Maximizing the Height of an Object If an object is thrown

Problem 71A Draw a sketch of the arch or culvert on coordinate axes, with the horizontal and vertical axes through the vertex of the parabola. Use the given information to label points on the parabola. Then give the equation of the parabola and answer the question. Parabolic Culvert A culvert is shaped like a parabola, 18 ft across the top and 12 ft deep. How wide is the culvert 8 ft from the top?

Problem 1E How does the value of a affect the graph of y = ax2? Discuss the case for a ? 1 and for 0 ?a ? 1.

Problem 2E How does the value of a affect the graph of y = ax2 if a? 0?

Problem 3E Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = x2 ? 3

Problem 7E Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = ?(3 ? x)2 + 2

Problem 5E Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = (x ? 3)2 + 2

Problem 6E Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = (x + 3)2 + 2

Problem 4E Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = (x ? 3)2

Problem 8E Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = ?(x + 3)2 + 2

Problem 12E Complete the square and determine the vertex for the following. y = ?5x2 ? 8x + 3

Problem 13E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept. y = x2+ 5x + 6

Problem 14E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept. y = x2+ 4x ? 5

Problem 9E Complete the square and determine the vertex for the following. y = 3x2 + 9x + 5

Problem 10E Complete the square and determine the vertex for the following. y = 4x2 ? 20x ? 7

Problem 11E Complete the square and determine the vertex for the following. y = ?2x2 + 8x ? 9

Problem 15E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept. y = ?2x2 ? 12x ? 16

Problem 18E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept. f(x) = ?x2+ 6x ? 6

Problem 21E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept. f(x) = ?2x2+ 16x ? 21

Problem 19E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept. f(x) = 2x2 ? 4x + 5

Problem 16E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept. y = ?3x2 ? 6x + 4

Problem 20E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept. f(x) = x2 + 6x + 24

Problem 17E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept. f(x) = 2x2 + 8x ? 8

Problem 22E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept. f(x) = x2 ? x ? 4

Problem 23E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept.

Problem 24E Graph the parabola and give its vertex, axis, x-intercepts, and y-intercept.

Problem 25E Follow the directions. Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = x2 ? 3

Problem 27E Follow the directions. Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = (x ? 3)2 + 2

Problem 26E Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = (x ? 3)2

Problem 28E Follow the directions. Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = (x + 3)2 + 2

Problem 30E Follow the directions. Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = ?(x + 3)2 + 2

Problem 31E Given the following graph, sketch by hand the graph of the function described, giving the new coordinates for the three points labeled on the original graph. y = ?f(x)

Problem 34E Given the following graph, sketch by hand the graph of the function described, giving the new coordinates for the three points labeled on the original graph. y = f(2 ? x) + 2

Problem 32E Given the following graph, sketch by hand the graph of the function described, giving the new coordinates for the three points labeled on the original graph. y = f(x ? 2) + 2

Problem 35E Use the ideas in this section to graph thefunction without a calculator.

Problem 33E Given the following graph, sketch by hand the graph of the function described, giving the new coordinates for the three points labeled on the original graph. y = f (?x)

Problem 29E Follow the directions. Match the correct graph A-F to the function without using your calculator. Then, if you have a graphing calculator, use it to check your answers. Each graph in this group shows x and y in [ ? 10, 10]. y = ?(3 ? x)2 + 2

Problem 36E Use the ideas in this section to graph thefunction without a calculator.

Problem 37E Use the ideas in this section to graph the function without a calculator.

Problem 38E Use the ideas in this section to graph thefunction without a calculator.

Problem 43E Using the graph of f(x) in Figure, show the graph of a f(x) where a satisfies the given condition. 0 <a <1

Problem 39E Using the graph of f(x) in Figure, show the graph of f(ax) where a satisfies the given condition. 0 <a <1

Problem 41E Using the graph of f(x) in Figure, show the graph of f(ax) where a satisfies the given condition. ? 1 <a < 0

Problem 40E Using the graph of f(x) in Figure, show the graph of f(ax) where a satisfies the given condition. 1 <a

Problem 44E Using the graph of f(x) in Figure, show the graph of a f(x) where a satisfies the given condition. 1 <a

Problem 42E Using the graph of f(x) in Figure, show the graph of f(ax) where a satisfies the given condition. a < ? 1

Problem 45E Using the graph of f(x) in Figure, show the graph of a f(x) where a satisfies the given condition. ? 1 < a < 0

Problem 46E Using the graph of f(x) in Figure, show the graph of a f(x) where a satisfies the given condition. a < ?1

Problem 47E If r is an x-intercept of the graph of y = f(x), what is an x-intercept of the graph of each of the following? a.y = ?f(x) ________________ b.y = f (?x) ________________ c.y = ?f (?x)

Problem 48E If b is the y-intercept of the graph of y = f(x), what is the y-intercept of the graph of each of the following? a.y = ?f(x) ________________ b.y = f (?x) ________________ c.y = ?f (?x)

Problem 51A Profit In Exercise, let C(x) be the cost to produce x batches of widgets, and let R(x) be the revenue in thousands of dollars. For exercise, (a) graph both functions, (b) find the minimum break-even quantity, (c) find the maximum revenue, and (d) find the maximum profit.

Problem 49A Profit In Exercise, let C(x) be the cost to produce x batches of widgets, and let R(x) be the revenue in thousands of dollars. For exercise, (a) graph both functions, (b) find the minimum break-even quantity, (c) find the maximum revenue, and (d) find the maximum profit. R(x ) = ?x2 + 8x, C(x ) = 2x + 5

Problem 50A Profit In Exercise, let C(x) be the cost to produce x batches of widgets, and let R(x) be the revenue in thousands of dollars. For exercise, (a) graph both functions, (b) find the minimum break-even quantity, (c) find the maximum revenue, and (d) find the maximum profit.

Problem 53A Maximizing Revenue The revenue of a charter bus company depends on the number of unsold seats. If the revenue R(x) is given by R(x) = 8000 + 70x ? x2, where x is the number of unsold seats, find the maximum revenue and the number of unsold seats that corresponds to maximum revenue.

Problem 54A Maximizing Revenue A charter flight charges a fare of $200 per person plus $4 per person for each unsold seat on the plane. The plane holds 100 passengers. Let x represent the number of unsold seats. a. Find an expression for the total revenue received for the flight R(x). (Hint: Multiply the number of people flying, 100 ? x, by the price per ticket.) ________________ b. Graph the expression from part a. ________________ c. Find the number of unsold seats that will produce the maximum revenue. ________________ d. What is the maximum revenue? ________________ e. Some managers might be concerned about the empty seats, arguing that it doesn’t make economic sense to leave any seats empty. Write a few sentences explaining why this is not necessarily so.

Problem 52A Profit In Exercise, let C(x) be the cost to produce x batches of widgets, and let R(x) be the revenue in thousands of dollars. For exercise, (a) graph both functions, (b) find the minimum break-even quantity, (c) find the maximum revenue, and (d) find the maximum profit. R(x) = ?4x2 + 36x, C (x) = 16x + 24

Problem 55A Maximizing Revenue The demand for a certain type of cosmetic is given by p = 500 ? x, where p is the price in dollars when x units are demanded. a. Find the revenue R(x) that would be obtained at a price p.(Hint: Revenue = Demand × Price) ________________ b. Graph the revenue function R(x). ________________ c. Find the price that will produce maximum revenue. ________________ d. What is the maximum revenue?

Problem 56A Revenue The manager of a peach orchard is trying to decide when to arrange for picking the peaches. If they are picked now, the average yield per tree will be 100 lb, which can be sold for 80¢ per pound. Past experience shows that the yield per tree will increase about 5 lb per week, while the price will decrease about 4¢ per pound per week. a. Let x represent the number of weeks that the manager should wait. Find the income per pound. ________________ b. Find the number of pounds per tree. ________________ c. Find the total revenue from a tree. ________________ d. When should the peaches be picked in order to produce maximum revenue? ________________ e. What is the maximum revenue?

Problem 57A Income The manager of an 80-unit apartment complex is trying to decide what rent to charge. Experience has shown that at a rent of $800, all the units will be full. On the average, one additional unit will remain vacant for each $25 increase in rent. a. Let x represent the number of $25 increases. Find an expression for the rent for each apartment. ________________ b. Find an expression for the number of apartments rented. ________________ c. Find an expression for the total revenue from all rented apartments. ________________ d. What value of x leads to maximum revenue? ________________ e. What is the maximum revenue?

Problem 58A Advertising A study done by an advertising agency reveals that when x thousands of dollars are spent on advertising, it results in a sales increase in thousands of dollars given by the function a. Find the increase in sales when no money is spent on advertising. ________________ b. Find the increase in sales when $10,000 is spent on advertising. ________________ c. Sketch the graph of S(x) without a calculator.

Problem 59A Length of Life According to recent data from the Teachers Insurance and Annuity Association (TIAA), the survival function for life after 65 is approximately given by S(x) = 1 ? 0.058x ? 0.076x2, where x is measured in decades. This function gives the probability that an individual who reaches the age of 65 will live at least x decades (10x years) longer. Source: Ralph DeMarr. a. Find the median length of life for people who reach 65, that is, the age for which the survival rate is 0.50. ________________ b. Find the age beyond which virtually nobody lives. (There are, of course, exceptions.)

Problem 60A Tooth Length The length (in mm) of the mesiodistal crown of the first molar for human fetuses can be approximated by L(t) = ?0.01t2 + 0.788t ? 7.048, where t is the number of weeks since conception. Source: American Journal of Physical Anthropology. a. What does this formula predict for the length at 14 weeks? 24 weeks? ________________ b. What does this formula predict for the maximum length, and when does that occur? Explain why the formula does not make sense past that time.

Problem 61A Splenic Artery Resistance Blood flow to the fetal spleen is of research interest because several diseases are associated with increased resistance in the splenic artery (the artery that goes to the spleen). Researchers have found that the index of splenic artery resistance in the fetus can be described by the function y = 0.057x ? 0.001x2, where x is the number of weeks of gestation. Source: American Journal of Obstetrics and Gynecology. a. At how many weeks is the splenic artery resistance a maximum? ________________ b. What is the maximum splenic artery resistance? ________________ c. At how many weeks is the splenic artery resistance equal to 0, according to this formula? Is your answer reasonable for this function? Explain.

Problem 65A Accident Rate According to data from the National Highway Traffic Safety Administration, the accident rate as a function of the age of the driver in years x can be approximated by the function f(x) = 60.0 ? 2.28x + 0.0232x2 for 16 ?x ? 85. Find the age at which the accident rate is a minimum and the minimum rate. Source: Ralph DeMarr.

Problem 62A Cancer From 1975 to 2007, the age-adjusted incidence rate of invasive lung and bronchial cancer among women can be closely approximated by f (t) = ?0.040194t2 + 2.1493t + 23.921, where t is the number of years since 1975. Source: National Cancer Institute. Based on this model, in what year did the incidence rate reach a maximum? On what years was the rate increasing? Decreasing?

Problem 70A Draw a sketch of the arch or culvert on coordinate axes, with the horizontal and vertical axes through the vertex of the parabola. Use the given information to label points on the parabola. Then give the equation of the parabola and answer the question. Parabolic Arch An arch is shaped like a parabola. It is 30 m wide at the base and 15 m high. How wide is the arch 10 m from the ground?

Problem 66A Maximizing the Height of an Object If an object is thrown upward with an initial velocity of 32 ft/second, then its height after t seconds is given by h = 32t ? 16t2. a. Find the maximum height attained by the object. ________________ b. Find the number of seconds it takes the object to hit the ground.

Problem 67A Stopping Distance According to data from the National Traffic Safety Institute, the stopping distance y in feet of a car traveling x mph can be described by the equation y = 0.056057x2 + 1.06657x. Source: National Traffic Safety Institute. a. Find the stopping distance for a car traveling 25 mph. ________________ b. How fast can you drive if you need to be certain of stopping within 150 ft?

Problem 69A Maximizing Area What would be the maximum area that could be enclosed by the college’s 380 ft of fencing if it decided to close the entrance by enclosing all four sides of the lot? (See Exercise.) Maximizing Area Glenview Community College wants to construct a rectangular parking lot on land bordered on one side by a highway. It has 380 ft of fencing to use along the other three sides. What should be the dimensions of the lot if the enclosed area is to be a maximum? (Hint:Let x represent the width of the lot, and let 380 ? 2x represent the length.)

Problem 68A Maximizing Area Glenview Community College wants to construct a rectangular parking lot on land bordered on one side by a highway. It has 380 ft of fencing to use along the other three sides. What should be the dimensions of the lot if the enclosed area is to be a maximum? (Hint:Let x represent the width of the lot, and let 380 ? 2x represent the length.)