Describe an asymptote, and explain when a rational

Problem 1E Explain how translations and reflections can be used to graph y = ?(x ? 1)4 + 2.

Problem 4E Use the principles of the previous section with the graphs of this section to sketch a graph of the given function. f(x) = (x + 1)3 ? 2

Problem 62A Coal Consumption The table gives U.S. coal consumption for selected years. Source: U.S. Department of Energy. Year Millions of Short Tons 1950 494.1 1960 398.1 1970 523.2 1980 702.7 1985 818.0 1990 902.9 1995 962.1 2000 1084.1 2005 1128.3 a. Draw a scatterplot, letting x = 0 represent 1950. ________________ b. Use the quadratic regression feature of a graphing calculator to get a quadratic function that approximates the data. ________________ c. Graph the function from part b on the same window as the scatterplot. ________________ d. Use cubic regression to get a cubic function that approximates the data. ________________ e. Graph the cubic function from part d on the same window as the scatterplot. ________________ f. Which of the two functions in parts b and d appears to be a better fit for the data? Explain your reasoning.

Problem 2E Describe an asymptote, and explain when a rational function will have (a) a vertical asymptote and (b) a horizontal asymptote.

Problem 3E Use the principles of the previous section with the graphs of this section to sketch a graph of the given function. f(x) = (x ? 2)3 + 3

Problem 6E Use the principles of the previous section with the graphs of this section to sketch a graph of the given function. f(x) = ?(x ? 1)4 + 2

Problem 7E Match the correct graph A-I to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph is plotted on [ ? 6, 6] by [ ? 50, 50]. y = x3 ? 7x ? 9

Problem 8E Match the correct graph A-I to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph is plotted on [ ? 6, 6] by [ ? 50, 50]. y = ?x3 + 4x2 + 3x ? 8

Problem 10E Match the correct graph A-I to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph is plotted on [ ? 6, 6] by [ ? 50, 50]. y = 2x3 + 4x + 5

Problem 13E Match the correct graph A-I to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph is plotted on [ ? 6, 6] by [ ? 50, 50]. y = ?x4 + 2x3+ 10x + 15

Problem 11E Match the correct graph A-I to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph is plotted on [ ? 6, 6] by [ ? 50, 50]. y = x4 ? 5x2+ 7

Problem 9E Match the correct graph A-I to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph is plotted on [ ? 6, 6] by [ ? 50, 50]. y = ?x3 ? 4x2+ x + 6

Problem 12E Match the correct graph A-I to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph is plotted on [ ? 6, 6] by [ ? 50, 50]. y = x4+ 4x3 ? 20

Problem 14E Match the correct graph A-I to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph is plotted on [ ? 6, 6] by [ ? 50, 50]. y = 0.7x5 ? 2.5x4 ? x3 + 8x2 + x + 2

Problem 15E Match the correct graph A-I to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph is plotted on [ ? 6, 6] by [ ? 50, 50]. y = ?x5 + 4x4 + x3 ? 16x2 + 12x + 5

Problem 16E Match the correct graph A-E to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph in this group is plotted on [ ? 6, 6] by [ ? 6, 6]. Hint: Consider the asymptotes. (If you try graphing these graphs in Connected mode rather than Dot mode, you will see some lines that are not part of the graph, but the result of the calculator connecting disconnected parts of the graph.)

Problem 18E Match the correct graph A-E to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph in this group is plotted on [ ? 6, 6] by [ ? 6, 6]. Hint: Consider the asymptotes. (If you try graphing these graphs in Connected mode rather than Dot mode, you will see some lines that are not part of the graph, but the result of the calculator connecting disconnected parts of the graph.)

Problem 20E Match the correct graph A-E to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph in this group is plotted on [ ? 6, 6] by [ ? 6, 6]. Hint: Consider the asymptotes. (If you try graphing these graphs in Connected mode rather than Dot mode, you will see some lines that are not part of the graph, but the result of the calculator connecting disconnected parts of the graph.)

Problem 17E Match the correct graph A-E to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph in this group is plotted on [ ? 6, 6] by [ ? 6, 6]. Hint: Consider the asymptotes. (If you try graphing these graphs in Connected mode rather than Dot mode, you will see some lines that are not part of the graph, but the result of the calculator connecting disconnected parts of the graph.)

Problem 19E Match the correct graph A-E to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph in this group is plotted on [ ? 6, 6] by [ ? 6, 6]. Hint: Consider the asymptotes. (If you try graphing these graphs in Connected mode rather than Dot mode, you will see some lines that are not part of the graph, but the result of the calculator connecting disconnected parts of the graph.)

Problem 21E The following is the graph of a polynomial function. Give the possible values for the degree of the polynomial, and give the sign (+ or ?) for the leading coefficient.

Problem 24E The following is the graph of a polynomial function. Give the possible values for the degree of the polynomial, and give the sign (+ or ?) for the leading coefficient.

Problem 25E The following is the graph of a polynomial function. Give the possible values for the degree of the polynomial, and give the sign (+ or ?) for the leading coefficient.

Problem 22E The following is the graph of a polynomial function. Give the possible values for the degree of the polynomial, and give the sign (+ or ?) for the leading coefficient.

Problem 26E The following is the graph of a polynomial function. Give the possible values for the degree of the polynomial, and give the sign (+ or ?) for the leading coefficient.

Problem 23E The following is the graph of a polynomial function. Give the possible values for the degree of the polynomial, and give the sign (+ or ?) for the leading coefficient.

Problem 27E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 28E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 30E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 31E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 29E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 34E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 32E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 35E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 33E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 36E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 37E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 38E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 41E Write an equation that defines a rational function with a vertical asymptote at x = 1 and a horizontal asymptote at y = 2.

Problem 40E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 42E Write an equation that defines a rational function with a vertical asymptote at x = ?2 and a horizontal asymptote at y = 0.

Problem 43E Consider the polynomial functions defined by f(x) = (x ? 1)(x ? 2)(x + 3), g(x) = x3 + 2x2 ? x ? 2, and h(x) = 3x3+ 6x2 ? 3x ? 6. a. What is the value of f(1)? ________________ b. For what values, other than 1, is f(x) = 0? ________________ c. Verify that g (?1) = g(1) = g(?2) = 0. ________________ d. Based on your answer from part c, what do you think is the factored form of g(x)? Verify your answer by multiplying it out and comparing with g(x). ________________ e. Using your answer from part d, what is the factored form of h(x)? ________________ f. Based on what you have learned in this exercise, fill in the blank: If f is a polynomial and f(a) = 0 for some number a, then one factor of the polynomial is __________.

Problem 39E Find any horizontal and vertical asymptotes and any holes that may exist for each rational function. Draw the graph of each function, including any x- and y-intercepts.

Problem 44E Consider the function defined by Source: The Mathematics Teacher. a. Graph the function on [ ? 6, 6] by [ ? 6, 6]. From your graph, estimate how many x-intercepts the function has and what their values are. ________________ b. Now graph the function on [ ? 1.5, ? 1.4] by [ ? 10?4, 10?4] and on [1.4, 1.5] by [ ? 10?5, 10?5]. From your graphs, estimate how many x-intercepts the function has and what their values are. ________________ c. From your results in parts a and b, what advice would you give a friend on using a graphing calculator to find x-intercepts?

Problem 45E Consider the function defined by Source: The Mathematics Teacher. a. Graph the function on [ ? 3.4, 3.4] by [ ? 3, 3]. From your graph, estimate how many vertical asymptotes the function has and where they are located. ________________ b. Now graph the function on [ ? 1.5, ? 1.4] by [ ? 10, 10] and on [1.4, 1.5] by [ ? 1000, 1000]. From your graphs, estimate how many vertical asymptotes the function has and where they are located. ________________ c. From your results in parts a and b, what advice would you give a friend on using a graphing calculator to find vertical asymptotes?

Problem 47A Cost Analysis In a recent year, the cost per ton, y, to build an oil tanker of x thousand deadweight tons was approximated by for x > 0. a. Find , and ________________ b. Find any asymptotes. ________________ c. Find any intercepts. ________________ d. Graph

Problem 46A Average Cost Suppose the average cost per unit , in dollars, to produce x units of yogurt is given by a. Find and ________________ b. Which of the intervals (0, ?) and [0, ?) would be a more reasonable domain for Why? ________________ c. Give the equations of any asymptotes. Find any intercepts. ________________ d. Graph

Problem 49A APPLY IT Tax Rates Exercises 48-50 refer to the Laffer curve, originated by the economist Arthur Laffer. An idealized version of this curve is shown here. According to this curve, decreasing a tax rate, say from x2 percent to x1 percent on the graph, can actually lead to an increase in government revenue. The theory is that people will work harder and earn more money if they are taxed at a lower rate, so the government ends up with more revenue than it would at a higher tax rate. All economists agree on the endpoints—0 revenue at tax rates of both 0% and 100%—but there is much disagreement on the location of the tax rate x1 that produces the maximum revenue. Find the equations of two quadratic functions that could describe the Laffer curve by having zeros at x = 0 and x = 100. Give the first a maximum of 100 and the second a maximum of 250, then multiply them together to get a new Laffer curve with a maximum of 25,000. Plot the resulting function.

Problem 50A APPLY IT Tax Rates Exercises 48-50 refer to the Laffer curve, originated by the economist Arthur Laffer. An idealized version of this curve is shown here. According to this curve, decreasing a tax rate, say from x2 percent to x1 percent on the graph, can actually lead to an increase in government revenue. The theory is that people will work harder and earn more money if they are taxed at a lower rate, so the government ends up with more revenue than it would at a higher tax rate. All economists agree on the endpoints—0 revenue at tax rates of both 0% and 100%—but there is much disagreement on the location of the tax rate x1 that produces the maximum revenue. An economist might argue that the models in the two previous exercises are unrealistic because they predict that a tax rate of 50% gives the maximum revenue, while the actual value is probably less than 50%. Consider the function where y is government revenue in millions of dollars from a tax rate of x percent, where 0 ?x ? 100. Source: Dana Lee Ling. a. Graph the function, and discuss whether the shape of the graph is appropriate. ________________ b. Use a graphing calculator to find the tax rate that produces the maximum revenue. What is the maximum revenue?

Problem 48A APPLY IT Tax Rates Exercises 48-50 refer to the Laffer curve, originated by the economist Arthur Laffer. An idealized version of this curve is shown here. According to this curve, decreasing a tax rate, say from x2 percent to x1 percent on the graph, can actually lead to an increase in government revenue. The theory is that people will work harder and earn more money if they are taxed at a lower rate, so the government ends up with more revenue than it would at a higher tax rate. All economists agree on the endpoints—0 revenue at tax rates of both 0% and 100%—but there is much disagreement on the location of the tax rate x1 that produces the maximum revenue. A function that might describe the entire Laffer curve is y = x(100 ? x)(x2 + 500), where y is government revenue in hundreds of thousands of dollars from a tax rate of x percent, with the function valid for 0 ?x ? 100. Find the revenue from the following tax rates. a. 10% ________________ b. 40% ________________ c. 50% ________________ d. 80% ________________ e. Graph the function.

Problem 52A Cost-Benefit Model Suppose a cost-benefit model is given by where y is the cost in thousands of dollars of removing x percent of a certain pollutant. a. Find the cost of removing each percent of pollutants: 0%; 50%; 80%; 90%; 95%; 99%; 100%. ________________ b. Graph the function.

Problem 51A Cost-Benefit Model Suppose a cost-benefit model is given by where y is the cost in thousands of dollars of removing x percent of a given pollutant. a. Find the cost of removing each percent of pollutants: 50%; 70%; 80%; 90%; 95%; 98%; 99%. ________________ b. Is it possible, according to this function, to remove all the pollutant? ________________ c. Graph the function.

Problem 54A Cardiac Output A technique for measuring cardiac output depends on the concentration of a dye after a known amount is injected into a vein near the heart. In a normal heart, the concentration of the dye at time x (in seconds) is given by the function a. Graph g(x) on [0, 6] by [0, 20]. ________________ b. In your graph from part a, notice that the function initially increases. Considering the form of g(x), do you think it can keep increasing forever? Explain. ________________ c. Write a short paragraph about the extent to which the concentration of dye might be described by the function g (x).

Problem 53A Contact Lenses The strength of a contact lens is given in units known as diopters, as well as in mm of arc. The following is taken from a chart used by optometrists to convert diopters to mm of arc. Source: Bausch&Lomb. Diopters mm of Arc 36.000 9.37 36.125 9.34 36.250 9.31 36.375 9.27 36.500 9.24 36.625 9.21 36.750 9.18 36.875 9.15 37.000 9.12 a. Notice that as the diopters increase, the mm of arc decrease. Find a value of k so the function a = f(d) = k/d gives a, the mm of arc, as a function of d, the strength in diopters. (Round k to the nearest integer. For a more accurate answer, average all the values of k given by each pair of data.) ________________ b. An optometrist wants to order 40.50 diopter lenses for a patient. The manufacturer needs to know the strength in mm of arc. What is the strength in mm of arc?

Problem 55A Alcohol Concentration The polynomial function A(x) = 0.003631x3 ? 0.03746x2 + 0.1012x + 0.009 gives the approximate blood alcohol concentration in a 170-lb woman x hours after drinking 2 oz of alcohol on an empty stomach, for x in the interval [0, 5]. Source: Medical Aspects of Alcohol Determination in Biological Specimens. a. Graph A(x) on 0 ?x ? 5. ________________ b. Using the graph from part a, estimate the time of maximum alcohol concentration. ________________ c. In many states, a person is legally drunk if the blood alcohol concentration exceeds 0.08%. Use the graph from part a to estimate the period in which this 170-lb woman is legally drunk.

Problem 56A Medical School For the years 1998 to 2009, the number of applicants to U.S. medical schools can be closely approximated by A(t) = ?6.7615t4 + 114.7t3 ? 240.1t2 ? 212t + 40,966, where t is the number of years since 1998. Source: Association of American Medical Colleges. a. Graph the number of applicants on 0 ? t ? 11. ________________ b. Based on the graph in part a, during what years did the number of medical school applicants increase?

Problem 57A Population Biology The function is used in population models to give the size of the next generation (f(x)) in terms of the current generation (x). Source: Models in Ecology. a. What is a reasonable domain for this function, considering what x represents? ________________ b. Graph this function for ? = a = b = 1. ________________ c. Graph this function for ? = a = 1 and b = 2. ________________ d. What is the effect of making b larger?

Problem 59A Brain Mass The mass (in grams) of the human brain during the last trimester of gestation and the first two years after birth can be approximated by the function where c is the circumference of the head in centimeters. Source: Early Human Development. a. Find the approximate mass of brains with a head circumference of 30, 40, or 50 cm. ________________ b. Clearly the formula is invalid for any values of c yielding negative values of w. For what values of c is this true? ________________ c. Use a graphing calculator to sketch this graph on the interval 20 ? c ? 50. ________________ d. Suppose an infant brain has mass of 700 g. Use features on a graphing calculator to find what the circumference of the head is expected to be.

Problem 58A Growth Model The function is used in biology to give the growth rate of a population in the presence of a quantity x of food. This is called Michaelis-Menten kinetics. Source: Mathematical Models in Biology. a. What is a reasonable domain for this function, considering what x represents? ________________ b. Graph this function for K = 5 and A = 2. ________________ c. Show that y = K is a horizontal asymptote. ________________ d. What do you think K represents? ________________ e. Show that A represents the quantity of food for which the growth rate is half of its maximum.

Problem 60A Head Start The enrollment in Head Start for some recent years is included in the table. Source: Administration for Children & Families. Year Enrollment 1966 733,000 1970 477,400 1980 376,300 1990 540,930 1995 750,696 2000 857,664 2005 906,993 a. Plot the points from the table using 0 for 1960, and so on. ________________ b. Use the quadratic regression feature of a graphing calculator to get a quadratic function that approximates the data. Graph the function on the same window as the scatterplot. ________________ c. Use cubic regression to get a cubic function that approximates the data. Graph the function on the same window as the scatterplot. ________________ d. Which of the two functions in part b and c appears to be a better fit for the data? Explain your reasoning.

Problem 61A Length of a Pendulum A simple pendulum swings back and forth in regular time intervals. Grandfather clocks use pendulums to keep accurate time. The relationship between the length of a pendulum L and the period (time)T for one complete oscillation can be expressed by the function L = kTn, where k is a constant and n is a positive integer to be determined. The data below were taken for different lengths of pendulums.* Source: Gary Rockswold. T (sec) L (ft) 1.11 1.0 1.36 1.5 1.57 2.0 1.76 2.5 1.92 3.0 2.08 3.5 2.22 4.0 a. Find the value of k for n = 1, 2, and 3, using the data for the 4-ft pendulum. ________________ b. Use a graphing calculator to plot the data in the table and to graph the function L = kTn for the three values of k (and their corresponding values of n) found in part a. Which function best fits the data? ________________ c. Use the best-fitting function from part a to predict the period of a pendulum having a length of 5 ft. ________________ d. If the length of pendulum doubles, what happens to the period? ________________ e. If you have a graphing calculator or computer program with a quadratic regression feature, use it to find a quadratic function that approximately fits the data. How does this answer compare with the answer to part b?

Problem 5E Use the principles of the previous section with the graphs of this section to sketch a graph of the given function. f(x) = ?(x + 3)4 + 1