the length x if the length of the rectangle is twice its width.

90. Geometry. Express the area A of an isosceles right triangle as a function of the length x of one of the two equal sides.

91. Constructing Functions. Express the gross salary G of a person who earns $10 per hour as a function of the number x of hours worked.

92. Constructing Functions. Tiffany, a commissioned salesperson, earns $100 base pay plus $10 per item sold. Express her gross salary G as a function of the number x of items sold.

95. Effect of Gravity on Earth. If a rock falls from a height of 20 meters on Earth, the height H (in meters) after x seconds is approximately H(x) = 20 – 4.9x2 (a) What is the height of the rock when x = 1 second? x = 1.1 seconds? (b) When is the height of the rock 15 meters? When is it 10 meters? When is it 5 meters? (c) When does the rock strike the ground?

96. Effect of Gravity on Jupiter. If a rock falls from a height of 20 meters on the planet Jupiter, its height H (in meters) after x seconds is approximately H(x) = 20 – 13x2 (a) What is the height of the rock when x = 1 second? x = 1.1 seconds? (b) When is the height of the rock 15 meters? When is it10 meters? When is it 5 meters? (c) When does the rock strike the ground?

97. Cost of Trans-Atlantic Travel. A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by C(x) = 100 + x/10 + 36000/x where x is the ground speed (airspeed ± wind). (a) What is the cost per passenger for quiescent (no wind) conditions? (b) What is the cost per passenger with a head wind of 50 miles per hour? (c) What is the cost per passenger with a tail wind of 100 miles per hour? (d) What is the cost per passenger with a head wind of100 miles per hour?

98. Cross-sectional Area. The cross-sectional area of a beam cut from a log with radius 1 foot is given by the function A(x) = 4x√(1 – x2), where x represents the length, in feet, of half the base of the beam. See the figure. Determine the cross-sectional area of the beam if the length of half the base of the beam is as follows: (a) One-third of a foot (b) One-half of a foot (c) Two-thirds of a foot.

99. Economics. The participation rate is the number of people in the labor force divided by the civilian population (excludes military). Let L(x) represent the size of the labor force in year x and P(x) represent the civilian population in year x. Determine a function that represents the participation rate R as a function of x.

100. Crimes. Suppose that V(x) represents the number of violent crimes committed in year x and P(x) represents the number of property crimes committed in year x. Determine a function T that represents the combined total of violent crimes and property crimes in year x.

101. Health Care. Suppose that P(x) represents the percentage of income spent on health care in year x and I(x) represents income in year x. Determine a function H that represents total health care expenditures in year x.

102. Income Tax. Suppose that I(x) represents the income of an individual in year x before taxes and T(x) represents the individual’s tax bill in year x. Determine a function N that represents the individual’s net income (income after taxes) in year x.

103. Profit Function. Suppose that the revenue R, in dollars, from selling x cell phones, in hundreds, is R(x) = -1.2x2 + 220x. The cost C, in dollars, of selling x cell phones is C(x) = 0.05x3 – 2x2 + 65x + 50. (a) Find the profit function, P(x) = R(x) – C(x) (b) Find the profit if x = 15 hundred cell phones are sold. (c) Interpret P(15).

104. Profit Function. Suppose that the revenue R, in dollars, from selling x clocks is R(x) = 30x. The cost C, in dollars, of selling x clocks is C(x) = 0.1x2 + 7x + 400. (a) Find the profit function, P(x) = R(x) – C(x) (b) Find the profit if clocks are sold. (c) Interpret P(30).

Section 2.2

In Problems 11–22, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

29. Free-throw Shots. According to physicist Peter Brancazio, the key to a successful foul shot in basketball lies in the arc of the shot. Brancazio determined the optimal angle of the arc from the free-throw line to be 45 degrees. The arc also depends on the velocity with which the ball is shot. If a player shoots a foul shot, releasing the ball at a 45-degree angle from a position 6 feet above the floor, then the path of the ball can be modeled by the function h(x) = -44x2/v2 + x + 6 where h is the height of the ball above the floor, x is the forward distance of the ball in front of the foul line, and v is the initial velocity with which the ball is shot in feet per second. Suppose a player shoots a ball with an initial velocity of 28 feet per second. (a) Determine the height of the ball after it has traveled 8 feet in front of the foul line. (b) Determine the height of the ball after it has traveled 12 feet in front of the foul line. (c) Find additional points and graph the path of the basketball. (d) The center of the hoop is 10 feet above the floor and15 feet in front of the foul line. Will the ball go through the hoop? Why or why not? If not, with what initial velocity must the ball be shot in order for the ball to go through the hoop?

33. Cost of Trans-Atlantic Travel. A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by C(x) = 100 + x/10 + 36000/x where x is the ground speed (airspeed ± wind). (a) Use a graphing utility to graph the function

(b) Create a TABLE with TblStart = 0 and ΔTbl = 50. (c) To the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?

34. Effect of Elevation on Weight. If an object weighs m pounds at sea level, then its weight W (in pounds) at a height of h miles above sea level is given approximately by W(h) = m(4000/(4000 + h)2 (a) If Amy weighs 120 pounds at sea level, how much will she weigh on Pike’s Peak, which is 14110 feet above sea level? (b) Use a graphing utility to graph the function W = W(h). Use m = 120 pounds. (c) Create a Table with TblStart = 0 and ΔTbl = 0.5 to see how the weight W varies as h changes from 0 to 5 miles. (d) At what height will Amy weigh 119.95 pounds? (e) Does your answer to part (d) seem reasonable? Explain.

41. Consider the following scenario: Barbara decides to take a walk. She leaves home, walks 2 blocks in 5 minutes at a constant speed, and realizes that she forgot to lock the door. So Barbara runs home in 1 minute. While at her doorstep, it takes her 1 minute to find her keys and lock the door. Barbara walks 5 blocks in 15 minutes and then decides to jog home. It takes her 7 minutes to get home. Draw a graph of Barbara’s distance from home (in blocks) as a function of time.

42. Consider the following scenario: Jayne enjoys riding her bicycle through the woods. At the forest preserve, she gets on her bicycle and rides up a 2000-foot incline in 10 minutes. She then travels down the incline in 3 minutes. The next 5000 feet is level terrain and she covers the distance in 20 minutes. She rests for 15 minutes. Jayne then travels 10000 feet in 30 minutes. Draw a graph of Jayne’s distance traveled (in feet) as a function of time.

43. The following sketch represents the distance d (in miles) that Kevin was from home as a function of time t (in hours). Answer the questions based on the graph. In parts (a)–(g), how many hours elapsed and how far was Kevin from home during this time?

Section 2.3

In Problems 21–28, the graph of a function is given. Use the graph to find: (a) The intercepts, if any (b) The domain and range (c) The intervals on which it is increasing, decreasing, or constant (d) Whether it is even, odd, or neither

In Problems 29–32, the graph of a function f is given. Use the graph to find: (a) The numbers, if any, at which f has a local maximum value. What are the local maximum values? (b) The numbers, if any, at which f has a local minimum value. What are the local minimum values?

In Problems 33–44, determine algebraically whether each function is even, odd, or neither.

33. f(x) = 4x3

In Problems 45–52, for each graph of a function y = f(x) find the absolute maximum and the absolute minimum, if they exist.

In Problems 53–60, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places.

53. f(x) = x3 – 3x + 2, (-2, 2)

61. Find the average rate of change of f(x) = -2x2 + 4. (a) From 0 to 2 (b) from 1 to 3.

75. Minimum Average Cost. The average cost per hour in dollars, C, of producing x riding lawn mowers can be modeled by the function C(x) = 0.3x2 + 21x – 251 + 2500/x. (a) Use a graphing utility to graph C = C(x) (b) Determine the number of riding lawn mowers to produce in order to minimize average cost. (c) What is the minimum average cost?

76. Medicine Concentration. The concentration C of a medication in the bloodstream t hours after being administered is modeled by the function C(t) = -0.002t4 + 0.039t3 – 0.285t2 + 0.766t + 0.085 (a) (a) After how many hours will the concentration be highest? (b) A woman nursing a child must wait until the concentration is below 0.5 before she can feed him. After taking the medication, how long must she wait before feeding her child?

76. E-coli Growth. A strain of E-coli Beu 397-recA441 is placed into a nutrient broth at 30° Celsius and allowed to grow. The data shown in the table are collected. The population is measured in grams and the time in hours. Since population P depends on time t and each input corresponds to exactly one output, we can say that population is a function of time; so P(t) represents the population at time t. (a) Find the average rate of change of the population from 0 to 2.5 hours. (b) Find the average rate of change of the population from 4.5 to 6 hours. (c) What is happening to the average rate of change as time passes?

78. e-Filing Tax Returns. The Internal Revenue Service Restructuring and Reform Act (RRA) was signed into law by President Bill Clinton in 1998. A major objective of the RRA was to promote electronic filing of tax returns. The data in the table that follows show the percentage of individual income tax returns filed electronically for filing years 2000–2008. Since the percentage P of returns filed electronically depends on the filing year y and each input corresponds to exactly one output, the percentage of returns filed electronically is a function of the filing year; so P(y) represents the percentage of returns filed electronically for filing year y. (a) Find the average rate of change of the percentage of e-filed returns from 2000 to 2002. (b) Find the average rate of change of the percentage of e-filed returns from 2004 to 2006. (c) Find the average rate of change of the percentage of e-filed returns from 2006 to 2008. (d) What is happening to the average rate of change as time passes?

Section 2.4

In Problems 29–40: (a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function. (d) Based on the graph, find the range. (e) Is f continuous on its domain?

In Problems 29–40: (a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function. (d) Based on the graph, find the range. (e) Is f continuous on its domain?

47. Cell Phone Service. Sprint PCS offers a monthly cellular phone plan for $39.99. It includes 450 anytime minutes and charges $0.45 per minute for additional minutes. The following function is used to compute the monthly cost for a subscriber:

C(x) = { 39.99 if 0 ≤ x ≤ 450

0.45x – 162.51 if x > 450

where x is the number of anytime minutes used. Compute the monthly cost of the cellular phone for use of the following number of anytime minutes: (a) 200 (b) 465 (c) 451

48. Parking at O’Hare International Airport. The short-term (no more than 24 hours) parking fee F (in dollars) for parking x hours at O’Hare International Airport’s main parking garage can be modeled by the function

3 if 0 < x ≤ 3

F(x) = {5 int(x + 1) + 1 if 3 < x < 9

50 if 9 ≤ x ≤ 24

Determine the fee for parking in the short-term parking garage for (a) 2 hours (b) 7 hours (c) 15 hours (d) 8 hours and 24 minutes

49. Cost of Natural Gas. In April 2009, Peoples Energy had the following rate schedule for natural gas usage in single-family residences:

Monthly service charge $15.95

Per therm service charge

1st 50 therms $0.33606/therm $0.3360/therm

Over 50 therms $0.10580/therm $0.10580/therm

Gas charge $0.3940/therm

(a) What is the charge for using 50 therms in a month? (b) What is the charge for using 500 therms in a month? (c) Develop a model that relates the monthly charge C for x therms of gas. (d) Graph the function found in part (c).

50. Cost of Natural Gas. In April 2009, Nicor Gas had the following rate schedule for natural gas usage in single-family residences:

Monthly customer charge $8.40

Distribution charge

1st 20 therms $0.1473/therm

Next 30 therms $0.0579/therm

Over 50 therms $0.0519/therm

Gas supply charge $0.43/therm

(a) What is the charge for using 40 therms in a month? (b) What is the charge for using 150 therms in a month? (c) Develop a model that gives the monthly charge C for x therms of gas. (d) Graph the function found in part (c).

52. Federal Income Tax. Refer to the revised 2009 tax rate schedules. If x equals taxable income and y equals the tax due, construct a function y = f(x) for Schedule Y-1.

53. Cost of Transporting Goods. A trucking company transports goods between Chicago and New York, a distance of 960 miles. The company’s policy is to charge, for each pound, $0.50 per mile for the first 100 miles, $0.40 per mile for the next 300 miles, $0.25 per mile for the next 400 miles, and no charge for the remaining 160 miles. (a) Graph the relationship between the cost of transportation in dollars and mileage over the entire 960-mile route. (b) Find the cost as a function of mileage for hauls between 100 and 400 miles from Chicago. (c) Find the cost as a function of mileage for hauls between 400 and 800 miles from Chicago.

54. Car Rental Costs. An economy car rented in Florida from National Car Rental ® on a weekly basis costs $95 per week. Extra days cost $24 per day until the day rate exceeds the weekly rate, in which case the weekly rate applies. Also, any part of a day used counts as a full day. Find the cost C of renting an economy car as a function of the number x of days used, where 7 ≤ x ≤ 14. Graph this function.

55. Minimum Payments for Credit Cards. Holders of credit cards issued by banks, department stores, oil companies, and so on, receive bills each month that state minimum amounts that must be paid by a certain due date. The minimum due depends on the total amount owed. One such credit card company uses the following rules: For a bill of less than $10, the entire amount is due. For a bill of at least $10 but less than $500, the minimum due is $10. A minimum of $30 is due on a bill of at least $500 but less than $1000, a minimum of $50 is due on a bill of at least $1000 but less than $1500, and a minimum of $70 is due on bills of $1500 or more. Find the function f that describes the minimum payment due on a bill of x dollars. Graph f.

56. Interest Payments for Credit Cards. Refer to Problem 55. The card holder may pay any amount between the minimum due and the total owed. The organization issuing the card charges the card holder interest of 1.5% per month for the first $1000 owed and 1% per month on any unpaid balance over $1000. Find the function g that gives the amount of interest charged per month on a balance of x dollars. Graph g.

57. Wind Chill. The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is

t 0 ≤ v < 1.79

W ={ 33 – (10.45 + 10√v – v)(33 – t) 1.79 ≤ v ≤ 20

33 – 1.5958(33 – t) v > 20

where v represents the wind speed (in meters per second) and t represents the air temperature (°C). Compute the wind chill for the following: (a) An air temperature of 10°C and a wind speed of 1 meter per second (b) An air temperature of 10°C and a wind speed of 5 m/sec (c) An air temperature of 10°C and a wind speed of 15 m/sec (d) An air temperature of 10°C and a wind speed of 25 m/sec (e) Explain the physical meaning of the equation corresponding to 1.79 ≤ v ≤ 20. (f) Explain the physical meaning of the equation corresponding to v > 20

58. Wind Chill. Redo Problem 57(a)–(d) for an air temperature of -10 °C.

59. First-class Mail. In 2009 the U.S. Postal Service charged $1.17 postage for first-class mail retail flats (such as an 8.5″ by 11″ envelope) weighing up to 1 ounce, plus $0.17 for each additional ounce up to 13 ounces. First-class rates do not apply to flats weighing more than 13 ounces. Develop a model that relates C, the first-class postage charged, for a flat weighing x ounces. Graph the function.

Section 2.5

In Problems 7–18, match each graph to one of the following functions:

In Problems 19–26, write the function whose graph is the graph of y = x3, but is:

19. Shifted to the right 4 units

20. Shifted to the left 4 units

21. Shifted up 4 units

22. Shifted down 4 units

23. Reflected about the y-axis

24. Reflected about the x-axis

25. Vertically stretched by a factor of 4

26. Horizontally stretched by a factor of 4

In Problems 27–30, find the function that is finally graphed after each of the following transformations is applied to the graph of y = √x in the order stated.

27. (1) Shift up 2 units (2) Reflect about the x-axis (3) Reflect about the y-axis

In Problems 39–62, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, y = x2) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.

39. f(x) = x2 – 1

77. Thermostat Control. Energy conservation experts estimate that homeowners can save 5% to 10% on winter heating bills by programming their thermostats 5 to10 degrees lower while sleeping. In the given graph, the temperature T (in degrees Fahrenheit) of a home is given as a function of time t (in hours after midnight) over a 24-hour period. (a) At what temperature is the thermostat set during daytime hours? At what temperature is the thermostat set overnight? (b) The homeowner reprograms the thermostat to y = T(t) – 2. Explain how this affects the temperature in the house. Graph this new function. (c) The homeowner reprograms the thermostat to y = T(t + 1). Explain how this affects the temperature in the house. Graph this new function.

78. Digital Music Revenues. The total projected worldwide digital music revenues R, in millions of dollars, for the years 2005 through 2010 can be estimated by the function R(x) = 170.7x2 + 1373x + 1080 where x is the number of years after 2005. (a) Find R(0), R(3) and R(5) and explain what each value represents. (b) Find r = R(x – 5). (c) Find r(5), r(8) and r(10) and explain what each value represents. (d) In the model r, what does x represent? (e) Would there be an advantage in using the model r when estimating the projected revenues for a given year instead of the model R?

79. Temperature Measurements. The relationship between the Celsius (°C) and Fahrenheit (°F) scales for measuring temperature is given by the equation F = (9/5)C + 32. The relationship between the Celsius (°C) and Kelvin (K) scales is K = C + 273. Graph the equation F = (9/5)C + 32 using degrees Fahrenheit on the y-axis and degrees Celsius on the x-axis. Use the techniques introduced in this section to obtain the graph showing the relationship between Kelvin and Fahrenheit temperatures.

80. Period of a Pendulum. The period T (in seconds) of a simple pendulum is a function of its length l (in feet) defined by the equation T = 2π√(l/g) where g ≈ 32.2 feet per second per second is the acceleration of gravity. (a) Use a graphing utility to graph the function T = T(l) (b) Now graph the functions T = T(l + 1), T = T(l + 2), and T = T(l + 3) (c) Discuss how adding to the length l changes the period T. (d) Now graph the functions T = T(2l), T = T(3l), and T = T(4l). (e) Discuss how multiplying the length l by factors of 2, 3, and 4 changes the period T.

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