In Exercises 14, find the domain of the rational function

Functions of the form where and are polynomials and is not the zero polynomial, are called ________ ________.

When as from the left or the right, is a ________ ________ of the graph of f

When as is a ________ ________ of the graph of f

The graph of is called a ________.

In Exercises 512, find the domain of the function, and discuss the behavior of near any excluded -values. f x1 x 1 f

In Exercises 512, find the domain of the function, and discuss the behavior of near any excluded -values. f x4 x 3 f

In Exercises 512, find the domain of the function, and discuss the behavior of near any excluded -values. fx5x x 2

In Exercises 512, find the domain of the function, and discuss the behavior of near any excluded -values. f x6x x 3 fx

In Exercises 512, find the domain of the function, and discuss the behavior of near any excluded -values. fx3x2 x2 1 f

In Exercises 512, find the domain of the function, and discuss the behavior of near any excluded -values. fx2x x2 4 fx

In Exercises 512, find the domain of the function, and discuss the behavior of near any excluded -values. fxx2 3x 2 x2 2x 1 f

In Exercises 512, find the domain of the function, and discuss the behavior of near any excluded -values. fxx2 x 20 x2 8x 16 fx

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. f x4 x 2

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. f x1 x 23 f x

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. f x5 x 5 x f

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. f x3 7x 3 2x f

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. f xx 3 x 2 1 f

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. f x2x 2 x 1 f

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. f x3x 2 1 x 2 x 9

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. f x3x 2 x 5 x 2 1 f

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. fxx 4 x2 16 f

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. xx 3 x2 9 fx

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. fxx2 1 x2 2x 3 fx

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. fxx2 4 x2 3x 2 fx

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. fxx2 3x 4 2x2 x 1 fx

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. fxx2 x 2 2x2 5x 2 f

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. fx6x2 5x 6 3x2 8x 4 fx

In Exercises 1328, find all vertical and horizontal asymptotes of the graph of the function. fx6x2 11x 3 6x2 7x 3 fx

In Exercises 2936, match the rational function with its graph. [The graphs are labeled (a)(h).]f x4 x 5 x

In Exercises 2936, match the rational function with its graph. [The graphs are labeled (a)(h).] f x5 x 2 f

In Exercises 2936, match the rational function with its graph. [The graphs are labeled (a)(h).] f xx 1 x 4 f

In Exercises 2936, match the rational function with its graph. [The graphs are labeled (a)(h).] f xx 2 x 4 f

In Exercises 2936, match the rational function with its graph. [The graphs are labeled (a)(h).] fx2x2 x2 1 f

In Exercises 2936, match the rational function with its graph. [The graphs are labeled (a)(h).] x2x x2 1 fx2

In Exercises 2936, match the rational function with its graph. [The graphs are labeled (a)(h).] fx3 x 22 fx

In Exercises 2936, match the rational function with its graph. [The graphs are labeled (a)(h).] fx3x x 22 fx

In Exercises 3740, (a) determine the domains of and (b) simplify and find any vertical asymptotes of (c) complete the table, and (d) explain how the two functions differ f xgxx 2 x 2 4 x 2 , f f,

In Exercises 3740, (a) determine the domains of and (b) simplify and find any vertical asymptotes of (c) complete the table, and (d) explain how the two functions differ f xgxx x 2x 3 x 2 3x , f x

In Exercises 3740, (a) determine the domains of and (b) simplify and find any vertical asymptotes of (c) complete the table, and (d) explain how the two functions differ f x2x 1 2x 2 x , f gx1 x

In Exercises 3740, (a) determine the domains of and (b) simplify and find any vertical asymptotes of (c) complete the table, and (d) explain how the two functions differ f x2x 8 x 2 9x 20 , g gx2 x 5 f

The cost (in dollars) of supplying recycling bins to of the population of a rural township is given by (a) Use a graphing utility to graph the cost function. (b) Find the costs of supplying bins to 15%, 50%, and 90% of the population. (c) According to this model, would it be possible to supply bins to 100% of the residents? Explain.

Pollution The cost (in millions of dollars) of removing of the industrial and municipal pollutants discharged into a river is given by (a) Use a graphing utility to graph the cost function. (b) Find the costs of removing 10%, 40%, and 75% of the pollutants. (c) According to this model, would it be possible to remove 100% of the pollutants? Explain.

Population Growth The game commission introduces 100 deer into newly acquired state game lands. The population of the herd is modeled by where is the time in years. (a) Use a graphing utility to graph this model. (b) Find the populations when and (c) What is the limiting size of the herd as time increases?

Food Consumption A biology class performs an experiment comparing the quantity of food consumed by a certain kind of moth with the quantity supplied. The model for the experimental data is where is the quantity (in milligrams) of food supplied and is the quantity (in milligrams) of food consumed. (a) Use a graphing utility to graph this model. (b) At what level of consumption will the moth become satiated?

Human Memory Model Psychologists have developed mathematical models to predict memory performance as a function of the number of trials of a certain task. Consider the learning curve where is the fraction of correct responses after trials. (a) Complete the table for this model. What does it suggest? (b) According to this model, what is the limiting percent of correct responses as increases?

. Data Analysis: Physics Experiment Consider a physics laboratory experiment designed to determine an unknown mass. A flexible metal meter stick is clamped to a table with 50 centimeters overhanging the edge (see figure). Known masses ranging from 200 grams to 2000 grams are attached to the end of the meter stick. For each mass, the meter stick is displaced vertically and then allowed to oscillate. The average time (in seconds) of one oscillation for each mass is recorded in the table. A model for the data that can be used to predict the time of one oscillation is (a) Use this model to create a table showing the predicted time for each of the masses shown in the table. (b) Compare the predicted times with the experimental times. What can you conclude? (c) Use the model to approximate the mass of an object for which seconds.

In Exercises 47 and 48, determine whether the statement is true or false. Justify your answer. The graph of a polynomial function can have infinitely many vertical asymptotes.

In Exercises 47 and 48, determine whether the statement is true or false. Justify your answer. fxx3 2x2 5x 6 t  is a rational function

In Exercises 4952, (a) determine the value that the function approaches as the magnitude of increases. Is greater than or less than this functional value when (b) is positive and large in magnitude and (c) is negative and large in magnitude? f x4 1 x x

In Exercises 4952, (a) determine the value that the function approaches as the magnitude of increases. Is greater than or less than this functional value when (b) is positive and large in magnitude and (c) is negative and large in magnitude? f x2  1 x 3 f

In Exercises 4952, (a) determine the value that the function approaches as the magnitude of increases. Is greater than or less than this functional value when (b) is positive and large in magnitude and (c) is negative and large in magnitude? f x2x 1 x 3 f

In Exercises 4952, (a) determine the value that the function approaches as the magnitude of increases. Is greater than or less than this functional value when (b) is positive and large in magnitude and (c) is negative and large in magnitude? f x2x 1 x 2 1 f

In Exercises 53 and 54, write a rational function that has the specified characteristics. (There are many correct answers.) Vertical asymptote: None Horizontal asymptote y 2

In Exercises 53 and 54, write a rational function that has the specified characteristics. (There are many correct answers.) Vertical asymptotes: Horizontal asymptote: x  2, x 1 None

Think About It Give an example of a rational function whose domain is the set of all real numbers. Give an example of a rational function whose domain is the set of all real numbers except x 15.

HOW DO YOU SEE IT? The graph of a rational function is shown below. Determine which of the statements about the function is false. Justify your answer. (a) When (b) The degrees of and are equal. (c) The ratio of the leading coefficients of and is 1.

For the rational function if the degree of is exactly one more than the degree of then the graph of has a ________ (or oblique) ________.

The graph of has a ________ asymptote at

In Exercises 3 6, use the graph of to sketch the graph of g gx2 x 4 f

In Exercises 3 6, use the graph of to sketch the graph of g gx2 x 5 4

In Exercises 3 6, use the graph of to sketch the graph of g gx 2 x gx

In Exercises 3 6, use the graph of to sketch the graph of g gx1 x 2 g

In Exercises 710, use the graph of to sketch the graph of g gx3 x 2 1 fx

In Exercises 710, use the graph of to sketch the graph of g gx 3 x 2 2

In Exercises 710, use the graph of to sketch the graph of g gx3 x 12 gx 3

In Exercises 710, use the graph of to sketch the graph of g gx1 x 2 g

In Exercises 1114, use the graph of to sketch the graph of g gx4 x 23 fx

In Exercises 1114, use the graph of to sketch the graph of g gx4 x 3 2 4

In Exercises 1114, use the graph of to sketch the graph of g gx 4 x 3 gx

In Exercises 1114, use the graph of to sketch the graph of g gx2 x 3

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f x1 x 2 g

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f x1 x 3 f x

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. hx1 x 4 f

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. gx1 6 x hx

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. Cx7 2x 2 x g

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. x1 3x 1 x Cx7

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.gx1 x 2 2 P

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f x2 3 x 2 gx1

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. xx 2 x 2 9 f

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f t1 2t t f x

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. hxx 2 x 2 9 f

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. gxx x 2 9 hx

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. s4s s 2 4 g

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f x 1 x 22 gs4s

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. gx4x 1 xx 4 f x 1

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. hx2 x 2x 2gx4x 

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f x2x x 2 3x 4 hx2

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f x3x x 2 2x 3 f x

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. hxx2 5x 4 x2 4 f x

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. gxx2 2x 8 x2 9 hxx2 5

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx6x x2 5x 14 gxx

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx3x2 1 x2 2x 15 fx6x

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx2x2 5x 3 x 3 2x2 x 2 fx3x2

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fxx2 x 2 x 3 2x2 5x 6 fx2x2

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fxx2 3x x2 x 6 fx

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx5x 4 x2 x 12 fxx2

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx2x2 5x 2 2x2 x 6 fx5x

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx3x2 8x 4 2x2 3x 2 fx2x2

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. ftt2 1 t 1 fx3x

In Exercises 15 44, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fxx2 36 x 6 ftt

In Exercises 45 and 46, (a) determine the domains of and (b) use a graphing utility to graph and in the same viewing window, and (c) explain why the graphing utility may not show the difference in the domains of f and g f xgxx 1 x 2 1 x 1 , f g.

In Exercises 45 and 46, (a) determine the domains of and (b) use a graphing utility to graph and in the same viewing window, and (c) explain why the graphing utility may not show the difference in the domains of f and g f xx 2 x 2 2x , f x gx1 x f

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. hxx2 9 x gx

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. gxx2 5 x h

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx2x2 1 x g

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx1 x2 x fx

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. gxx2 1 x f

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. hxx2 x 1 gxx

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f t t 2 1 t 5 hx

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f xx2 3x 1 f

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f xx 3 x 2 4 f

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. gxx 3 2x 2 8 f

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f xx3 1 x2 x gxx

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f xx4 x x3 f

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f xx 2 x 1 x 1 f xx

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f x2x 2 5x 5 x 2 f x

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx2x3 x2 2x 1 x2 3x 2 f x

In Exercises 4762, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx2x3 x2 8x 4 x2 3x 2 fx2x

In Exercises 6366, use a graphing utility to graph the rational function. Give the domain of the function and find any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line f xx 2 5x 8 x 3 fx

In Exercises 6366, use a graphing utility to graph the rational function. Give the domain of the function and find any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line fx2x 2 x x 1 f

In Exercises 6366, use a graphing utility to graph the rational function. Give the domain of the function and find any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line gx1 3x 2 x 3 x 2 fx

In Exercises 6366, use a graphing utility to graph the rational function. Give the domain of the function and find any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line hx12 2x x 2 24 x gx1

In Exercises 6770, (a) use the graph to determine any -intercepts of the graph of the rational function and (b) set and solve the resulting equation to confirm your result in part (a).

In Exercises 6770, (a) use the graph to determine any -intercepts of the graph of the rational function and (b) set and solve the resulting equation to confirm your result in part (a).

In Exercises 6770, (a) use the graph to determine any -intercepts of the graph of the rational function and (b) set and solve the resulting equation to confirm your result in part (a).

In Exercises 6770, (a) use the graph to determine any -intercepts of the graph of the rational function and (b) set and solve the resulting equation to confirm your result in part (a).

In Exercises 7174, (a) use a graphing utility to graph the rational function and determine any -intercepts of the graph and (b) set and solve the resulting equation to confirm your result in part (a). y 1 x 5  4 x

In Exercises 7174, (a) use a graphing utility to graph the rational function and determine any -intercepts of the graph and (b) set and solve the resulting equation to confirm your result in part (a). y 202 x 1 3 x

In Exercises 7174, (a) use a graphing utility to graph the rational function and determine any -intercepts of the graph and (b) set and solve the resulting equation to confirm your result in part (a). y x 6 x 1 y

In Exercises 7174, (a) use a graphing utility to graph the rational function and determine any -intercepts of the graph and (b) set and solve the resulting equation to confirm your result in part (a). y x 9 x

Concentration of a Mixture A 1000-liter tank contains 50 liters of a 25% brine solution. You add liters of a 75% brine solution to the tank. (a) Show that the concentration , the proportion of brine to total solution, in the final mixture is (b) Determine the domain of the function based on the physical constraints of the problem. (c) Sketch a graph of the concentration function. (d) As the tank is filled, what happens to the rate at which the concentration of brine is increasing? What percent does the concentration of brine appear to approach?

Geometry A rectangular region of length and width has an area of 500 square meters. (a) Write the width as a function of (b) Determine the domain of the function based on the physical constraints of the problem. (c) Sketch a graph of the function and determine the width of the rectangle when meters.

Page Design A page that is inches wide and inches high contains 30 square inches of print. The top and bottom margins are each 1 inch deep and the margins on each side are 2 inches wide (see figure). (a) Show that the total area on the page is (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper is used. Verify your answer numerically using the table feature of the graphing utility.

Page Design A rectangular page is designed to contain 64 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on each side are inches wide. What should the dimensions of the page be so that the least amount of paper is used?

In Exercises 79 and 80, use a graphing utility to graph the function and locate any relative maximum or minimum points on the graph. f x3x 1 x 2 x 1 1

In Exercises 79 and 80, use a graphing utility to graph the function and locate any relative maximum or minimum points on the graph. Cxx  32 x f

Minimum Cost The ordering and transportation cost (in thousands of dollars) for the components used in manufacturing a product is given by where is the order size (in hundreds). Use a graphing utility to graph the cost function. From the graph, estimate the order size that minimizes cost.

Minimum Cost The cost of producing units of a product is given by and the average cost per unit is given by Sketch the graph of the average cost function and estimate the number of units that should be produced to minimize the average cost per unit.

Average Speed A driver averaged 50 miles per hour on the round trip between two cities 100 miles apart. The average speeds for going and returning were and miles per hour, respectively. (a) Show that (b) Determine the vertical and horizontal asymptotes of the graph of the function. (c) Use a graphing utility to graph the function. (d) Complete the table. (e) Are the results in the table what you expected? Explain. (f) Is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip? Explain.

The concentration of a chemical in the bloodstream hours after injection into muscle tissue is given by (a) Determine the horizontal asymptote of the graph of the function and interpret its meaning in the context of the problem. (b) Use a graphing utility to graph the function and approximate the time when the bloodstream concentration is greatest. (c) Use the graphing utility to determine when the concentration is less than 0.345

In Exercises 8588, determine whether the statement is true or false. Justify your answer. When the graph of a rational function has a vertical asymptote at it is possible to sketch the graph without lifting your pencil from the paper

In Exercises 8588, determine whether the statement is true or false. Justify your answer.The graph of a rational function can never cross one of its asymptotes.

In Exercises 8588, determine whether the statement is true or false. Justify your answer. The graph of has a slant asymptote.

In Exercises 8588, determine whether the statement is true or false. Justify your answer. Every rational function has a horizontal asymptote

In Exercises 89 and 90, use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function may indicate that there should be one. hx6 2x 3 x fx2

In Exercises 89 and 90, use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function may indicate that there should be one. gxx 2 x 2 x 1 hx6

Writing (a) Given a rational function how can you determine whether has a slant asymptote? (b) You determine that a rational function has a slant asymptote. How do you find it?

Writing and Graphing a Rational Function Write a rational function satisfying the following criteria. Then sketch a graph of your function. Vertical asymptote: Slant asymptote: Zero of the function:

Think About It Is it possible for a rational function to have all three types of asymptotes (vertical, horizontal, and slant)? Why or why not?

HOW DO YOU SEE IT? For each statement below, determine which graph(s) show that the statement is incorrect. (i) (ii) (iii) (iv) (a) Every rational function has a vertical asymptote. (b) Every rational function has at least one vertical, horizontal, or slant asymptote. (c) A rational function can have at most one vertical asymptote. (d) The graph of a rational function with a slant asymptote cannot cross the slant asymptote.

A ________ is the intersection of a plane and a double-napped cone.

The equation is the standard form of the equation of a ________ with center ________ and radius ________.

A ________ is the set of all points in a plane that are equidistant from a fixed line, called the ________, and a fixed point, called the ________, not on the line.

The ________ of a parabola is the midpoint between the focus and the directrix.

The line that passes through the focus and the vertex of a parabola is called the ________ of the parabola.

An ________ is the set of all points in a plane the sum of whose distances from two distinct fixed points, called ________, is constant

The chord joining the vertices of an ellipse is called the ________ ________, and its midpoint is the ________ of the ellipse

The chord perpendicular to the major axis at the center of an ellipse is the ________ ________ of the ellipse.

. A ________ is the set of all points in a plane the difference of whose distances from two distinct fixed points, called ________, is a positive constant.

The line segment connecting the vertices of a hyperbola is called the ________ ________, and the midpoint of the line segment is the ________ of the hyperbola.

In Exercises 1116, match the equation with its graph. [The graphs are labeled (a)(f).] x 2 2x 2  2y

In Exercises 1116, match the equation with its graph. [The graphs are labeled (a)(f).] y x 2 2x

In Exercises 1116, match the equation with its graph. [The graphs are labeled (a)(f).] x 2 9 2 9y 2 9

In Exercises 1116, match the equation with its graph. [The graphs are labeled (a)(f).] 9x 2 y x 2 9

In Exercises 1116, match the equation with its graph. [The graphs are labeled (a)(f).] y 2 16 2 9x 2 9

In Exercises 1116, match the equation with its graph. [The graphs are labeled (a)(f).] x2 y y 2 16

In Exercises 1722, find the focus and directrix of the parabola. Then sketch the parabola. y 1 2x 2

In Exercises 1722, find the focus and directrix of the parabola. Then sketch the parabola. y  4x 2

In Exercises 1722, find the focus and directrix of the parabola. Then sketch the parabola. y 2 3x 2  6x

In Exercises 1722, find the focus and directrix of the parabola. Then sketch the parabola. y y 2 3x

In Exercises 1722, find the focus and directrix of the parabola. Then sketch the parabola. x 2 0 2 12y 0

In Exercises 1722, find the focus and directrix of the parabola. Then sketch the parabola. x y x 2 0

In Exercises 2328, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 2 2, 0 x y

In Exercises 2328, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 0, 1 2 2, 0 x y

In Exercises 2328, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. y  2

In Exercises 2328, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: y  2 x 4

In Exercises 2328, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Passes through the point 2, 14; vertical axis

In Exercises 2328, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin.

In Exercises 29 and 30, find the standard form of the equation of the parabola and determine the coordinates of the focus.

In Exercises 29 and 30, find the standard form of the equation of the parabola and determine the coordinates of the focus.

Flashlight The light bulb in a flashlight is at the focus of the parabolic reflector, 1.5 centimeters from the vertex of the reflector (see figure). Write an equation for a cross section of the flashlights reflector with its focus on the positive -axis and its vertex at the origin.

Satellite Antenna Write an equation for a cross section of the parabolic satellite dish antenna shown in the figure

Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cables touch the roadway at the midpoint between the towers. (a) Sketch the bridge in a rectangular coordinate system with the origin at the center of the roadway. Label the coordinates of known points. (b) Write an equation that models the cables. (c) Complete the table by finding the height of the suspension cables over the roadway at a distance of meters from the center of the bridge

. Beam Deflection A simply supported beam (see figure) is 64 feet long and has a load at the center. The deflection of the beam at its center is 1 inch. The shape of the deflected beam is parabolic.a) Find an equation of the parabola. (Assume that the origin is at the center of the beam.) (b) How far from the center of the beam is the deflection inch?.

Exercises 3544, find the standard form of the equation of the ellipse with the given characteristics and center at the origin.

Exercises 3544, find the standard form of the equation of the ellipse with the given characteristics and center at the origin.

In Exercises 3544, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 5, 0; 2, 0 x

In Exercises 3544, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 0, 8; 0, 4 5,

In Exercises 3544, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 5, 0; 0 14

In Exercises 3544, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 2, 0; 10

In Exercises 3544, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 9, 0; 6

In Exercises 3544, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 0, 10; 2

In Exercises 3544, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 0, 5; 4, 2 0,

In Exercises 3544, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 0, 42, 0 0,

In Exercises 4554, find the vertices of the ellipse. Then sketch the ellipse. x2 25 y 2 16 1

In Exercises 4554, find the vertices of the ellipse. Then sketch the ellipse. x2 121 y 2 144 1

In Exercises 4554, find the vertices of the ellipse. Then sketch the ellipse. 25 9 y2 16 9 1

In Exercises 4554, find the vertices of the ellipse. Then sketch the ellipse. x 2 4 y 2 1 4 1

In Exercises 4554, find the vertices of the ellipse. Then sketch the ellipse. x 2 36 y 2 7 1

In Exercises 4554, find the vertices of the ellipse. Then sketch the ellipse. x 2 28 y 2 64 1

In Exercises 4554, find the vertices of the ellipse. Then sketch the ellipse.4x 2 36 2 y2 1

In Exercises 4554, find the vertices of the ellipse. Then sketch the ellipse. 4x2 9y 4x 2 36 2

In Exercises 4554, find the vertices of the ellipse. Then sketch the ellipse. 16x2 1 81y2 1

In Exercises 4554, find the vertices of the ellipse. Then sketch the ellipse. 1 100x2 1 49y 2 1 1

Architecture A fireplace arch is to be constructed in the shape of a semiellipse. The opening is to have a height of 2 feet at the center and a width of 6 feet along the base (see figure). The contractor draws the outline of the ellipse on the wall by the method shown in Figure 4.22. Give the required positions of the tacks and the length of the string

Architecture A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 50 feet and a height at the center of 10 feet. (a) Sketch the arch of the tunnel on a rectangular coordinate system with the center of the road entering the tunnel at the origin. Identify the coordinates of the known points. (b) Find an equation of the semielliptical arch over the tunnel. (c) You are driving a moving truck that has a width of 8 feet and a height of 9 feet. Will the moving truck clear the opening of the arch?

Architecture Repeat Exercise 56 for a semielliptical arch with a major axis of 40 feet and a height at the center of 15 feet. The dimensions of the truck are 10 feet wide by 14 feet high.

Geometry A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major axis is called a latus rectum of the ellipse. An ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it yields other points on the curve (see figure). Show that the length of each latus rectum

In Exercises 5962, sketch the ellipse, using the latera recta (see Exercise 58). x2 4 y 2 1 1

In Exercises 5962, sketch the ellipse, using the latera recta (see Exercise 58). x2 9 y 2 16 1

In Exercises 5962, sketch the ellipse, using the latera recta (see Exercise 58). 9x 2 15 2 4y 2 36

In Exercises 5962, sketch the ellipse, using the latera recta (see Exercise 58). 9x 2 15 2 4y 2 36

In Exercises 6370, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 0, 2; 0, 6 5x

In Exercises 6370, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 4, 0; 5, 0 0,

In Exercises 6370, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 1, 0; y 3x

In Exercises 6370, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 0, 3; y 3x

In Exercises 6370, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 0, 8; y 4x 0

In Exercises 6370, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 4 10, 0; x 0

In Exercises 6370, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin.0, 3; 2, 5 y 

In Exercises 6370, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 2, 0; 3, 3 0,

In Exercises 7180, find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. x 1 2 y 2 1

In Exercises 7180, find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. x 2 9 y 2 16 x 1 2

In Exercises 7180, find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. y 2 1 x2 4 1

In Exercises 7180, find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. y 2 9 x2 1 1 y

In Exercises 7180, find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. y 2 49 x2 196 1

In Exercises 7180, find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. x2 36 y 2 4 1

In Exercises 7180, find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. 4y 2 36 2 x2 1

In Exercises 7180, find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. 4y2 9x 4y 2 36

In Exercises 7180, find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. 36y2 1 100x2 1

In Exercises 7180, find the vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. 144x2 1 169y2 1 1

Art A sculpture has a hyperbolic cross section (see figure). (a) Write an equation that models the curved sides of the sculpture. (b) Each unit on the coordinate plane represents 1 foot. Find the width of the sculpture at a height of 5 feet.

. Optics A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at the focus will be reflected to the other focus. The focus of a hyperbolic mirror (see figure) has coordinates Find the vertex of the mirror when its mount at the top edge of the mirror has coordinates

Aeronautics When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. If the airplane is flying parallel to the ground, then the sound waves intersect the ground in a hyperbola with the airplane directly above its center. A sonic boom is heard along the hyperbola. You hear a sonic boom that is audible along a hyperbola with the equation where and are measured in miles. What is the shortest horizontal distance you could be from the airplane?

Navigation Long-distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (186,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations 300 miles apart are positioned on a rectangular coordinate system at points with coordinates and and that a ship is traveling on a path with coordinates , as shown in the figure. Find the -coordinate of the position of the ship when the time difference between the pulses from the transmitting stations is 1000 micro-seconds (0.001 second).

In Exercises 8588, determine whether the statement is true or false. Justify your answer. The equation represents a circle

In Exercises 8588, determine whether the statement is true or false. Justify your answer. The major axis of the ellipse is vertical.

In Exercises 8588, determine whether the statement is true or false. Justify your answer. It is possible for a parabola to intersect its directrix

In Exercises 8588, determine whether the statement is true or false. Justify your answer. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical.

Area of an Ellipse Consider the ellipse (a) The area of the ellipse is given by Write the area of the ellipse as a function of (b) Find the equation of an ellipse with an area of 264 square centimeters. (c) Complete the table using your equation from part (a), and make a conjecture about the shape of the ellipse with maximum area. (d) Use a graphing utility to graph the area function and use the graph to support your conjecture in part (c).

HOW DO YOU SEE IT? In parts (a)(d), describe in words how a plane could intersect the double-napped cone to form the conic section (see figure). (a) Circle (b) Ellipse (c) Parabola (d) Hyperbola

Think About It How can you tell if an ellipse is a circle from the equation?

Think About It Is the graph of an ellipse? Explain

Think About It The graph of is a degenerate conic. Sketch this graph and identify the degenerate conic.

Think About It Which part of the graph of the ellipse is represented by each equation? (Do not graph.) x  3 24 y2 y 2 39 x2

Writing At the beginning of this section, you learned that each type of conic section can be formed by the intersection of a plane and a double-napped cone. Write a short paragraph describing examples of physical situations in which hyperbolas are formed.

Writing Write a paragraph discussing the changes in the shape and orientation of the graph of the ellipse as increases from 1 to 8.

Using a Definition Use the definition of an ellipse to derive the standard form of the equation of an ellipse

Using a Definition Use the definition of a hyperbola to derive the standard form of the equation of a hyperbola.

Drawing Ellipses An ellipse can be drawn using two thumbtacks placed at the foci of the ellipse, a string of fixed length (greater than the distance between the tacks), and a pencil, as shown in Figure 4.22. Try doing this. Vary the length of the string and the distance between the thumbtacks. Explain how to obtain ellipses that are almost circular. Explain how to obtain ellipses that are long and narrow.

Match the description of the conic with its standard equation. The equations are labeled (a)(f). Hyperbola with horizontal transverse axis

Match the description of the conic with its standard equation. The equations are labeled (a)(f). Ellipse with vertical major axis

Match the description of the conic with its standard equation. The equations are labeled (a)(f). Parabola with vertical axis

Match the description of the conic with its standard equation. The equations are labeled (a)(f). Hyperbola with vertical transverse axis

Match the description of the conic with its standard equation. The equations are labeled (a)(f). Ellipse with horizontal major axis

Match the description of the conic with its standard equation. The equations are labeled (a)(f). Parabola with horizontal axis"In Exercises 714, identify the conic. Then describe the translation of the graph of the conic. x 22 42x 22 y 12 4

97811339"

In Exercises 714, identify the conic. Then describe the translation of the graph of the conic. y 1x 22 42x 22 y 12 4"In Exercises 714, identify the conic. Then describe the translation of the graph of the conic. y 2 41x 12 6 2

In Exercises 714, identify the conic. Then describe the translation of the graph of the conic. x 12 y 3y 2 41x 12 6 2

x

y

-2"

In Exercises 714, identify the conic. Then describe the translation of the graph of the conic. y 32 4 x 12 1 x y

In Exercises 714, identify the conic. Then describe the translation of the graph of the conic. x 22 9 y 12 4 1 y 32

In Exercises 714, identify the conic. Then describe the translation of the graph of the conic. x 42 9 y 22 16 1 x y

In Exercises 714, identify the conic. Then describe the translation of the graph of the conic. x 22 4 y 32 9 1 x 42

In Exercises 1520, find the center and radius of the circle. x 2 1 2 y2 49

In Exercises 1520, find the center and radius of the circle. x2 y x 2 1 2

In Exercises 1520, find the center and radius of the circle. x 42 (y 52 36 x2

In Exercises 1520, find the center and radius of the circle. x 82 y 12 144 x

In Exercises 1520, find the center and radius of the circle. x 12 24 2 y2 10 x 

In Exercises 1520, find the center and radius of the circle. x2 y 12x 12 24 2 y2

In Exercises 2126, write the equation of the circle in standard form, and then find its center and radius. x2 y2 2x 6y 9 0 x

In Exercises 2126, write the equation of the circle in standard form, and then find its center and radius. x2 y2 10x 6y 25 0 x2

In Exercises 2126, write the equation of the circle in standard form, and then find its center and radius. x2 y 2 8x 0 x

In Exercises 2126, write the equation of the circle in standard form, and then find its center and radius.2x2 2y2 2x 2y 7 0 x2

In Exercises 2126, write the equation of the circle in standard form, and then find its center and radius. 4x2 4y2 12x 24y 41 0 2

In Exercises 2126, write the equation of the circle in standard form, and then find its center and radius. 9x2 9y2 54x 36y 17 0 4

In Exercises 2734, find the vertex, focus, and directrix of the parabola. Then sketch the parabola. x 12 8y 20 9x2 

In Exercises 2734, find the vertex, focus, and directrix of the parabola. Then sketch the parabola. x 2y 42 0 x 12

In Exercises 2734, find the vertex, focus, and directrix of the parabola. Then sketch the parabola. x 2y 42 0 x 12

In Exercises 2734, find the vertex, focus, and directrix of the parabola. Then sketch the parabola. x 1 2 2 y 4y 31 2

In Exercises 2734, find the vertex, focus, and directrix of the parabola. Then sketch the parabola. y 2 2y 33 0 1 4x 2 2x 5 x 1

In Exercises 2734, find the vertex, focus, and directrix of the parabola. Then sketch the parabola. x y y 2 2y 33 0 1 4

In Exercises 2734, find the vertex, focus, and directrix of the parabola. Then sketch the parabola. y 2 6y 8x 25 0

In Exercises 2734, find the vertex, focus, and directrix of the parabola. Then sketch the parabola. y 2 4y 4x 0 y

In Exercises 3540, find the standard form of the equation of the parabola with the given characteristics. 3, 2; 1, 2 y 2

In Exercises 3540, find the standard form of the equation of the parabola with the given characteristics. 1, 2; 1, 0 3, 2;

In Exercises 3540, find the standard form of the equation of the parabola with the given characteristics. 0, 4; y 2 1,

In Exercises 3540, find the standard form of the equation of the parabola with the given characteristics. 2, 1; x 1 0,

In Exercises 3540, find the standard form of the equation of the parabola with the given characteristics. 4, 4; x 4 2,

In Exercises 3540, find the standard form of the equation of the parabola with the given characteristics. 40. Focus: directrix: 0, 0; y 4 4,

A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour (see figure). When this velocity is multiplied by the satellite has the minimum velocity necessary to escape Earths gravity and follow a parabolic path with the center of Earth as the focus. (a) Find the escape velocity of the satellite. (b) Find an equation of its path (assume that the radius of Earth is 4000 miles)

Fluid Flow Water is flowing from a horizontal pipe 48 feet above the ground. The falling stream of water has the shape of a parabola whose vertex is at the end of the pipe (see figure). The stream of water strikes the ground at the point Find the equation of the path taken by the water.

In Exercises 43 and 44, consider the path of an object projected horizontally with a velocity of feet per second at a height of feet, where the model for the path is In this model (in which air resistance is disregarded), is the height (in feet) of the projectile and is the horizontal distance (in feet) the projectile travels. A ball is thrown from the top of a 100-foot tower with a velocity of 28 feet per second. (a) Find the equation of the parabolic path. (b) How far does the ball travel horizontally before striking the ground?

In Exercises 43 and 44, consider the path of an object projected horizontally with a velocity of feet per second at a height of feet, where the model for the path is In this model (in which air resistance is disregarded), is the height (in feet) of the projectile and is the horizontal distance (in feet) the projectile travels.A cargo plane is flying at an altitude of 500 feet and a speed of 135 miles per hour. A supply crate is dropped from the plane. How many feet will the crate travel horizontally before it hits the ground?

In Exercises 4552, find the center, foci, and vertices of the ellipse. Then sketch the ellipse. x 12 9 y 52 25 1 x y

In Exercises 4552, find the center, foci, and vertices of the ellipse. Then sketch the ellipse. x 62 4 y 72 16 1 x 12

In Exercises 4552, find the center, foci, and vertices of the ellipse. Then sketch the ellipse. x 22 y 42 1 4 1 x 6

In Exercises 4552, find the center, foci, and vertices of the ellipse. Then sketch the ellipse. x 32 25 9 y 82 1 x 22

In Exercises 4552, find the center, foci, and vertices of the ellipse. Then sketch the ellipse. 9x 2 25y 2 36x 50y 52 0 x

In Exercises 4552, find the center, foci, and vertices of the ellipse. Then sketch the ellipse. 16x 2 25y 2 32x 50y 16 0 9

In Exercises 4552, find the center, foci, and vertices of the ellipse. Then sketch the ellipse. 9x 2 4y 2 36x 16y 16 0 1

In Exercises 4552, find the center, foci, and vertices of the ellipse. Then sketch the ellipse. 9x 2 4y 2 36x 8y 31 0 9

In Exercises 5360, find the standard form of the equation of the ellipse with the given characteristics. Vertices: minor axis of length 2

In Exercises 5360, find the standard form of the equation of the ellipse with the given characteristics.54. Vertices: minor axis of length 6In Exercises 5360, find the standard form of the equation of the ellipse with the given characteristics. 55. Foci: major axis of length 8In Exercises 5360, find the standard form of the equation of the ellipse with the given characteristics. 56. Foci: major axis of length 16"In Exercises 5360, find the standard form of the equation of the ellipse with the given characteristics. 0, 4; a 2c; 4, 4, 4, 4

0, 0, 0"

In Exercises 5360, find the standard form of the equation of the ellipse with the given characteristics. 3, 2; a 3c; 1, 2, 5, 2 0, 4;

In Exercises 5360, find the standard form of the equation of the ellipse with the given characteristics. 3, 0, 7, 0; 0, 0, 4, 0 3, 2; a

In Exercises 5360, find the standard form of the equation of the ellipse with the given characteristics. 2, 0, 2, 4; 2, 1, 2, 3 3, 0, 7

In Exercises 61 and 62, use the following information. The flatness of an ellipse is measured by its eccentricity defined by where When an ellipse is nearly circular, is close to 0. When an ellipse is elongated, is close to 1 (see figures). Find the standard form of the equation of the ellipse with vertices and eccentricity e 3

In Exercises 61 and 62, use the following information. The flatness of an ellipse is measured by its eccentricity defined by where When an ellipse is nearly circular, is close to 0. When an ellipse is elongated, is close to 1 (see figures). Find the standard form of the equation of the ellipse with vertices and eccentricity e 1 2 0, 8. e

Planetary Motion The dwarf planet Pluto moves in an elliptical orbit with the sun at one of the foci, as shown in the figure. The length of half of the major axis, is miles, and the eccentricity is 0.249. Find the least distance (perihelion) and the greatest distance (aphelion) of Pluto from the center of the sun.

Australian Football In Australia, football by Australian Rules (or rugby) is played on elliptical fields. The field can be a maximum of 155 meters wide and a maximum of 185 meters long. Let the center of a field of maximum size be represented by the point Find the standard form of the equation of the ellipse that represents this field. (Source: Australian Football League)

In Exercises 6572, find the center, foci, and vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. x 22 16 y 12 9 1 0, 77

In Exercises 6572, find the center, foci, and vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. x 12 144 y 42 25 1 x 22

In Exercises 6572, find the center, foci, and vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. y 62 x 22 1 x 12

In Exercises 6572, find the center, foci, and vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. y 12 1 4 x 32 1 9 1 y 62

In Exercises 6572, find the center, foci, and vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. x 2 9y 2 2x 54y 85 0 y 1

In Exercises 6572, find the center, foci, and vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. 16y 2 x 2 2x 64y 62 0 x

In Exercises 6572, find the center, foci, and vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. 9x 2 y 2 36x 6y 18 0 16y

In Exercises 6572, find the center, foci, and vertices of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. x 2 9y 2 36y 72 0 9x

In Exercises 7380, find the standard form of the equation of the hyperbola with the given characteristics. 73. Vertices: foci: 0, 2, 0, 0; 0, 3, 0, 1 x 2 9y 2

In Exercises 7380, find the standard form of the equation of the hyperbola with the given characteristics. 74. Vertices: foci: 1, 2, 5, 2; 0, 2, 6, 2 0, 2, 0

In Exercises 7380, find the standard form of the equation of the hyperbola with the given characteristics. 75. Vertices: foci: 2, 0, 6, 0; 0, 0, 8, 0 1, 2, 5

In Exercises 7380, find the standard form of the equation of the hyperbola with the given characteristics. 76. Vertices: foci: 2, 3, 2, 3; 2, 5, 2, 5 2, 0, 6,

In Exercises 7380, find the standard form of the equation of the hyperbola with the given characteristics. 77. Vertices: passes through the point 2, 3, 2, 3; 0, 5 2, 3,

In Exercises 7380, find the standard form of the equation of the hyperbola with the given characteristics. 78. Vertices: passes through the point 2, 1, 2, 1; 4, 3 2, 3,

In Exercises 7380, find the standard form of the equation of the hyperbola with the given characteristics. 79. Vertices: asymptotes: 3 0, 2, 6, 2; x 2,

In Exercises 7380, find the standard form of the equation of the hyperbola with the given characteristics. 80. Vertices: asymptotes: 3 3, 0, 3, 4; x y 2

In Exercises 8190, identify the conic by writing its equation in standard form. Then sketch its graph.x 2 y 2 6x 4y 9 0 y

In Exercises 8190, identify the conic by writing its equation in standard form. Then sketch its graph. x 2 4y 2 6x 16y 21 0 x

In Exercises 8190, identify the conic by writing its equation in standard form. Then sketch its graph. y 2 4y 4x 0 2 x2 4y 0 x 2

In Exercises 8190, identify the conic by writing its equation in standard form. Then sketch its graph. y y 2 4y 4x 0 2

In Exercises 8190, identify the conic by writing its equation in standard form. Then sketch its graph. 16y2 128x 8y 7 0 y

In Exercises 8190, identify the conic by writing its equation in standard form. Then sketch its graph. 4x 2 y 2 4x 3 0 16y

In Exercises 8190, identify the conic by writing its equation in standard form. Then sketch its graph. 9x2 16y 2 36x 128y 148 0

In Exercises 8190, identify the conic by writing its equation in standard form. Then sketch its graph.25x 2 10x 200y 119 0 9x2

In Exercises 8190, identify the conic by writing its equation in standard form. Then sketch its graph. 16x2 16y2 16x 24y 3 0 25

In Exercises 8190, identify the conic by writing its equation in standard form. Then sketch its graph. 4x 2 3y 2 8x 24y 51 0 1

In Exercises 91 and 92, determine whether the statement is true or false. Justify your answer. The conic represented by the equation is an ellipse

In Exercises 91 and 92, determine whether the statement is true or false. Justify your answer. The graphs of and do not intersect.

Alternate Form of the Equation of an Ellipse Consider the ellipse (a) Show that the equation of the ellipse can be written as where is the eccentricity (see Exercises 61 and 62). (b) Use a graphing utility to graph the ellipse for 0.75, 0.5, 0.25, and 0. Make a conjecture about the change in the shape of the ellipse as approaches 0.

HOW DO YOU SEE IT? Consider the ellipse shown. (a) Find the center, vertices, and foci of the ellipse. (b) Find the standard form of the equation of the ellipse.

In Exercises 14, find the domain of the rational function and discuss the behavior of near any excluded -values. fx3x x 10 f

In Exercises 14, find the domain of the rational function and discuss the behavior of near any excluded -values. fx4x3 2 5x fx

In Exercises 14, find the domain of the rational function and discuss the behavior of near any excluded -values. fx8 x2 10x 24 fx

In Exercises 14, find the domain of the rational function and discuss the behavior of near any excluded -values. fxx2 x 2 x2 4x 4 fx8

In Exercises 510, find all vertical and horizontal asymptotes of the graph of the function. fx4 x 3 f

In Exercises 510, find all vertical and horizontal asymptotes of the graph of the function. fx2x2 5x 3 x2 2 fx4

In Exercises 510, find all vertical and horizontal asymptotes of the graph of the function. gxx2 x2 4 fx

In Exercises 510, find all vertical and horizontal asymptotes of the graph of the function. gx1 x 32 gxx2

In Exercises 510, find all vertical and horizontal asymptotes of the graph of the function. hx5x 20 x2 2x 24 gx1

In Exercises 510, find all vertical and horizontal asymptotes of the graph of the function. hxx3 4x2 x2 3x 2 hx5

Average Cost A business has a cost function of where is measured in dollars and is the number of units produced. The average cost per unit function is given by Determine the average cost per unit as increases without bound. (Find the horizontal asymptote.)

Seizure of Illegal Drugs The cost (in millions of dollars) for the federal government to seize of an illegal drug as it enters the country is given by (a) Use a graphing utility to graph the cost function. (b) Find the costs of seizing 25%, 50%, and 75% of the drug. (c) According to this model, would it be possible to seize 100% of the drug?

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx3 2x 2 4.

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx4 x f

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. gx2 x 1 x fx

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. hxx 4 x 7 gx2

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. px5x2 4x2 1 h

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx2x x 2 4 px

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fxx x 2 1 fx

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. hx9 x 32 fxx

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx6x2 x 2 1 hx

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. y 2x 2 x 2 4 f

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx6x2 11x 3 3x2 x y 2

In Exercises 1324, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx6x2 7x 2 4x2 1 fx6x2

In Exercises 2530, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx2x3 x2 1 f

In Exercises 2530, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fxx2 1 x 1 fx

In Exercises 2530, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fxx2 3x 10 x 2 fx

In Exercises 2530, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fxx3 x2 25 fxx

In Exercises 2530, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx3x3 2x2 3x 2 3x2 x 4 fxx3

In Exercises 2530, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. fx3x3 4x2 12x 16 3x2 5x 2 fx3x

Minimum Cost The cost of producing units of a product is and the average cost per unit is given by (a) Sketch the graph of the average cost function. (b) Find the average costs of producing 10,000, and 100,000 units. (c) By increasing the level of production, what is the least average cost per unit you can obtain? Explain your reasoning.

Page Design A page that is inches wide and inches high contains 30 square inches of print. The top and bottom margins are each 2 inches deep and the margins on each side are 2 inches wide. (a) Show that the total area of the page is (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper is used. Verify your answer numerically using the table feature of the graphing utility.

Photosynthesis The amount of uptake (in milligrams per square decimeter per hour) at optimal temperatures and with the natural supply of is approximated by the model where is the light intensity (in watts per square meter). Use a graphing utility to graph the function and determine the limiting amount of uptake.

Medicine The concentration of a chemical in the bloodstream hours after injection into muscle tissue is given by (a) Determine the horizontal asymptote of the graph of the function and interpret its meaning in the context of the problem. (b) Use a graphing utility to graph the function and approximate the time when the bloodstream concentration is greatest.

In Exercises 3542, identify the conic. y 2 16 2 16x 4

In Exercises 3542, identify the conic. 16x2 y y 2 16

In Exercises 3542, identify the conic. x2 64 y2 4 1

In Exercises 3542, identify the conic. x2 1 y2 36 1 x

In Exercises 3542, identify the conic. x 2 400 2 20y 0

In Exercises 3542, identify the conic. x2 y x 2 400

In Exercises 3542, identify the conic. y2 49 x2 144 1 x

x2 49 y2 144 1 y

In Exercises 4348, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus 6, 0 x2

In Exercises 4348, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus 0, 7 6

In Exercises 4348, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: y 3 0

In Exercises 4348, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix x 3

In Exercises 4348, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 47. Passes through the point horizontal axis

In Exercises 4348, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 48. Passes through the point vertical axis

Satellite Antenna A cross section of a large parabolic antenna (see figure) is modeled by The receiving and transmitting equipment is positioned at the focus. Find the coordinates of the focus.

Suspension Bridge Each cable of a suspension bridge is suspended (in the shape of a parabola) between two towers (see figure). (a) Find the coordinates of the focus. (b) Write an equation that models the cables.

In Exercises 5156, find the standard form of the equation of the ellipse with the given characteristics and center at the origin.

In Exercises 5156, find the standard form of the equation of the ellipse with the given characteristics and center at the origin.

In Exercises 5156, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 7, 06, 0 x

In Exercises 5156, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 3, 0 major axis of length 12

In Exercises 5156, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 14, 0 minor axis of length 10

In Exercises 5156, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 0, 62, 2 14

Architecture A semielliptical archway is to be formed over the entrance to an estate (see figure). The arch is to be set on pillars that are 10 feet apart and is to have a height (atop the pillars) of 4 feet. Where should the foci be placed in order to sketch the arch?

Wading Pool You are building a wading pool that is in the shape of an ellipse. Your plans give an equation for the elliptical shape of the pool measured in feet as Find the longest distance across the pool, the shortest distance, and the distance between the foci.

In Exercises 5962, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 0, 10, 5 978

In Exercises 5962, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 4, 06, 0 0,

1, 0y 2x 4

In Exercises 5962, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 5 0, 2x

In Exercises 6366, find the standard form of the equation of the parabola with the given characteristics. 4, 2; 4, 0 4.4

In Exercises 6366, find the standard form of the equation of the parabola with the given characteristics. 2, 0; 0, 0 4,

In Exercises 6366, find the standard form of the equation of the parabola with the given characteristics. 8, 8; y 1 2,

In Exercises 6366, find the standard form of the equation of the parabola with the given characteristics. 0, 5; x 6 8,

In Exercises 6770, find the standard form of the equation of the ellipse with the given characteristics. 0, 2, 4, 2; 0, minor axis of length 2

In Exercises 6770, find the standard form of the equation of the ellipse with the given characteristics. 5, 0, 5, 12; 0, minor axis of length 10

In Exercises 6770, find the standard form of the equation of the ellipse with the given characteristics. 2, 2, 2, 8; 2, 0, 2, 6 5, 0, 5,

In Exercises 6770, find the standard form of the equation of the ellipse with the given characteristics. 9, 4, 11, 4; 5, 4, 7, 4 2, 2, 2, 8; 2

In Exercises 7176, find the standard form of the equation of the hyperbola with the given characteristics. 10, 3, 6, 3; 12, 3, 8, 3 9, 4, 11,

In Exercises 7176, find the standard form of the equation of the hyperbola with the given characteristics. 2, 2, 2, 2; 4, 2, 4, 2 10, 3, 6,

In Exercises 7176, find the standard form of the equation of the hyperbola with the given characteristics. 0, 0, 0, 4; 2, 2 2, 25 1 0, 0

In Exercises 7176, find the standard form of the equation of the hyperbola with the given characteristics. 6, 7; 2 y x 7 1

In Exercises 7176, find the standard form of the equation of the hyperbola with the given characteristics. 0, 0, 8, 0; y 2x 4 y 1 2

In Exercises 7176, find the standard form of the equation of the hyperbola with the given characteristics. 3, 2; y 2x 3 0, 0,

In Exercises 7784, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation. x 2 6x 2y 9 0 3

In Exercises 7784, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation. y 2 12y 8x 20 0 x

In Exercises 7784, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation.x 2 y 2 2x 4y 5 0 y

In Exercises 7784, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation. 16x 2 16y 2 16x 24y 3 0 x

In Exercises 7784, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation. x 2 9y 2 10x 18y 25 0 1

In Exercises 7784, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation. 4x 2 y 2 16x 15 0 x

In Exercises 7784, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation. 9x2 y2 72x 8y 119 0 4x

In Exercises 7784, identify the conic by writing its equation in standard form. Then sketch its graph and describe the translation. x 2 9y 2 10x 18y 7 0 9

Architecture A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters (see figure). How wide is the archway at ground level?

Architecture A church window (see figure) is bounded above by a parabola and below by the arc of a circle. (a) Find equations for the parabola and the circle. (b) Complete the table by filling in the vertical distance between the circle and the parabola for each given value of

Running Path Let represent a water fountain located in a city park. Each day you run through the park along a path given by where and are measured in meters. (a) What type of conic is your path? Explain your reasoning. (b) Write the equation of the path in standard form. Sketch a graph of the equation. (c) After you run, you walk to the water fountain. If you stop running at how far must you walk for a drink of water?

In Exercises 88 and 89, determine whether the statement is true or false. Justify your answer. The domain of a rational function can never be the set of all real numbers.

In Exercises 88 and 89, determine whether the statement is true or false. Justify your answer. The graph of the equation can be a single point