 4.4.4.1: Match the description of the conic with its standard equation. The ...
 4.4.1.1: Functions of the form where and are polynomials and is not the zero...
 4.4.2.1: For the rational function if the degree of is exactly one more than...
 4.1: In Exercises 14, find the domain of the rational function and discu...
 4.4.3.1: A ________ is the intersection of a plane and a doublenapped cone.
 4.4.4.2: Match the description of the conic with its standard equation. The ...
 4.4.1.2: When as from the left or the right, is a ________ ________ of the g...
 4.4.2.2: The graph of has a ________ asymptote at
 4.2: In Exercises 14, find the domain of the rational function and discu...
 4.4.3.2: The equation is the standard form of the equation of a ________ wit...
 4.4.4.3: Match the description of the conic with its standard equation. The ...
 4.4.1.3: When as is a ________ ________ of the graph of f
 4.4.2.3: In Exercises 3 6, use the graph of to sketch the graph of g gx2x4f
 4.3: In Exercises 14, find the domain of the rational function and discu...
 4.4.3.3: A ________ is the set of all points in a plane that are equidistant...
 4.4.4.4: Match the description of the conic with its standard equation. The ...
 4.4.1.4: The graph of is called a ________.
 4.4.2.4: In Exercises 3 6, use the graph of to sketch the graph of g gx2x 54
 4.4: In Exercises 14, find the domain of the rational function and discu...
 4.4.3.4: The ________ of a parabola is the midpoint between the focus and th...
 4.4.4.5: Match the description of the conic with its standard equation. The ...
 4.4.1.5: In Exercises 512, find the domain of the function, and discuss the ...
 4.4.2.5: In Exercises 3 6, use the graph of to sketch the graph of g gx 2xgx
 4.5: In Exercises 510, find all vertical and horizontal asymptotes of th...
 4.4.3.5: The line that passes through the focus and the vertex of a parabola...
 4.4.4.6: Match the description of the conic with its standard equation. The ...
 4.4.1.6: In Exercises 512, find the domain of the function, and discuss the ...
 4.4.2.6: In Exercises 3 6, use the graph of to sketch the graph of g gx1x 2g
 4.6: In Exercises 510, find all vertical and horizontal asymptotes of th...
 4.4.3.6: An ________ is the set of all points in a plane the sum of whose di...
 4.4.4.7: 97811339"
 4.4.1.7: In Exercises 512, find the domain of the function, and discuss the ...
 4.4.2.7: In Exercises 710, use the graph of to sketch the graph of g gx3x 2 1fx
 4.7: In Exercises 510, find all vertical and horizontal asymptotes of th...
 4.4.3.7: The chord joining the vertices of an ellipse is called the ________...
 4.4.4.8: In Exercises 714, identify the conic. Then describe the translation...
 4.4.1.8: In Exercises 512, find the domain of the function, and discuss the ...
 4.4.2.8: In Exercises 710, use the graph of to sketch the graph of g gx 3x 22
 4.8: In Exercises 510, find all vertical and horizontal asymptotes of th...
 4.4.3.8: The chord perpendicular to the major axis at the center of an ellip...
 4.4.4.9: In Exercises 714, identify the conic. Then describe the translation...
 4.4.1.9: In Exercises 512, find the domain of the function, and discuss the ...
 4.4.2.9: In Exercises 710, use the graph of to sketch the graph of g gx3x 12...
 4.9: In Exercises 510, find all vertical and horizontal asymptotes of th...
 4.4.3.9: . A ________ is the set of all points in a plane the difference of ...
 4.4.4.10: x
 4.4.1.10: In Exercises 512, find the domain of the function, and discuss the ...
 4.4.2.10: In Exercises 710, use the graph of to sketch the graph of g gx1x 2g
 4.10: In Exercises 510, find all vertical and horizontal asymptotes of th...
 4.4.3.10: The line segment connecting the vertices of a hyperbola is called t...
 4.4.4.11: y
 4.4.1.11: In Exercises 512, find the domain of the function, and discuss the ...
 4.4.2.11: In Exercises 1114, use the graph of to sketch the graph of g gx4x 23fx
 4.11: Average Cost A business has a cost function of where is measured in...
 4.4.3.11: In Exercises 1116, match the equation with its graph. [The graphs a...
 4.4.4.12: 2"
 4.4.1.12: In Exercises 512, find the domain of the function, and discuss the ...
 4.4.2.12: In Exercises 1114, use the graph of to sketch the graph of g gx4x 3 24
 4.12: Seizure of Illegal Drugs The cost (in millions of dollars) for the ...
 4.4.3.12: In Exercises 1116, match the equation with its graph. [The graphs a...
 4.4.4.13: In Exercises 714, identify the conic. Then describe the translation...
 4.4.1.13: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.13: In Exercises 1114, use the graph of to sketch the graph of g gx 4x 3gx
 4.13: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.13: In Exercises 1116, match the equation with its graph. [The graphs a...
 4.4.4.14: In Exercises 714, identify the conic. Then describe the translation...
 4.4.1.14: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.14: In Exercises 1114, use the graph of to sketch the graph of g gx2x 3
 4.14: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.14: In Exercises 1116, match the equation with its graph. [The graphs a...
 4.4.4.15: In Exercises 714, identify the conic. Then describe the translation...
 4.4.1.15: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.15: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.15: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.15: In Exercises 1116, match the equation with its graph. [The graphs a...
 4.4.4.16: In Exercises 714, identify the conic. Then describe the translation...
 4.4.1.16: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.16: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.16: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.16: In Exercises 1116, match the equation with its graph. [The graphs a...
 4.4.4.17: In Exercises 1520, find the center and radius of the circle. x 2 1 ...
 4.4.1.17: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.17: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.17: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.17: In Exercises 1722, find the focus and directrix of the parabola. Th...
 4.4.4.18: In Exercises 1520, find the center and radius of the circle. x2 y x...
 4.4.1.18: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.18: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.18: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.18: In Exercises 1722, find the focus and directrix of the parabola. Th...
 4.4.4.19: In Exercises 1520, find the center and radius of the circle. x 42 (...
 4.4.1.19: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.19: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.19: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.19: In Exercises 1722, find the focus and directrix of the parabola. Th...
 4.4.4.20: In Exercises 1520, find the center and radius of the circle. x 82 y...
 4.4.1.20: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.20: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.20: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.20: In Exercises 1722, find the focus and directrix of the parabola. Th...
 4.4.4.21: In Exercises 1520, find the center and radius of the circle. x 12 2...
 4.4.1.21: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.21: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.21: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.21: In Exercises 1722, find the focus and directrix of the parabola. Th...
 4.4.4.22: In Exercises 1520, find the center and radius of the circle. x2 y 1...
 4.4.1.22: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.22: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.22: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.22: In Exercises 1722, find the focus and directrix of the parabola. Th...
 4.4.4.23: In Exercises 2126, write the equation of the circle in standard for...
 4.4.1.23: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.23: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.23: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.23: In Exercises 2328, find the standard form of the equation of the pa...
 4.4.4.24: In Exercises 2126, write the equation of the circle in standard for...
 4.4.1.24: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.24: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.24: In Exercises 1324, (a) state the domain of the function, (b) identi...
 4.4.3.24: In Exercises 2328, find the standard form of the equation of the pa...
 4.4.4.25: In Exercises 2126, write the equation of the circle in standard for...
 4.4.1.25: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.25: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.25: In Exercises 2530, (a) state the domain of the function, (b) identi...
 4.4.3.25: In Exercises 2328, find the standard form of the equation of the pa...
 4.4.4.26: In Exercises 2126, write the equation of the circle in standard for...
 4.4.1.26: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.26: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.26: In Exercises 2530, (a) state the domain of the function, (b) identi...
 4.4.3.26: In Exercises 2328, find the standard form of the equation of the pa...
 4.4.4.27: In Exercises 2126, write the equation of the circle in standard for...
 4.4.1.27: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.27: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.27: In Exercises 2530, (a) state the domain of the function, (b) identi...
 4.4.3.27: In Exercises 2328, find the standard form of the equation of the pa...
 4.4.4.28: In Exercises 2126, write the equation of the circle in standard for...
 4.4.1.28: In Exercises 1328, find all vertical and horizontal asymptotes of t...
 4.4.2.28: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.28: In Exercises 2530, (a) state the domain of the function, (b) identi...
 4.4.3.28: In Exercises 2328, find the standard form of the equation of the pa...
 4.4.4.29: In Exercises 2734, find the vertex, focus, and directrix of the par...
 4.4.1.29: In Exercises 2936, match the rational function with its graph. [The...
 4.4.2.29: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.29: In Exercises 2530, (a) state the domain of the function, (b) identi...
 4.4.3.29: In Exercises 29 and 30, find the standard form of the equation of t...
 4.4.4.30: In Exercises 2734, find the vertex, focus, and directrix of the par...
 4.4.1.30: In Exercises 2936, match the rational function with its graph. [The...
 4.4.2.30: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.30: In Exercises 2530, (a) state the domain of the function, (b) identi...
 4.4.3.30: In Exercises 29 and 30, find the standard form of the equation of t...
 4.4.4.31: In Exercises 2734, find the vertex, focus, and directrix of the par...
 4.4.1.31: In Exercises 2936, match the rational function with its graph. [The...
 4.4.2.31: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.31: Minimum Cost The cost of producing units of a product is and the av...
 4.4.3.31: Flashlight The light bulb in a flashlight is at the focus of the pa...
 4.4.4.32: In Exercises 2734, find the vertex, focus, and directrix of the par...
 4.4.1.32: In Exercises 2936, match the rational function with its graph. [The...
 4.4.2.32: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.32: Page Design A page that is inches wide and inches high contains 30 ...
 4.4.3.32: Satellite Antenna Write an equation for a cross section of the para...
 4.4.4.33: In Exercises 2734, find the vertex, focus, and directrix of the par...
 4.4.1.33: In Exercises 2936, match the rational function with its graph. [The...
 4.4.2.33: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.33: Photosynthesis The amount of uptake (in milligrams per square decim...
 4.4.3.33: Each cable of the Golden Gate Bridge is suspended (in the shape of ...
 4.4.4.34: In Exercises 2734, find the vertex, focus, and directrix of the par...
 4.4.1.34: In Exercises 2936, match the rational function with its graph. [The...
 4.4.2.34: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.34: Medicine The concentration of a chemical in the bloodstream hours a...
 4.4.3.34: . Beam Deflection A simply supported beam (see figure) is 64 feet l...
 4.4.4.35: In Exercises 2734, find the vertex, focus, and directrix of the par...
 4.4.1.35: In Exercises 2936, match the rational function with its graph. [The...
 4.4.2.35: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.35: In Exercises 3542, identify the conic. y 2 16 2 16x4
 4.4.3.35: Exercises 3544, find the standard form of the equation of the ellip...
 4.4.4.36: In Exercises 2734, find the vertex, focus, and directrix of the par...
 4.4.1.36: In Exercises 2936, match the rational function with its graph. [The...
 4.4.2.36: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.36: In Exercises 3542, identify the conic. 16x2 y y 2 16
 4.4.3.36: Exercises 3544, find the standard form of the equation of the ellip...
 4.4.4.37: In Exercises 3540, find the standard form of the equation of the pa...
 4.4.1.37: In Exercises 3740, (a) determine the domains of and (b) simplify an...
 4.4.2.37: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.37: In Exercises 3542, identify the conic. x264 y24 1
 4.4.3.37: In Exercises 3544, find the standard form of the equation of the el...
 4.4.4.38: In Exercises 3540, find the standard form of the equation of the pa...
 4.4.1.38: In Exercises 3740, (a) determine the domains of and (b) simplify an...
 4.4.2.38: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.38: In Exercises 3542, identify the conic. x21y236 1 x
 4.4.3.38: In Exercises 3544, find the standard form of the equation of the el...
 4.4.4.39: In Exercises 3540, find the standard form of the equation of the pa...
 4.4.1.39: In Exercises 3740, (a) determine the domains of and (b) simplify an...
 4.4.2.39: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.39: In Exercises 3542, identify the conic. x 2 400 2 20y 0
 4.4.3.39: In Exercises 3544, find the standard form of the equation of the el...
 4.4.4.40: In Exercises 3540, find the standard form of the equation of the pa...
 4.4.1.40: In Exercises 3740, (a) determine the domains of and (b) simplify an...
 4.4.2.40: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.40: In Exercises 3542, identify the conic. x2 y x 2 400
 4.4.3.40: In Exercises 3544, find the standard form of the equation of the el...
 4.4.4.41: In Exercises 3540, find the standard form of the equation of the pa...
 4.4.1.41: The cost (in dollars) of supplying recycling bins to of the populat...
 4.4.2.41: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.41: In Exercises 3542, identify the conic. y249 x2144 1x
 4.4.3.41: In Exercises 3544, find the standard form of the equation of the el...
 4.4.4.42: In Exercises 3540, find the standard form of the equation of the pa...
 4.4.1.42: Pollution The cost (in millions of dollars) of removing of the indu...
 4.4.2.42: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.42: x249y2144 1 y
 4.4.3.42: In Exercises 3544, find the standard form of the equation of the el...
 4.4.4.43: A satellite in a 100milehigh circular orbit around Earth has a ve...
 4.4.1.43: Population Growth The game commission introduces 100 deer into newl...
 4.4.2.43: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.43: In Exercises 4348, find the standard form of the equation of the pa...
 4.4.3.43: In Exercises 3544, find the standard form of the equation of the el...
 4.4.4.44: Fluid Flow Water is flowing from a horizontal pipe 48 feet above th...
 4.4.1.44: Food Consumption A biology class performs an experiment comparing t...
 4.4.2.44: In Exercises 15 44, (a) state the domain of the function, (b) ident...
 4.44: In Exercises 4348, find the standard form of the equation of the pa...
 4.4.3.44: In Exercises 3544, find the standard form of the equation of the el...
 4.4.4.45: In Exercises 43 and 44, consider the path of an object projected ho...
 4.4.1.45: Human Memory Model Psychologists have developed mathematical models...
 4.4.2.45: In Exercises 45 and 46, (a) determine the domains of and (b) use a ...
 4.45: In Exercises 4348, find the standard form of the equation of the pa...
 4.4.3.45: In Exercises 4554, find the vertices of the ellipse. Then sketch th...
 4.4.4.46: In Exercises 43 and 44, consider the path of an object projected ho...
 4.4.1.46: . Data Analysis: Physics Experiment Consider a physics laboratory e...
 4.4.2.46: In Exercises 45 and 46, (a) determine the domains of and (b) use a ...
 4.46: In Exercises 4348, find the standard form of the equation of the pa...
 4.4.3.46: In Exercises 4554, find the vertices of the ellipse. Then sketch th...
 4.4.4.47: In Exercises 4552, find the center, foci, and vertices of the ellip...
 4.4.1.47: In Exercises 47 and 48, determine whether the statement is true or ...
 4.4.2.47: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.47: In Exercises 4348, find the standard form of the equation of the pa...
 4.4.3.47: In Exercises 4554, find the vertices of the ellipse. Then sketch th...
 4.4.4.48: In Exercises 4552, find the center, foci, and vertices of the ellip...
 4.4.1.48: In Exercises 47 and 48, determine whether the statement is true or ...
 4.4.2.48: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.48: In Exercises 4348, find the standard form of the equation of the pa...
 4.4.3.48: In Exercises 4554, find the vertices of the ellipse. Then sketch th...
 4.4.4.49: In Exercises 4552, find the center, foci, and vertices of the ellip...
 4.4.1.49: In Exercises 4952, (a) determine the value that the function approa...
 4.4.2.49: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.49: Satellite Antenna A cross section of a large parabolic antenna (see...
 4.4.3.49: In Exercises 4554, find the vertices of the ellipse. Then sketch th...
 4.4.4.50: In Exercises 4552, find the center, foci, and vertices of the ellip...
 4.4.1.50: In Exercises 4952, (a) determine the value that the function approa...
 4.4.2.50: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.50: Suspension Bridge Each cable of a suspension bridge is suspended (i...
 4.4.3.50: In Exercises 4554, find the vertices of the ellipse. Then sketch th...
 4.4.4.51: In Exercises 4552, find the center, foci, and vertices of the ellip...
 4.4.1.51: In Exercises 4952, (a) determine the value that the function approa...
 4.4.2.51: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.51: In Exercises 5156, find the standard form of the equation of the el...
 4.4.3.51: In Exercises 4554, find the vertices of the ellipse. Then sketch th...
 4.4.4.52: In Exercises 4552, find the center, foci, and vertices of the ellip...
 4.4.1.52: In Exercises 4952, (a) determine the value that the function approa...
 4.4.2.52: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.52: In Exercises 5156, find the standard form of the equation of the el...
 4.4.3.52: In Exercises 4554, find the vertices of the ellipse. Then sketch th...
 4.4.4.53: In Exercises 4552, find the center, foci, and vertices of the ellip...
 4.4.1.53: In Exercises 53 and 54, write a rational function that has the spec...
 4.4.2.53: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.53: In Exercises 5156, find the standard form of the equation of the el...
 4.4.3.53: In Exercises 4554, find the vertices of the ellipse. Then sketch th...
 4.4.4.54: In Exercises 4552, find the center, foci, and vertices of the ellip...
 4.4.1.54: In Exercises 53 and 54, write a rational function that has the spec...
 4.4.2.54: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.54: In Exercises 5156, find the standard form of the equation of the el...
 4.4.3.54: In Exercises 4554, find the vertices of the ellipse. Then sketch th...
 4.4.4.55: In Exercises 5360, find the standard form of the equation of the el...
 4.4.1.55: Think About It Give an example of a rational function whose domain ...
 4.4.2.55: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.55: In Exercises 5156, find the standard form of the equation of the el...
 4.4.3.55: Architecture A fireplace arch is to be constructed in the shape of ...
 4.4.4.56: In Exercises 5360, find the standard form of the equation of the el...
 4.4.1.56: HOW DO YOU SEE IT? The graph of a rational function is shown below....
 4.4.2.56: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.56: In Exercises 5156, find the standard form of the equation of the el...
 4.4.3.56: Architecture A semielliptical arch over a tunnel for a oneway road...
 4.4.4.57: 0, 0, 0"
 4.4.2.57: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.57: Architecture A semielliptical archway is to be formed over the entr...
 4.4.3.57: Architecture Repeat Exercise 56 for a semielliptical arch with a ma...
 4.4.4.58: In Exercises 5360, find the standard form of the equation of the el...
 4.4.2.58: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.58: Wading Pool You are building a wading pool that is in the shape of ...
 4.4.3.58: Geometry A line segment through a focus of an ellipse with endpoint...
 4.4.4.59: In Exercises 5360, find the standard form of the equation of the el...
 4.4.2.59: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.59: In Exercises 5962, find the standard form of the equation of the hy...
 4.4.3.59: In Exercises 5962, sketch the ellipse, using the latera recta (see ...
 4.4.4.60: In Exercises 5360, find the standard form of the equation of the el...
 4.4.2.60: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.60: In Exercises 5962, find the standard form of the equation of the hy...
 4.4.3.60: In Exercises 5962, sketch the ellipse, using the latera recta (see ...
 4.4.4.61: In Exercises 61 and 62, use the following information. The flatness...
 4.4.2.61: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.61: 1, 0y 2x4
 4.4.3.61: In Exercises 5962, sketch the ellipse, using the latera recta (see ...
 4.4.4.62: In Exercises 61 and 62, use the following information. The flatness...
 4.4.2.62: In Exercises 4762, (a) state the domain of the function, (b) identi...
 4.62: In Exercises 5962, find the standard form of the equation of the hy...
 4.4.3.62: In Exercises 5962, sketch the ellipse, using the latera recta (see ...
 4.4.4.63: Planetary Motion The dwarf planet Pluto moves in an elliptical orbi...
 4.4.2.63: In Exercises 6366, use a graphing utility to graph the rational fun...
 4.63: In Exercises 6366, find the standard form of the equation of the pa...
 4.4.3.63: In Exercises 6370, find the standard form of the equation of the hy...
 4.4.4.64: Australian Football In Australia, football by Australian Rules (or ...
 4.4.2.64: In Exercises 6366, use a graphing utility to graph the rational fun...
 4.64: In Exercises 6366, find the standard form of the equation of the pa...
 4.4.3.64: In Exercises 6370, find the standard form of the equation of the hy...
 4.4.4.65: In Exercises 6572, find the center, foci, and vertices of the hyper...
 4.4.2.65: In Exercises 6366, use a graphing utility to graph the rational fun...
 4.65: In Exercises 6366, find the standard form of the equation of the pa...
 4.4.3.65: In Exercises 6370, find the standard form of the equation of the hy...
 4.4.4.66: In Exercises 6572, find the center, foci, and vertices of the hyper...
 4.4.2.66: In Exercises 6366, use a graphing utility to graph the rational fun...
 4.66: In Exercises 6366, find the standard form of the equation of the pa...
 4.4.3.66: In Exercises 6370, find the standard form of the equation of the hy...
 4.4.4.67: In Exercises 6572, find the center, foci, and vertices of the hyper...
 4.4.2.67: In Exercises 6770, (a) use the graph to determine any intercepts o...
 4.67: In Exercises 6770, find the standard form of the equation of the el...
 4.4.3.67: In Exercises 6370, find the standard form of the equation of the hy...
 4.4.4.68: In Exercises 6572, find the center, foci, and vertices of the hyper...
 4.4.2.68: In Exercises 6770, (a) use the graph to determine any intercepts o...
 4.68: In Exercises 6770, find the standard form of the equation of the el...
 4.4.3.68: In Exercises 6370, find the standard form of the equation of the hy...
 4.4.4.69: In Exercises 6572, find the center, foci, and vertices of the hyper...
 4.4.2.69: In Exercises 6770, (a) use the graph to determine any intercepts o...
 4.69: In Exercises 6770, find the standard form of the equation of the el...
 4.4.3.69: In Exercises 6370, find the standard form of the equation of the hy...
 4.4.4.70: In Exercises 6572, find the center, foci, and vertices of the hyper...
 4.4.2.70: In Exercises 6770, (a) use the graph to determine any intercepts o...
 4.70: In Exercises 6770, find the standard form of the equation of the el...
 4.4.3.70: In Exercises 6370, find the standard form of the equation of the hy...
 4.4.4.71: In Exercises 6572, find the center, foci, and vertices of the hyper...
 4.4.2.71: In Exercises 7174, (a) use a graphing utility to graph the rational...
 4.71: In Exercises 7176, find the standard form of the equation of the hy...
 4.4.3.71: In Exercises 7180, find the vertices of the hyperbola. Then sketch ...
 4.4.4.72: In Exercises 6572, find the center, foci, and vertices of the hyper...
 4.4.2.72: In Exercises 7174, (a) use a graphing utility to graph the rational...
 4.72: In Exercises 7176, find the standard form of the equation of the hy...
 4.4.3.72: In Exercises 7180, find the vertices of the hyperbola. Then sketch ...
 4.4.4.73: In Exercises 7380, find the standard form of the equation of the hy...
 4.4.2.73: In Exercises 7174, (a) use a graphing utility to graph the rational...
 4.73: In Exercises 7176, find the standard form of the equation of the hy...
 4.4.3.73: In Exercises 7180, find the vertices of the hyperbola. Then sketch ...
 4.4.4.74: In Exercises 7380, find the standard form of the equation of the hy...
 4.4.2.74: In Exercises 7174, (a) use a graphing utility to graph the rational...
 4.74: In Exercises 7176, find the standard form of the equation of the hy...
 4.4.3.74: In Exercises 7180, find the vertices of the hyperbola. Then sketch ...
 4.4.4.75: In Exercises 7380, find the standard form of the equation of the hy...
 4.4.2.75: Concentration of a Mixture A 1000liter tank contains 50 liters of ...
 4.75: In Exercises 7176, find the standard form of the equation of the hy...
 4.4.3.75: In Exercises 7180, find the vertices of the hyperbola. Then sketch ...
 4.4.4.76: In Exercises 7380, find the standard form of the equation of the hy...
 4.4.2.76: Geometry A rectangular region of length and width has an area of 50...
 4.76: In Exercises 7176, find the standard form of the equation of the hy...
 4.4.3.76: In Exercises 7180, find the vertices of the hyperbola. Then sketch ...
 4.4.4.77: In Exercises 7380, find the standard form of the equation of the hy...
 4.4.2.77: Page Design A page that is inches wide and inches high contains 30 ...
 4.77: In Exercises 7784, identify the conic by writing its equation in st...
 4.4.3.77: In Exercises 7180, find the vertices of the hyperbola. Then sketch ...
 4.4.4.78: In Exercises 7380, find the standard form of the equation of the hy...
 4.4.2.78: Page Design A rectangular page is designed to contain 64 square inc...
 4.78: In Exercises 7784, identify the conic by writing its equation in st...
 4.4.3.78: In Exercises 7180, find the vertices of the hyperbola. Then sketch ...
 4.4.4.79: In Exercises 7380, find the standard form of the equation of the hy...
 4.4.2.79: In Exercises 79 and 80, use a graphing utility to graph the functio...
 4.79: In Exercises 7784, identify the conic by writing its equation in st...
 4.4.3.79: In Exercises 7180, find the vertices of the hyperbola. Then sketch ...
 4.4.4.80: In Exercises 7380, find the standard form of the equation of the hy...
 4.4.2.80: In Exercises 79 and 80, use a graphing utility to graph the functio...
 4.80: In Exercises 7784, identify the conic by writing its equation in st...
 4.4.3.80: In Exercises 7180, find the vertices of the hyperbola. Then sketch ...
 4.4.4.81: In Exercises 8190, identify the conic by writing its equation in st...
 4.4.2.81: Minimum Cost The ordering and transportation cost (in thousands of ...
 4.81: In Exercises 7784, identify the conic by writing its equation in st...
 4.4.3.81: Art A sculpture has a hyperbolic cross section (see figure). (a) Wr...
 4.4.4.82: In Exercises 8190, identify the conic by writing its equation in st...
 4.4.2.82: Minimum Cost The cost of producing units of a product is given by a...
 4.82: In Exercises 7784, identify the conic by writing its equation in st...
 4.4.3.82: . Optics A hyperbolic mirror (used in some telescopes) has the prop...
 4.4.4.83: In Exercises 8190, identify the conic by writing its equation in st...
 4.4.2.83: Average Speed A driver averaged 50 miles per hour on the round trip...
 4.83: In Exercises 7784, identify the conic by writing its equation in st...
 4.4.3.83: Aeronautics When an airplane travels faster than the speed of sound...
 4.4.4.84: In Exercises 8190, identify the conic by writing its equation in st...
 4.4.2.84: The concentration of a chemical in the bloodstream hours after inje...
 4.84: In Exercises 7784, identify the conic by writing its equation in st...
 4.4.3.84: Navigation Longdistance radio navigation for aircraft and ships us...
 4.4.4.85: In Exercises 8190, identify the conic by writing its equation in st...
 4.4.2.85: In Exercises 8588, determine whether the statement is true or false...
 4.85: Architecture A parabolic archway is 12 meters high at the vertex. A...
 4.4.3.85: In Exercises 8588, determine whether the statement is true or false...
 4.4.4.86: In Exercises 8190, identify the conic by writing its equation in st...
 4.4.2.86: In Exercises 8588, determine whether the statement is true or false...
 4.86: Architecture A church window (see figure) is bounded above by a par...
 4.4.3.86: In Exercises 8588, determine whether the statement is true or false...
 4.4.4.87: In Exercises 8190, identify the conic by writing its equation in st...
 4.4.2.87: In Exercises 8588, determine whether the statement is true or false...
 4.87: Running Path Let represent a water fountain located in a city park....
 4.4.3.87: In Exercises 8588, determine whether the statement is true or false...
 4.4.4.88: In Exercises 8190, identify the conic by writing its equation in st...
 4.4.2.88: In Exercises 8588, determine whether the statement is true or false...
 4.88: In Exercises 88 and 89, determine whether the statement is true or ...
 4.4.3.88: In Exercises 8588, determine whether the statement is true or false...
 4.4.4.89: In Exercises 8190, identify the conic by writing its equation in st...
 4.4.2.89: In Exercises 89 and 90, use a graphing utility to graph the functio...
 4.89: In Exercises 88 and 89, determine whether the statement is true or ...
 4.4.3.89: Area of an Ellipse Consider the ellipse (a) The area of the ellipse...
 4.4.3.90: HOW DO YOU SEE IT? In parts (a)(d), describe in words how a plane c...
 4.4.4.90: In Exercises 8190, identify the conic by writing its equation in st...
 4.4.2.90: In Exercises 89 and 90, use a graphing utility to graph the functio...
 4.4.3.91: Think About It How can you tell if an ellipse is a circle from the ...
 4.4.4.91: In Exercises 91 and 92, determine whether the statement is true or ...
 4.4.2.91: Writing (a) Given a rational function how can you determine whether...
 4.4.3.92: Think About It Is the graph of an ellipse? Explain
 4.4.4.92: In Exercises 91 and 92, determine whether the statement is true or ...
 4.4.2.92: Writing and Graphing a Rational Function Write a rational function ...
 4.4.3.93: Think About It The graph of is a degenerate conic. Sketch this grap...
 4.4.4.93: Alternate Form of the Equation of an Ellipse Consider the ellipse (...
 4.4.2.93: Think About It Is it possible for a rational function to have all t...
 4.4.3.94: Think About It Which part of the graph of the ellipse is represente...
 4.4.4.94: HOW DO YOU SEE IT? Consider the ellipse shown. (a) Find the center,...
 4.4.2.94: HOW DO YOU SEE IT? For each statement below, determine which graph(...
 4.4.3.95: Writing At the beginning of this section, you learned that each typ...
 4.4.3.96: Writing Write a paragraph discussing the changes in the shape and o...
 4.4.3.97: Using a Definition Use the definition of an ellipse to derive the s...
 4.4.3.98: Using a Definition Use the definition of a hyperbola to derive the ...
 4.4.3.99: Drawing Ellipses An ellipse can be drawn using two thumbtacks place...
Solutions for Chapter 4: Rational Functions and Conics
Full solutions for College Algebra  9th Edition
ISBN: 9781133963028
Solutions for Chapter 4: Rational Functions and Conics
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra, edition: 9. Since 432 problems in chapter 4: Rational Functions and Conics have been answered, more than 39653 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9781133963028. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4: Rational Functions and Conics includes 432 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.